Rings with several objects

ADVANCES

IN MATHEMATICS

8,

l-161

(1972)

Rings with

Department

of Mathematics,

Several

BARRY

MITCHELL

Dalhousie

University,

Communicated

Objects

Halifax,

Nova

Introduction . . . 1. Preliminaries ............... ....... 2. Tensor Products of Categories 3. Functor Categories ............ 4. Semisimple Categories ........... 5. Adjoint Functors ............... .......... 6. Additive Kan Extensions. The Ring of an Additive Category ....... I. 8. Functor Categories which are Module Categories .......... 9. Homological Dimension Projective and Flat Functors ........ 10. I I. K-Categories .............. Dimension of K-Categories ....... 12. ....... 13. The Subadditivity Theorem Change of Base ............. 14. 15. Cohomological Dimension of Groups ... Derived Functors of the Limit Functors 16. ......... 17. The Standard Complex Homology of Simplicial Complexes .... 18. 19. The 2-Category of a Set of Relations ... ........ 20. Generators and Relations 21. Length ................ 22. Deltas ................ 23. Projectives over Deltas. ......... 24. Rigid Categories ............ 25. ...... Categories of Iterated Fractions 26. The Word Problem for G(,) ....... 27. Derivations .............. ....... 28. The Four Term Resolution 29. The Dimension Raising Lemma ..... 30. B-Subcategories ............ 31. Bridge Categories ............ 32. Dimension of Deltas .......... ........ 33. Deltas of Dimension ~2 34. The n-gem ................. 35. An Upper Bound on the Dimension of a Weak Poset 36. Directed Functors .............. 31. Direct Limits of Functors with Varying Domain .

1 0

1972 by Academic

607/8/1-r

Press,

Inc.

Scotia,

Catzada

by S. Eilenberg

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The purpose of this paper is to indicate that most of noncommutative homological ring theory generalizes to (pre)additive category theory, as well as to provide some reasons for wanting it to do so. One of the first hints of the possibility of such generalization came in Freyd’s thesis, where the ring theoretic notion of injective envelope was linked up with the abelian categorical notion of exactness of a functor to yield Freyd’s striking proof of the abelian group valued imbedding theorem for small abelian categories. Since then there have been several papers concerned with replacing theorems about rings by theorems about additive categories. What does not seem to be generally realized is the degree of completeness to which the program can be carried out, providing that one recognizes the notion of an additive category as an entirely natural generalization of that of a ring. Indeed, if it came initially as a surprise that certain ring theoretic facts generalize to such an apparently far-out setting, it now comes as even more of a surprise when they don’t. Historically, additive categories were abstracted from some of the more unwieldy examples, such as the category of all modules over a given ring, rather than the well established example of the ring itself. In fact, in one of the more important early papers on the subject, it is stated, “While it is easy enough to construct preadditive categories which fail to (have finite products), such examples all seem sufficiently artificial to suggest that the notion of a preadditive category can for the most part be by-passed”. This remark should not be interpreted as a slight against rings, but rather only as indicative of the belief prevalent at that time that the only categories with any decent properties must necessarily be abelian, or at least close to it. The approach in the present paper will be to imitate as closely as possible the classical proofs of ring theory. Although something new may be called for on occasion owing to the fact that some phenomenon or other may have collapsed in the one object case, the sailing is relatively smooth, thanks mainly to the additive Yoneda lemma, which in the one object case is nothing but the familiar natural isomorphism HomR(R, M) B M of right R-modules. In the early sections of the paper we have tried to provide a complete enough sampling of the techniques involved so as to enable US later to

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send the reader directly to the ring theoretic sources for some of the generalized ring theoretic facts which we shall need. Many of the constructions dealt with in these early sections, as well as the fibred category construction of the last section, belong properly to the realm of closed categories. However, such a general treatment would be unwarranted in view of our rather specialized goals. Among the early sections, we might mention Section 7, not because of its importance in the sequel, but because of a misconception which it may give rise to. We consider a matrix ring [V] associated with a small additive category 9?, and we find that the category of right modules Abl’l is related to the category of covariant additive functors Ab’ via exact functors

such that R and S are full, faithful, right and left adjoints, respectively, for T. This is the additive counterpart of a theorem of Lawvere [31]. Now when V has only a finite number of objects, T is an equivalence, and so on the basis of upbringing, one may be tempted to abandon the functor category in favor of the module category. However, before making such a choice, one should realize that there are sometimes graph theoretic aspects of a category % which are obscured when one passes to the one object category [%?I. For example, if 7~ is a finite poset and Zn is the free additive category associated with 7~, then the effect of passing to the matrix ring [ZX] is to reduce all the vertices in the Hasse diagram of n to a single point. This does not strike us as contributing to optimum clarity. We shall now summarize what can be considered as new in the paper. Let Q! be an abelian category and let %? be a small additive category. If F E U”, then one can ask for an upper bound on the amount by which the homological dimension of F can exceed the sup of the homological dimensions of the values F(p), p E 1%?1. Let us consider the example where F is the horn functor %?itself, considered as an object in the functor category Ab V*oV = (AbQ*)V. Here the category /lb’* is playing the role of 6Z. The homological dimension of 9? in this case is called the cohomological (or Hochschild) dimension of the small additive category %‘, and is denoted by dim 59. The values of F are the contravariant horn functors V( ) p) E Ab%:*, which are all projective. Thus %’ provides us with an example of a functor whose homological dimension is precisely dim % greater than the sup of the homological dimensions of its values.

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Now under the assumption that $? is Z-projective (that is, V( p, q) is a projective abelian group for allp, q E 1%?I) and 0L has exact coproducts (AB4), one can prove that for any FE 6YV we have (see Section 13) h.d.F < dim T? + sup h.d.F(p). Z)EIYI

(1)

(Of course, projective abelian group means free, but one can work in the more general setting of K-categories for a fixed commutative ring K, where oTV is interpreted as the category of all K-functors from 27 to 6Y.) One can also weaken the hypothesis on %?to the assumption that it be Z-flat (torsion free), providing that one strengthens the hypothesis on CPIto the assumption that it have exact direct limits (A&5). We are mainly interested in the case where %?is of the form Zx, where 7~is a small, nonadditive category. This is the additive category whose objects are those of n, and where Z+ p, q) is the free abelian group on n( p, q). Thus the above hypothesis on V is satisfied with a vengeance. In this case dim %’ is denoted simply by dim r, and the category of additive functors Q?‘” is the same as the category of nonadditive functors GP. Another invariant which one can study when V is of the form Zn is the homological dimension of OZ, the constant functor at Z in A@. This is the same as the degree of the highest nonvanishing derived functor of the inverse limit functor b, : A@ -+ Ab. By the inequality (l), we know that h.d. AZ < dim rr

always. If r is a group then equality holds. This is proved in Cartan and Eilenberg [I 1, p. 1951, b ut in the interest of self-containment, we have included a proof in Section 15. For more general categories the inequality is usually strict. For example, if 7~is a category with an initial object, then dZ is representable, and so h.d. OZ = 0. On the other hand, it is easy to show that if 7r is a poset, then dim r = 0 if and only if x is discrete (Section 33). One defines the global dimensionof an abelian category as the sup of the homological dimensions of its objects. If V is a small additive category, one then defines the right (left) global dimension of V as gl dim AbV(gl dim Abv*). From the inequality (1) we have gl dim IT

< dim 9 + gl dim 0,

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valid under the hypothesis that %? is Z-projective (flat) and C! is AB4(AB5). Again the inequality can be strict, and several of our results center around this question. Some of these results were obtained in [38] in disguised form, since there we had not introduced the notion of cohomological dimension. We have been able to make the present exposition independent of those papers without too much repetition. For quite some time we knew of no finite poset 7~ such that the difference gl dim OZ” - gl dim GY depends on a. The first such example was found by Spears [47], who subsequently found several others. In Section 34 we consider a class of such examples, the smallest of which is that of Spears. These examples, which resulted from a conversation with Albrecht Dold, are obtained by ordering the cells of a certain family of regular cell complexes by inclusion. They enable us furthermore to exhibit finite posets 7~, T/ such that the inequality dim n x r’ < dim rr + dim 7~’ (which is valid for all pairs of small categories) is strict. We have devoted considerable space to the question of determining categories of cohomological dimension 62. We shall exhibit a large class of such categories, using a certain exact sequence in a functor category Q? which is a generalization of a sequence introduced by Roger Lyndon in his paper “Cohomology theory of groups with a single defining relation” [34]. The description is in terms of the triviality of a certain 2-category which can be associated with a presentation of a category in terms of generators and relations. In the case of locally finite posets (or more generally, locally finite deltas, where a delta is a category with no nonidentity endomorphisms), we show in Section 33 that this description is complete. This can be considered as one of our main results. It uses a technique of Eilenberg and Moore [15] concerned with the question of when projectives in a functor category Ab’ are free. Among the categories of dimension 62 mentioned above, there is a subclass of categories of dimension < 1, the categories of “iterated fractions” of free categories. Some of these, the socalled “bridge categories” (S ec t ion 31), will be shown to have the property that tensoring with any one of them raises dimension by one. This result embraces several of Eilenberg, Rosenberg, and Zelinsky [14], including the Hilbert syzygy theorem. If V is an additive category retract of 9 and if 9 is Z-flat, then using the inequality (1) it is not difficult to show that dim % < dim B (Section 13). In particular, this applies to Zn’ C Zr where X’ is a

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(nonadditive) retract of n. In some cases this provides an effective tool for getting a lower bound on dim PT.For example, it applies whenever 7~ is a poset and 77’ is a full subcategory which is a complete lattice. Using this, we show in Section 36 that a poset has cohomological dimension 61 if and only if it is the free category generated by an oriented graph. The same result is obtained for the more general deltas in Section 33, but only under the assumption of local finiteness. In this case the retract argument does not seem to work. There has been some interest in the vanishing of the derived functors bLk) where 7~* is a directed set of cardinal number x1, . Various partial results were obtained by Jensen [27], John Moore, and others, before RCmi Goblot [19] (and, I believe, independently, Alex Heller) showed that they are all zero for K > n + 1. In Section 16, we deduce this as an easy consequence of a well-known generalization of Osofsky [43] of a theorem of Berstein [6] concerning the homological dimension of a direct limit. As a by-product, we reobtain Osofsky’s theorem itself for general AB5 categories (not necessarily with enough projectives). Later in Section 36 we show that the above upper bound can be attained. Explicitly, we show that if n is any totally ordered set, then the greatest integer K such that l&, ck) # 0 is n + 1 where the smallest cardinality of a cofinal subset of r* is N, .l Here the range category can be any functor category Ab’ where $? is a nonzero additive category (in particular, a nonzero ring). This uses another theorem of Osofsky [44] concerning the homological dimension of a directed module (which of course must first be generalized to directed functors). One can further deduce that if rr is the ordered set of ordinals less than or equal to N, (that is, r = K, + I), and if K is a division ring, then the right global dimension of Kn is one (true for any ordinal n), whereas the left global dimension is n + 2. Examples of this nature were actually found by Jategaonkar [25] in the case where the additive category is a ring. We have also tried to determine the cohomological dimension of totally ordered sets (Section 37). Here we bring the fibred category of all abelian group valued additive functors into play, so as to deduce the above theorem of Osofsky-Berstein in the original form where the domain categories (rings, classically) vary throughout the direct system. We find that if 7r is a totally ordered set of N, elements, then dim r < n + 2. However, we have not been able to determine the 1 We have a forthcoming

recently paper.

extended

this

result

to all

directed

sets

71. This

will

be

treated

in

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dimension of all totally ordered sets. For example, assuming the continuum hypothesis, the ordered set of real numbers R has dimension ~3 by the theorem just quoted, and >,2 since it is not free. We don’t know which it is. We suspect that if R has X, elements, then dim R = n + 2. I would like to thank Bill Lawvere for many conversations relating to this paper. The genuine interest which he is capable of lending to a field which is not his own has enabled him to make several useful suggestions. I would also like to express my appreciation to Beno Eckmann for a very enjoyable year spent at the Forschungsinstitut in Zurich, during the course of which a substantial portion of this material was developed. Finally, my hat is off to Arnold Tingley of the Dalhousie Mathematics department, whose initiative and foresight have brought about the stimulating atmosphere and the excellent working circumstances with which I have been favored during the preparation of this manuscript. 1.

PRELIMINARIES

Throughout this paper, the composition fg of two morphisms is to be read as first f and then g. However this will not prevent us from sometimes writingf(X) in pl ace of xf when f happens to be a function. When this is done, one must take into account the switch (fg)(x) = g(f(x)). If a is a category, the membership notation x E 6Z will usually denote that x is a morphism of GZ, and the class of objects of GZ will be denoted by 16!? (. However, in certain cases when there can be no confusion, we shall write A E GZ to denote that A is an object of a. Occasionally we shall identify an object A with its identity morphism 1, . If x E C!!, then dom x and cod x will denote, respectively, the domain and codomain (range) of x. If A, A’ E ) OT 1, th en 0&4, A’) is the class of morphisms from A to A’. Let a:F-+F’ and /3:G --t G’ be natural transformations, where F, F’ : C! -+ S? and G, G’ : 9’ 4 %? are (covariant) functors. This situation can be represented by the following diagram:

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The Godement product of 01 and 6 is the natural transformation (Y . /3 : FG -+ F’G’ whose value at an object A E / Q! 1 is given by either of the compositions in the commutative diagram AFG OPAAFG,

cube, from which it A morphism A + A’ induces a commutative follows that 01. /3 is indeed a natural transformation. We often write F . /3 in place of 1, * p and (II . G in place of 01* 1, . This product of natural transformations is governed by the following rules: (9

1~ - a=a=a’l~,

(ii) (iii)

(a * P) * y = a * (P - y), lF * 1, = lFG ,

(iv)

In the situation

we have (4

(PP’) = (a . Ma’ . P’)

The notion of a 2-category (or a category enriched by Cat) is had by abstracting the above situation. Thus a 2-category is a category V (whose composition we denote by .) together with a category structure on each such that 1, is an identity in the category homset~(p,q),p,qE/~l, %7(p, p) for each p E ( %?(, and such that the composition of an identity in %?(p, 9) with one in %?(q, r) is an identity in V( p, r). Furthermore, denoting the “vertical” composition of two morphisms 01,01’ in V( p, q) by 011 cy’ (when defined), we require that in the situation where 011 0~’

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and /3 1 /I’ are defined in %Y(p, q) and ‘%‘(q, Y), respectively, the composition (a * /I) 1 (01’ * p’) be defined in %?(p, r), and, moreover, (a I 4 . (B -L B’> = cm . B) I (01’ P’).

(1)

We shall encounter another example of a 2-category in connection with cohomological dimension two (Section 19). This 2-category will have the further property that each of the categories V( p, q) is a groupoid, or in other words a category in which every morphism is an isomorphism. By an additive category we shall mean a category together with an abelian group structure on each of its horn sets such that composition is bilinear. We do not impose the condition that the category have finite products, so that what we are calling “additive” has sometimes been called “preadditive”. A category with finite products is additive in at most one way. A ring (with identity) is an additive category with precisely one object. If the category K is additive, then we can sum natural transformations of functors -8 + %, and the Godement rules can be supplemented by the rule (4

4 * t+ + ?u = 4 * # + + * #‘.

Likewise, if g and V are both additive are additive, then we have the rule (4

and the functors

G, G’ : .g + $5

(4 + 6’) f # = 4 ’ fj + 4’ . Q.

If V is a small category and Cn is an arbitrary category, then GZ’ will denote the category of (covariant) functors F : ?? ---f QY(additive functors if ?? and 02 are additive) with natural transformations as morphisms. In speaking of functor categories @ we shall always assume that V is small, even though we may neglect to say so explicitly each time. We shall denote the dual (opposite) category of %? by V*. Thus GF?‘* is the category of contravariant functors from % to @!. The category of abelian groups will be denoted by Ab. If /l is. a ring, then Ab” is the category of right fl-modules, and A&‘* ’IS the category of left fl-modules. The one morphism category is denoted by 1. If 6!! is any category, then CP can be identified with 0’. Similarly if 67 is an additive category and Z denotes the ring of integers, then MZ = 0’. Let R = OncZ R, be a graded ring. Then R can be converted into the additive category V whose objects are the integers, and where The %(m,n) = R,-,,,. Composition is given by ring multiplication. functor category Abv can then be identified with the category of graded R-modules with degree zero homomorphisms.

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Let T : G!! + J% and V : 9 + V be functors, where small categories. Then there is an induced functor

+? and 9 are

defined by FTV = VFT on objects (functors) F, and by 4TV = V * 4 * T on morphisms (natural transformations) $. It follows from rules (v) and (vi) above that if T is additive, then so is TV. If T and V are both additive, then TV is defined between the categories of additive functors. then there If rl : T + T’ and 8 : V + V’ are natural transformations, is an induced natural transformation qs : TV --+ T’V’ defined at F E Ol’ as either composition in the commutative diagram VFT -

1

VFT’ --

V’FT

1

V%T’

When V is the identity functor on V’, TV will be denoted by T’, and when T is the identity functor on G!, TV will be denoted by GP. Recall that a functor T : 6X + a is an equivalence if it is full and faithful, and if every object in 93 is isomorphic to AT for some A E / Q! j. Alternatively, T is an equivalence if there is a functor S : $3 + Q? such that the compositions ST and TS are naturally equivalent to identity functors. An equivalence which is a one to one correspondence on objects is an &morphism. The following lemma is simply a restatement of the first definition of equivalence in a particular case. LEMMA 1.l. Let q be a subcategory of a small category 8, and let &? be a full subcategory of O!c. Then the composition

is an equivalence if and only tf 1. Every functor F : $7 -+ GZ extends to somefunctor F : @ + r3 withFEIg1, and 2. If F, G E 198 / extend F, G : %?+ GZ, then any natural transformation F ---f G extends uniquely to a natural transformation P --) G. Remark. One can replace “equivalence” by “isomorphism” in the lemma if in condition (I) the functor F is required to be unique. In any case it is easy to see that condition (2) implies that the functor F of

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condition (1) is unique up to natural equivalence. The category g need not be small here. If 7~ is any category, we let Zrr denote the additive category whose objects are those of n, and where Z~T( p, q) is the free abelian group on S$ p, q). Composition is defined in the unique way so as to be bilinear and so as to make the inclusion 7r + ZX a functor. If n is a monoid, then Zr is the monoid ring of 7 with integral coefficients. If r is small and a is additive, then the inclusion m + Zn induces an isomorphism of categories azr = Q?n* where, according to our convention, the left side is the category of additive functors from Zr to a and the right side is the category of all functors from n to a. This is an easy application of Lemma 1.1, with the additive functor category playing the role of 33. If %? is an additive category, then we can form the category Mat $5 whose objects are finite sequences of objects of V, and where a morphism [riJ with from (PI ,.-, Pm> to (ql ,..., qn) is an m x n matrix yij E %( pi , qj). Composition is defined by ordinary matrix multiplication. Mat 97 is an additive category with finite products, and contains V as a full additive subcategory. If % is small and cpl is additive with finite products, then one uses Lemma I. 1 again to see that the inclusion %?+ Mat % induces an equivalence of categories

Let n be a category, and let 01E n( p, q) and p E “(9, p) be such that a/3 = 1,. Then p is called a retraction (or split epimorphism) and LY is called a coretraction (or split monomorphism). We shall also say that a is a left inverse for ,5 or that ,5 is a right inverse for 01.The object p is called a retract of the object q. The endomorphism 0 = pa! is called a split idempotent. A category is idempotent complete if every idempotent is split, or in other words can be written as /3a where 0#3 = 1. If 71 has equalizers, then n is idempotent complete. For if 19is idempotent, then we can take a to be the equalizer of 0 and 1. A full subcategory STC ii will be called a cover for ii if every object of ii is a retract of one in 37. It is a finite cover if / 77 j is finite. If 57 is a cover for ii, and if r2’ is idempotent complete, then it is easy to extend any functor F : rr -+ C!! to a functor F : + 4 G! (additive functor if n is an additive subcategory and F is additive), and any natural transformation F + G to a unique natural transformation F + G. Thus

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it follows once more from Lemma 1.1 that when ii is small, the inclusion 7~+ 7i induces an equivalence of categories a* e OP. If n is a cover for 73and if 73is idempotent complete, then ii will be called an idempotent completion for r. Idempotent completions are unique up to equivalence of categories. For if i3 and ii are both idempotent completions for rr, then we can find functors 7j + 73and 73+ 75 extending the inclusions 7~+ 73 and TI + +. Then 5 + 73+ + is naturally equivalent to the identity functor on ii since both extend the inclusion n -+ ii, and similarly 73+ ii + 73 is naturally equivalent to the identity functor on 73. The following construction of an idempotent completion for n is due to Freyd [17, p. 61, Exercise B]. Let 73 denote the category whose objects are the idempotents in rr, where a morphism from the idempotent 8 to the idempotent 0’ is a triple (0, X, 0’) with x a morphism in 7~ satisfying ex = x = x8’. Composition is defined by (0, x, cl’)(O’,y, e”) = (e, xy, ey. Composition is clearly associative, and (0, 0, 19)is an identity for 0. It is easy to see that 7; is idempotent complete, and z can be considered as a full subcategory of 73by identifying the object p of 7~with the idempotent I,. Also if OEn(p,p) is idempotent, then considered as an object of 73, 8 is a retract of 1, . Thus 7~is a cover for i3, and so 73is an idempotent completion for r. If n is additive, then there is an obvious additive structure on 73,and if n has (finite) products, then so does 6. An idempotent complete, additive category with finite products has been called amenable by Freyd. Summarizing the discussion of this section we see that every category rr is a subcategory of an amenable category %?= gt?%, and if n is small and GZis amenable, then the inclusion 7r -+ 9 induces an equivalence of categories

We conclude this section with a proposition concerning retracts of categories which will be useful in getting lower bounds on dimension. PROPOSITION 1.2. Let T be a complete lattice, and supposethat r is a subcategory of a category T in such a way that whenever r( p, q) = 4 we have m( p, q) = 4. Then 7 is a retract of T.

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Proof. Composing the inclusion T -% 7~with the partially ordered reflection of 7~,we see that we may assume that 7~is a poset, and T is a full subcategory. We construct r : 7~---f -r by taking Y(P) to be the inf of all elements of T greater than or equal to p. Then Y is clearly a functor (order preserving map), and using the fact that T is full we see that UY = 1.

2. TENSOR

PRODUCTS OF CATEGORIES

If %i and g2 are categories, then we can form the product category Vi x %?ain the obvious way, and we have the identification

If 5~7~ and V2 are additive, then there is an obvious additive structure on Vi x V2 . However, if @ is also additive, then (1) is no longer valid since the functor categories must then be interpreted as categories of additive functors. This leads us to consider the notion of the tensor product Vi @ %‘aof two additive categories. This is the category whose class of objects is 1%‘i 1 x 1??a/, where the abelian group of morphisms from (pi , pa) to (qi , q2) is the ordinary tensor product of abelian groups

Using the associativity and commutativity of the tensor product of abelian groups, together with the bilinear compositions in Vi and %?a, it is straightforward to define a bilinear composition in Vi @$%Y2 . Explicitly the composition is given by the rule

The additive counterpart of the isomorphism (1) is

where @ is an additive category. We also have the identifications Z@U=%?,

%T~@e2 =9?2@vl,

WI 0 U2) 0 v, = %I 0 (@20 %A.

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If rr is any category and ‘27 is an additive %Z@ ZT by Vrr. Thus we have Qy’” = @yFZr Observe

= (@)Zw

category,

then we denote

= (@),.

also the identity Z7rl @ z7T, = Z(T1 x Tiy?),

which

should

be compared

with

zTrl x zi-rz = (Z x Z)(Trl x 7fJ. Let %?r ,..., GF?~be additive &2Yi

categories.

we define

inductively

= (&)o%T,.

i=l

An additive functor

Then

i=l

of n variables

is an additive

functor

If we fix all of the variables except the k-th at objects of the respective categories, we obtain an additive functor %?k+ CY, called a k-th variable partial functor of T. T is called limit preserving in the k-th variable if all of its k-th variable partial functors are limit preserving. If this is true for each k, then T is called a limit preserving functor of n variables. This is not the same as saying that T is limit preserving as a one variable functor from the tensor product category. This is one reason why the theory of n variable functors does not reduce entirely to the theory of one variable functors. Consider additive functors Fi : Wi --f 6Yi , 1 < i < n. These induce the additive functor &Fi i=l which

=F&&+&& i=l

i=l

is defined by Fh

0 ... 0 rrJ = F,(Y,) 0 ... OK&J.

This defines objectwise

the functor

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which takes the morphism c$~@ **’ @ &, into the natural transformation whose value at (p, ,..., p,), pi E / Vi j , is the morphism MA)

0 ‘.. 0 cnw.

Now if T : O’, @ ... @ G!, + .9?is an additive functor, then composing (2) with T’~I G,. Pvn, we obtain a functor @I @ ... @@fL+gp&~~P~n~ This is the process whereby the values of a several variable functor T pick up operators when operators are added to the arguments. For example, if M is a left r, right A-bimodule and N is a left A, right LAbimodule, then M On N is a left r, right Q-bimodule. If (Vi ) i ~1) is a family of categories, we can form the disjoint union (J1V( , and we have the identity

If 02 and the %?iare additive, then (3) is not valid, since the disjoint union is not additive. However, it can easily be made additive by adding zero morphisms between objects from distinct categories. If we take this as the definition of the disjoint union in the additive case, then (3) is valid. We note further the isomorphism

3. FUNCTOR

CATEGORIES

If GYis an additive category and A E 16Y /, then Q!, will denote the covariant horn functor OL(A, ) represented by A. Thus GZ,* = 6Y*(A, ) = a( , A) is the contravariant horn functor represented by A. The symbol CZwill sometimes be used to denote the two variable horn functor GP @ OL+ Ab. Recall that an object A is projective if @,, preserves epimorphisms, and is small if @, preserves coproducts. A is faithful if CPI, is faithful. More generally a family 9 of objects is faithful if XAEY Q?, is faithful. (We shall reserve the “generating” terminology for other purposes.) Module categories were characterized independently by Gabriel

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and Mitchell as abelian categories with coproducts and a faithful small projective. Using a different proof, Freyd characterized additive functor categories Ab” as abelian categories with coproducts and a faithful set of small projectives. This theorem can be proved more simply along the lines of the original proof for module categories (with a slight assistance from Bass [5]), and this we now proceed to do. Recall that if V is an additive category, then the Yoneda lemma provides an isomorphism Ab%(%‘, F) = F which

is natural

in FE AbV.

THEOREM 3.1 (Freyd). A category LZ is equivalent to a category of additive functors Ab’ if and only if a is abelian with coproducts and a faithful set of small projectives.

Proof. It is an easy consequence of the Yoneda lemma that the family {VP I P E I v I> is a faithful set of small projectives for AbV. Conversely, suppose that I?( is abelian with coproducts and a faithful set L? of small projectives. Considering B as a full subcategory of a, let T : 12 + Ab”’ be the functor defined by (P) AT = LZ(P, A),

A~!edl.

PElW,

We shall show that T is an equivalence. Since the P’s are projective, small, and a faithful set, it follows, respectively, that T is exact, coproduct preserving, and faithful. For fixed B E 1 Q’ I, consider the natural transformation induced by T G&4, B) -% Ab+(AT,

BT)

of abelian group valued functors of A. To show that T is full, we must show that t? is a bijection. When A E [ 9 1, the range of 8 is Ab;“*(ga*,

BT) = (A) BT = @4, B)

by the Yoneda lemma, and under this identification 0 is just the identity, function. Since T preserves coproducts, it follows that 6 is an isomorphism whenever A is a coproduct of P’s. For general A we use the fact that B is a faithful set to write an exact sequence in G!

RINGS WITH

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17

Applying 0 to this sequence and using exactness of T, we see that 8 is a bijection at A. Finally, if FE Ab9*, we must show that F is isomorphic to AT for some A E 1Q?I. Since the representables are a faithful set for Ab”‘, we can write an exact sequence

Since PT = .!YP* for P E 19’ 1 and since T preserves coproducts, this sequence may be rewritten

Since T is full, we can write f = aT, and so if A is the cokernel of LY, then by exactness of T we find F - AT. This proves that T is an equivalence. COROLLARY 3.2. A category is equivalent to a module category ;f and only if it is abelian with coproducts and a faithful, small projective.

Let $9 be a small additive category, and let ME AbV. It will be convenient notationally to pretend that the abelian groups (p&W, p E 1?2 /, are mutually disjoint. Thus if x E (p)M for some p, then we shall refer to x simply as an element of M, and we shall denote p by 1x (. If X E V(p, q) and x E (p)M, then we shall denote (x)(h)&17 simply by xX. Thus xX is defined whenever x is a member of M, h is a morphism of %‘, and / x 1 = dom h, and we have 1xX 1 = cod h. Then if one writes down explicitly what it means for M to be an additive abelian group valued functor, one obtains the following rules: (i)

(x + x’)h = xX + x’h

(ii)

x(h + h’) = xX + xh’

(iii)

x(hp) = (xX)p

(iv)

xl,

= x.

A family of elements xi , i E I, of M is called a family of generators for M if every element x E M can be written m = 1 Xi& , &I 607/8/I-Z

4 E%(I xi 1,I x I>,

(1)

18

MITCHELL

where all but a finite number of the hi are zero. Equivalently, family is a family of generators if the natural transformation

the above

(2) which takes 1 ,si, to xi is an epimorphism in AbV. The family is a basis for A4 if the representations (1) are unique, or equivalently the morphism (2) is an isomorphism. M is free if it has a basis. Thus a free functor is simply a coproduct of representables. A functor is jkitely generated if it has a finite set of generators. A singly generated functor (that is, a quotient of a representable) is called cyclic. If GJ?has finite products, then every finitely generated functor is cyclic. If M is free and is finitely generated, then any basis for M is necessarily finite, and we shall call M simply a finite free. A jinitely presentable functor is a cokernel of a morphism of finite frees. Any functor ME AbW is a direct limit of finitely presentable functors (Bourbaki [7, Chap. I, Section 2, Exert. lo]). It will be convenient in many formulas of this paper to adopt the convention that K, stands for any finite cardinal. If N is any cardinal number, then we say more generally that M is &generated if it has a family of generators of cardinal number X. Likewise M is N-presentable if it is a cokernel of a morphism between N-generated frees. then the full Yoneda embedding If %? is idempotent complete, %?* --t Ab’ shows that any retract of a representable is representable. From this one easily deduces in the case where %? is amenable that the small projectives (which are the same as the finitely generated projectives) in Ab’ are precisely the representable functors, and hence the full subcategory of small projectives is equivalent to %*. This is Freyd’s version of Morita equivalence: If Abv is equivalent to Ab” and %?and 59’ are amenable, then V and %?’ are equivalent. A right ideal in an additive category V is a subfunctor of VP for some p E / 9 1, and a left ideal is a subfunctor of V,*. A two sided ideal is a subfunctor of the two variable functor 5??.If I is a two-sided ideal, then we can form the quotient category V/I whose objects are those of V, and where w/w,

4) = %(A d/W

4).

Composition is induced by that of G?. Recall that an object of an abelian category (artinian) if it has the ascending (descending)

is called noetherian chain condition on

RINGS WITH

SEVERAL OBJECTS

19

subobjects. An additive category ‘%Tis right noetherian (artinian) if YP is noetherian (artinian) in Ab’ for all p E / V 1. Similarly V is left noetherian (artinian) if VP* is noetherian (artinian) in Ab”* for all p E / %?/. If 9 is right noetherian (artinian) and if M is finitely generated in Ab’, then M is noetherian (artinian). The following example shows that a left artinian category need not be left noetherian. EXAMPLE. If n is a poset and K is a ring, then a subfunctor I C Kv( p, ) is completely determined by a family I(p), 4 E 17~[, of right ideals of K, such that if q < r, then I(q) C 1(r), and I(q) = 0 for p z& q. In particular, if K is a division ring, then 1 must have the property that if I(q) # 0 for some q, then I(r) = K for all Y > q. In this case it follows easily that Kn(p, ) is artinian if and only if there is no infinite chainp
4. SEMISIMPLE

CATEGORIES

The Jacobson radical of an additive category has been defined by Kelly [29], and used to discuss semisimplicity in additive categories by Leduc [33]. W e give here a brief discussion of these matters which differ in some aspects from the above treatments. A nonzero object A in an abelian category GI?is simple if it has no proper, nonzero subobjects, and is semisimpleif it is a coproduct of simple objects. Schur’s lemma asserts that if A is simple, then GZ(A, A) is a division ring, and if A and B are simple but not isomorphic, then 6Y(A, B) = 0. A small additive category GT? is left semisimpleif qD* is semisimple for all p E 1V I. Suppose that %?is left semisimple. Because of the identity morphisms, one sees that the VP* are actually finite coproducts of simple ideals. Let Y be a family of objects of Abv* consisting of one object from each

20 isomorphism write

MITCHELL

class of simple

ideals.

Then

for each p E 1 5~91 we can

where c 4% P) < CQ. SE9

(1)

For each S E 9’ let K, denote the division ring of endomorphisms of S. Then by the Yoneda embedding and Schur’s lemma, we can write, for P>!lEI~l,

where in general M,llXn (K) denotes the abelian group of m x n matrices with entries in the ring K. Thus we see that every semisimple category is of the following form. Let {K,l SET] b e a family of division rings, and let / %?1 be a set. Let n be a function from Y x I %?( to the nonnegative integers satisfying (1) for each p E / %?I. For each S E Y, let V be the category whose set of objects is 1V /, and where a morphism from p to p is an n(S, p) X n(S, Q) matrix over KS , with composition defined by matrix multiplication. (Equivalently, Vs is a category which is equivalent to a full subcategory of the category of finitely generated left K modules, such that 1%?S1 = / 9? I.) Let %?be the full subcategory of the product category )(sE,4pVs consisting of the objects on the diagonal. We have shown above that every left semisimple category is isomorphic to a category of this form. Conversely, one can see that any category of this form is left semisimple. In fact the primitive idempotents on the diagonal in V( p, p), considered as a large matrix, generate simple left ideals such that qD* is their coproduct. This is the Artin-Wedderburn structure theorem for semisimple categories, from which it follows that the notions of left and right semisimplicity coincide. It is not difficult to see that %?is amenable if and only if for each family of nonnegative integers {m, / S E 9} such that CsE9 m, < oc), there is an object p E 1%?/ such that n(S, p) = m, for all S. In this case %?will actually be abelian.

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WITH

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OBJECTS

21

The reader may wish to check that the various other equivalent formulations of semisimplicity given in Cartan and Eilenberg [ 11, Chap. I, Section 41, are valid in the present generality. We now take up the Jacobson radical. If V is small additive category and I C VP is a proper right ideal (that is, 1, $ I(C)), then Zorn’s lemma shows as usual that I is contained in some maximal right ideal in VP . We define J, as the intersection of all maximal right ideals in “e, . LEMMA 4.1. Jp(q) = (~3 E V(p, q) j 1, - C+ has a right inverse for all /3 E V(q, p)> = {a E %( p, q) 1 1, - $3 has a two-sided inverse for all P (5 %(QY P>>.

Proof. Suppose that 1, - c@ has no right inverse. Then it generates a proper right ideal in VP , which must be contained in some maximal ideal M. If 01E M(q), then c& E M(p), and so 1, E M(p), a contradiction. Conversely if 01E Jp(q), then a $ M(q) for some maximal ideal M, and so (CX) + M = VD , where (a) is the right ideal generated by 01. Hence $3 + m = 1, for some m E M(p), and so 1, - UP has no right inverse. Now suppose that oi E JD(q) and /3 E %?(q, p). By what we have shown, we have (1, - U&Y = 1, for some X. Hence x = 1, + c@x), and so again by what we have shown, x has a right inverse. This means that 1, - a$ has x as a two-sided inverse. LEMMA 4.2. right inverse.

Proqf.

1, -

If (1, -

C$ has a right inverse if and only q 1, - pa has a

a/3)x = 1,) , then (1, - PJl)il, + b)

= 1,.

From Lemmas 4.1 and 4.2 and their duals, we see that if we define J,* to be the intersection of all maximal left ideals in Vq*, then IIlk)

= J,*(P).

Thus if we define J(p, q) = J,(q), then J is a two-sided ideal, called the Jacobson radical of the additive category %‘. The quotient category %?/J has zero radical. By Lemma 4.1, we see that J( p, p) is the Jacobson radical of the endomorphism ring Y( p, p) for each p E / V (. If V(q, p) = 0, then J(A d = VP, d.

22

MITCHELL

An endomorphism x in 9? is nilpotent if xn = 0 for some n. A right ideal I C VP is nil if every element of I(p) is nilpotent. LEMMA

4.3.

Ij I C ‘i”?, and I is nil, then I C J, .

Proof Let 01E I(g) and /3 E ??(q, p). Then $ E 1(p), and so ($)n = 0 for some n. The usual identity then shows that 1, - afl has an inverse, and so 01E Jr,(q). THEOREM 4.4. Let p be an object of the small additive category %?. Then VP is semisimple ij and only ij it is artinian and J, = 0. Consequently, Sf?is semisimple if and oni$ if it is right artinian and J = 0.

Proof. If an object of an abelian category is a finite coproduct of simple objects, then it follows from the usual composition series argument that the object is artinian, and furthermore it is clear that the intersection of the maximal subobjects is zero. In particular this applies to Yp . Conversely, suppose that VP is artinian and J, = 0. Suppose that we have constructed orthogonal primitive idempotents e, ,..., e, in V(p, p), but 13= 1, - zy=, ei # 0. Let I be any simple right ideal in (0). Such exists since VD is artinian. Since Jz, = 0 we must have 01~# 0 for some 01E 1(p) by Lemma 4.3. Hence left composition with 01induces an automorphism of I since I is simple. Therefore oLe = 01 for some e E r(p), which implies that oI(e2 - e) = 0, and so e2 = e. Also e # 0 since ale = CY.# 0, and so (e) = I C (0). Thus we can take e,,, = e and add it to our list of primitive orthogonal idempotents. Since VP is artinian, this process must terminate for some n, and so VP is semisimple. If I is a two-sided ideal in 9? and F E Ab’, then we let FI denote the subfunctor of F generated by elements of the form x01 with x E F(p) for some p and 01E I(p, q) for some q. Relative to this operation, Nakayama’s lemma is valid: If F is finitely generated and FJ = 0, then F = 0. If I is a two-sided ideal, we can define In as the two-sided ideal generated by al1 compositions of n morphisms of I. If I% = 0 for some n, then I is nilpotent. If V is artinian, it does not follow that J” = 0 for some n unless 1 9 / is finite. This is what prevents us from proving that a right artinian category is right noetherian. (See the example of the previous section.) Of course the condition that J be nilpotent can always be imposed on the category, and, in particular, one can define %? to be semiprimary if $91J is semisimple and J is nilpotent.

RINGS WITH

23

SEVERAL OBJECTS

5. ADJOINT

FUNCTORS

Recall that a functor S : GY+ S? is a left adjoint for a functor T : 9? + GZ (or that T is a right adjoint for S) if there is a natural equivalence c&4, BT) e qm,

B)

(1)

of two variable set valued functors. It is equivalent to be given natural transformations (called adjunctions) E:TS+I~,

T: I,+

ST

(2)

such that (T. dk . T) = 17.7

(‘i.S)(S.E)

= I,.

(3)

We begin by giving an easy proof of a well-known proposition. First note that a natural transformation of horn functors gX + LZ?~is a monomorphism if and only if the morphism Y + X which induces it is an epimorphism (by definition of epimorphism), and is an epimorphism if and only if Y + X is a split monomorphism (evaluate at Y). PROPOSITION 5.1. In the adjoint situation (2), the adjunction E is an epimorphism if and only if T is faithful, and is pointwise a split monomorphism if and only if T is full. Hence E is a natural equivalence if and only if T is full and faithful.

Proof.

Consider the composition of natural transformations

g’fB> B’) 2

@(BT, B’T) -id

c%‘(BTS,B’),

where 0 is induced by T and q5 is (1). For fixed B, the morphism BTS -+ B inducing the composition is found by setting B’ = B and taking the image of 1, , and is thus seen to be es. Since 4 is an isomorphism, we see that 04 is a monomorphism (epimorphism) if and only if 8 is. The first statement of the proposition now follows from the remark immediately preceding the proposition. The second statement follows from the fact that an isomorphism can be characterized as a split monomorphism which is also an epimorphism. COROLLARY 5.2. If T isfull, then each of T . 7, E * T, 7 . S, and S . E is a natural equivalence.

24

MITCHELL

Proof. morphism.

This follows

from Eqs. (3) since E is pointwise

a split mono-

Remark. By duality, the corollary is true also if S is full. Hence the corollary has no converse, since there are full functors whose adjoints are not full. By the dual of Proposition 5.1, S is full and faithful if and only if 7 is a natural equivalence. Hence if T and S are both full and faithful, then T is an equivalence. Conversely, one can show that any equivalence has a full and faithful left adjoint. If S is a left adjoint for T, then T preserves limits and S preserves colimits. Consequently if OT and a are additive with finite products, then T and S are both additive. In this case it follows from the way that (I) is constructed from (2) that (1) is also additive. Let S : I?? --f 9? be a left adjoint for T : 9? + GZ, and let F : 7~ -+ rr’ be a left adjoint for U : n’ --+ r where n and n’ are small. Then using the second definition of adjointness it is an easy exercise in the Godement rules to verify that SF : 09’ + gn is a left adjoint for TU : .9@ -+ GV, and that TU is an equivalence if both T and U are equivalences. If U : n’ + 7r is any functor between small categories, then the functor Q?u : GZn + 0tn’ has a left adjoint provided that suitable colimits exist in Q!. This is the functor which assigns to F’ E GY its so-called left Kan extension along U. In the following section we shall construct a left adjoint for @J in the case where 0? is cocomplete and additive and U is an additive functor.

6.

ADDITIVE

KAN

We recall the following theorem can be found in [37, p. 1061.

EXTENSIONS

of Freyd

[18], the proof

of which

THEOREM 6.1. Let 99 be cocomplete abelian with a faithful set B of small projectives, and let O? be cocomplete and additive. Then, considering B as a full subcategory of 59, any additive functor F : B + Q! can be extended to a colimit preserving functor P : g + GZ. Furthermore, if FI and p2 are additive functors extending FI and F, and if FI is colimit preserving, then any natural transformation FI + F2 extends uniquely to a natural transformation FI + F2 .

Remarks.

1.

The hypothesis

on 9? implies

that it is equivalent

to

RINGS WITH

SEVERAL OBJECTS

25

the functor category Ab B* by Theorem 3.1. However, it is best to forget this in carrying out the proof. Freyd originally deduced Theorem 3.1 as a consequence of the above theorem. 2. It follows from the remark to Lemma 1.l that the extension F of the theorem is unique up to natural equivalence. COROLLARY 6.2. Let gi be a cocomplete abelian category with a faithful set Bi o.f small projectives for 1 < i < n, and let g,, and GZbe additive categories with C7 cocomplete. Then any additive functor F:930@8,@‘**@P’, ---t C? can be extended to an additive functor P: a0 @q 0.e. @.G?,, + Cl which is colimit preserving in the last n-variables. Furthermore, if FI and F, are additive functors extending F, and F, , and if PI is colimit preserving in the last n variables, then any natural transformation FI + F3 extends uniquely to a natural transformation FI ---f F, ,

Proof. First consider the case n = 1. For fixed B, E ! 3, 1, the theorem gives an extension P(B, , ) to F(B,, , ). Now a morphism B, --+ B,’ induces a natural transformation P(B, , ) -+ F(SY’,‘, ), which again by the theorem extends uniquely to a natural transformation F(BO , ) + F(B,‘, ). In this way F becomes a two variable additive functor. Now consider FI + F, . By the theorem, FI(BO , ) + F,(B, , ) extends uniquely to &(B, , ) -+ Pz(B, , ). Relative to a morphism 4, - Bo’, we must show that the diagram ~ljl(BO?1-+-F,(B, 1 ~dB,,‘> ) -+

1 ~&,‘,

, ) )

is commutative. But this follows from the uniqueness part of the theorem, since both compositions yield the same thing when restricted to 8, . The general case follows by induction on n and the case n = 1 by appropriate groupings of the variables. To carry out the induction, it must also be checked in the case n = 1 that if J?+?,, is of the form @?@ S9, and if F is colimit preserving in 9, then the same is true of the extension F constructed above. But this follows since every object of 9Yi is a cokernel of a morphism between coproducts of members of 8,) and colimits in OTcommute with coproducts and cokernels.

26

MITCHELL

Now let V be a small additive category. Then the functors gP*, p E 1V I, constitute a faithful set B of small projectives in AbV*, and 9 is isomorphic to %‘. Thus if 67 is additive and cocomplete, then by Corollary 6.2, the evaluation functor CZV @ V 4 L7? can be extended uniquely (up to natural equivalence) to a functor

which is colimit preserving in the second variable. By definition we have the isomorphism F 0~ g,* which

is natural

in F and p. We now establish a(F &

which %* + of the When

= F(P) an isomorphism

G, A) = Ab**(G, CZ(F, A))

(2)

is natural in F, G, and A. Here OZ(F, A) denotes the functor Ab whose value at p is LZ?‘(F(p), A). Indeed, this is just an example, way a functor picks up operators, as explained in Section 2. G = qfl*, the identity (1) and the Yoneda lemma give Qc’(F &

VP*, A) = @(F(p), A) = Abq*(%TD*, Gl(F, A)).

(3)’

By Corollary 6.2, (3) can be extended uniquely to (2) since both sides of (2) are contravariant functors of G which take colimits to limits. We now see that @& is colimit preserving in the first variable as well as the second, for the right side of (2) takes colimits in the variable F into limits in Ab, and hence the same is true of the left side. If FE M”* and G E Abv, we define G&F=F&rG. When GZ = Ab, we thus obtain two functors & : Ab’ @ Ab”’ + Ab. That they agree follows from Corollary 6.2 since both are colimit. preserving in both variables, and both satisfy

In this case &

can be defined F Ou G =

alternatively

by

0 F(P) Oz G(P),‘M, P”l-++l

RINGS

where M is the subgroup of the form “YOY

-XOYY,

WITH

SEVERAL

of the numerator

where

27

OBJECTS

generated

Y E @(P, q),

by all elements

x E F(P),

Y E G(q).

The isomorphisms (1) and (2) can then be established by direct arguments, just as in the module theoretic case. We shall now imitate a familiar argument, due independently to B. Eckmann and H. A. Forrester, to construct a cofaithful (cogenerating) set of injectives for Abv*. Let Q be a cofaithful injective for Ab (for example, the rationals mod one). For p E 1V j, define Q,* = Ab(%YD , Q) EAbY;*. Then if G E Ab’*, we have using (2), Ab%*(G, Q,*) = Ab$*(G, Ab(V’, , Q)) = Ab(vD 01 G, Q) = Ab(G(p),

Q).

Using the fact that Q is a cofaithful injective for Ab, it follows that if&* I P E I q I>.1sa cofaithful family of injectives in Abv*. Dually, we see 0, = WV,*,

Q,,

PElW

yields a cofaithful family of injectives in Ab’. A useful notion in dealing with bifunctors of the form B E Ol’;“@’ is that of an end. An end for B is an object E E 1CZ1 together with a family of morphisms ~9 : E - B(P) P),

PCIVI

such that given any morphism m E %(p, q), the diagram B(P,P) /

Ly

E\

//cpl” B(q,q)

28

MITCHELL

is commutative, and such that given any other family 0~~’ : E’ -+ B(p, p: with this property, there is a unique morphism 4 : E’ -+ E satisfying for all p. Ends are unique up to canonical isomorphism. 4% = %’ When Q? is complete, the end of B can be constructed as the equalizer of the obvious pair of morphisms X

B(p,

Del

p) =z

)(

B(dom

?nPW

m,

cod m).

In this case the assignment of ends may be considered as a functor Gp@@ 4 Q?. An example of an end, and indeed the only example which we shall need, is given by the following situation. Let @ and 5.~7be additive, and consider F, G E Q?. Let B = ll(F, G) E Abe*@‘. Then the end of B is just the abelian group of natural transformations Q?‘(F, G). Now consider FE a’, G E Ab’*@“, and HE Gf9’, where CY iz cocomplete and additive. By (2) we have an isomorphism Q!(F 0% G, H) = Abv*(G, 6Y(F, H)) in Ab”*@. Since isomorphic the natural equivalence @‘(F &

bifunctors

have isomorphic

G, H) = Ab+@(G,

Let & and $7 be small additive categories, additive. We observe that the functor

ends, we obtain

cT(F, H)).

(4;

and let O? be cocomplete

can be defined either by using the identification O!‘@’ = (Q?)” and replacing @ by GZ’ in the usual definition of the tensor product, or by allowing the functor & : QP @ Ab’* - @ to pick up operators. This follows from Corollary 6.2, since both functors are colimit preserving in the second variable, and they agree when the second variable is of the form VP*. Thus we can replace @ by @* in (4) to obtain LJ?@(F 6Jv G, H) = AbQ* @(G, @(F, H)) for FE (X1@‘, G E Ab’*@, and HE O!“@“. We now consider associativity of the tensor establish a natural isomorphism FO~*~Q(GO~H)=(F~~“G)~~~~*H

product.

We

shall (5:

RINGS

WITH

SEVERAL

29

OBJECTS

for F E C??@‘, G E Ab’*OY, and HE AbQ*@‘. we have natural isomorphisms

By repeated

use of (4),

a(F @J~*~~ (G OS H), A) = Ab6’Q@(G @a H, LZ(F, A)) = AZI”*‘~)~(H, Ab@‘*(G, @(F, A))) = Ab”*@(H, = O?((F & By Yoneda, this composite must be induced (5), as required. Let U: %?+ 9 be an additive functor. yields an isomorphism in AbV*@’ Ab”(L@(U,

OT(F (jj& G, A)) G) @p;z8* H, A). by a natural transformation Then

the Yoneda

lemma

), a(F, H)) = LZ(F, UH)

(6)

relative to F E a” and H E OT”. Passing to ends, we obtain an isomorphism Abw* -(g( Taking

U, ), GT(F, H)) = 6P(F, UH).

G = 3( U, ) in (4) and composing

with

@cc 9( U, ), H) = @(F,

@(F

(7)

(7) yields UH).

This shows that the functor oT” : MQ + CPI’ has as its left adjoint functor S, : O?’ -+ @” given by

the

&(F)(q) = F 0~ g’(ut 4). The adjunction

morphism

is given by F oQ U, where U : 59 ---f a”( U, U) is considered as a morphism in Ab ‘*w . Thus if U is a coretraction in Ab’*@@, then 77 is a coretraction. Conversely, if qF is always a coretraction, then, in particular, taking O! = AbV* and F = %?,we see that U is a coretraction. The functor S,(F) is called the additive left Kan extension of F along U. In the terminology of Cartan and Eilenberg, it could also be called the covariant U-extension of F. When %? = 2, we can identify U with an object p E / 2~ 1 and F with

30

MITCHELL

an object A E 1 Q? 1. The functor GP in this case is the evaluation Tg : 6Z2 -+ a, and its left adjoint S,, is given by

functor

When 9 is of the form &-r, we shall denote A @ Zr simply by An. Thus Arr is an object of the category 6P*Xn, and the functor S,(A) is just A+, ). Wh en r is a monoid (that is, a category with a single object p), then A ny ASP, ), and 4 , P) will all be denoted ambiguously by An-. An explicit formula for the functor S, : G? + OP is given as follows:

Here u(y) : A --f @,+J A denotes the y-th coproduct injection. If one takes this as the definition of S, , then it is easy to verify directly that S, is a left adjoint for T, . The observation to be made here is that, whereas in the case of a general additive category %?we require that M be cocomplete in order to guarantee that the functors S, be defined, in the present case we need only assume that GZ have coproducts indexed by the sets n(p, 9) for all p E 1r /. If OL is abelian and these coproducts are exact, then S, is exact. In particular, if ~(p, Q) is finite for each q, and C!?is abelian, then S, exists and is exact. When n is a poset, S,(A) is just the diagram with A at q for all q > p and zeros elsewhere, with identity morphisms on A wherever possible. Now consider the case where OZ, instead of being co-complete and additive, is complete and additive. (By a theorem of Freyd, a cocomplete category with a generator is complete.) Then the construction of go dualizes to yield a functor horn% : (/NV)*

@ O? -

~2,

called the symbolic homfinctor. Thus horn, is the unique (up to natural equivalence) limit preserving functor satisfying homdeD , F) = F(P). When GZ = Ab, the Yoneda lemma gives us also AbYV,

, F) = F(P),

(1”)

RINGS

and so by Corollary

WITH

6.2 it follows

SEVERAL

31

OBJECTS

that in this case

home(G, F) = AbV(G, F). The dual of (4) reads @(F,

hom9(G, H)) = AM~*~:‘(G,

for FE Q!‘, GE A’*@, and HE C!?‘. Thus we obtain a natural isomorphism

Q!((F, H))

cd*)

combining

(4) and (4*),

CP(F @la G, H) = GP(F, hom9(G, H)), which reads

shows that &

G is a left adjoint

for horn&G,

). The dual of (5)

horn,@ .(G 0~ H, F) = homea p(G, horn&H, F)) for J’ E @@OQ, G E AbV@@, and H E Ab@@@. The right functor P : OP + OY is the functor R, defined by

%(F)(q) = homd~i(q, The functor R,(F) is the additive The right adjoint for the evaluation R, defined by %44(q)

adjoint

of the

W,F).

right Kan extension of F along U. functor T[, : aT’ ---f OTis the functor

= hom&%

~1~4.

Suppose now that F E Ab’, G E AbrmQ*, and H E @‘” where O! is bicomplete and additive. Then we have a natural transformation of a-valued functors F av hom+(G,

H) L+

hom~~(AbV(Fl

Gl, f0

(8)

This follows from Corollary 6.2 since the left side is colimit preserving in F, and both sides are just horn&G@, ), H) when F is of the form FP . Since u is an isomorphism for F representable, it is also an isomorphism for F a finite free, and hence for F any retract of such. In other words, u is an isomorphism whenever F is a finitely generated projective. Let us define the dual of F as the object F* = Abw(F, V) of AbV*. Then taking O? = AbV, 9 = V, and G and H both 5~7in the

32

MITCHELL

above,

the transformation D becomes the familiar transformation from the identity functor on Ab% to the double dual functor. One can also write down an explicit formula for u in this case. The dual of (8) yields a natural transformation

F 5 F**

Abv(F,

G) @g H--f

homW(F,G &, H)

for FE Ab@‘, G E AbvB9, and HE a’*. F is a finitely generated projective.

@*)

It is an isomorphism whenever

Let 6! be an abelian category with coproducts. we define

For FE @? and’

G E Ab’*,

TorRv(F, G) = H,(F @v Y) where Y is a projective resolution for G in Ab’“. Then Torny: is a two? variable additive functor from @ @ Ab’* to OT, and Tor’(F, ) is an exact connected sequence for each F. If GZhas exact coproducts, then & P is exact for P projective in Ab’* (see Section lo), and it follows, that tar’ is an exact biconnected sequence [l 1, Chap. V, Section 81.1 In this case TornW(F, G) can be computed as H,(X 0% Y) where Y is a projective resolution for G and X is any acyclic left complex over F. This is stated in Cartan and Eilenberg only for the case where X is a projective resolution of F, but the proof suggested on p. 105, exercise 6, requires only acyclicity. Note, however, that Tornq(F, G) cannot be computed as H,(X & G) w h ere X is a projective resolution for F unless projective objects in 0!’ are “%-flat”. For example, if Q?is of the form Ab9, then projectives in (Abq)’ are not necessarily v-flat unless 9 is Z-flat (Lemma 13.1). Dually, if CPIis cocomplete abelian, then for FE @ and G E Abv, we define Ext&G,

F) = Hn(homv(X, F))

where X is a projective resolution for G in Abv. If @ has exact products, Y)) where X is a then Ext”(G, F) can be computed as H”(hom,(X, projective resolution for G and Y is an acyclic right complex over F. The advantage of these more general definitions of Tor and Ext is that many of the formulas involving these functors given in Cartan and’ Eilenberg actually become categorical duals of each other. For example, the universal coefficient theorem for cohomology is the dual of the universal coefficient theorem for homology.

RINGS WITH

7. THE

RING

SEVERAL OBJECTS

OF AN ADDITIVE

33

CATEGORY

Let V be a small additive category, and let [V] be the set of 1G5/ x 1%?/ matrices of the form / olPq1where aPQE W(p, q), and each row and column has only a finite number of nonzero entries. Such matrices can be added and multiplied using addition and composition in V?, and in this way [%?I becomes a ring with identity. If y E V, we shall sometimes identify y with the matrix whose only nonzero entry is y. The identity morphisms 1, form a set of orthogonal idempotents when considered as members of [Vj. It is a complete set when 1%?1 is finite. EXAMPLES. 1. Let v be a finite poset, [Rn] is the set of j r 1 x 1n 1 matrices [r,J if p 4 4. These are the “tic tat toe” rings. incidence algebras by Rota [46]. In particular, set of n elements, [Rrr] is the ring of n x entries in R.

and let R be a ring. Then with rPq E R and rpq = 0 They have also been called when r is a totally ordered n triangular matrices with

2. Suppose that / %?1is finite, and that the condition %(p, q) # 0 with p # q implies W(q, p) = 0. Suppose also that %‘(p, p) is a semisimple ring for all p E 1%?1. Then [V] is a generalized triangular matrix ring in the sense of Chase [12]. 3. A category r is equivalent to the one morphism category 1 if and only if Z( p, q) has precisely one element for all p, q E 1n I. In this caseif R is a ring, then R is equivalent to Rm as an additive category, whereas (1 = [Rn] is the ring of row and column finite 17~j x / n 1 matrices over R in the usual sense.If I 7~I is infinite, it is easy to establish an isomorphism fl * fl @ /l as right cl-modules. From this it follows that any finitely generated right A-module is cyclic. The same applies to left fl-modules. THEOREM 7.1. Let %?be a small additive category, and let Q?be an amenable category with / %?I-fold coproducts. Then there are functors

such that S is a full and faithful left adjoint for T, and is an equivalence if 1%?1 is Jinite. If ~2 is abelian, then T is exact, and if the / 99 I-fold coproducts in Q?are exact, then S is also exact. 607/S/2-3

34

MITCHELL

Proof. If ME @‘I, then we denote the underlying object in 6Y also by M. Since ~2 is idempotent complete, the action of the idempotents lr, on M can be factored as M 2% T(M)(p) 211-t M, where pr,&, = 1. A morphism diagram

y E %?(p, q) gives rise to a commutative

M 2% T(M)(p) 1”“s M

Y1

1y

M 2%

T(M)(g) i”q

M

and so there is a unique morphism T(M)(y) making the diagram; commutative. In this way T(M) becomes an object of M’. A morphism f : M + N in 6Vl induces a commutative diagram

M ----f T(M)(p) --+ f

1

1 N

M

-4

T(N)(p)

__f

f

N

and so we obtain a morphism T(f)(p) which is the value at p of a natural transformation T(f). This defines T as a functor. On the other hand, starting with FE CYv, we define

with

01 = [CXJ acting on S(F) by

where u, is the coproduct injection. This makes sense because 01is row finite. Using additive functoriality of F, it is easily verified that this defines a right ring action of [V] on S(F). It is clear how a morphism F + F’ induces a morphism S(F) + S(F’) so as to convert S to a functor. We now define functions 4, $5,

@(F, T(M)) +

a[el(S(F),

M)

RINGS

as follows. If 0 : F ---t T(M), such that the diagrams

WITH

SEVERAL

35

OBJECTS

we define $(O) as the unique

morphism

are commutative for all p E 1 $7 [. To see that d(O) commutes [Fj-operators, observe first that if 01 = [cY~,J E [V], then

with

where olPq and 1, are regarded as single entry matrices. Since relative to the action on M we have hpplj = l,, and A, is an epimorphism, we obtain

Then

we compute,

using the fact that 0 is natural,

= pk7) %w = %Pd(~) for all p E 1% /. Hence $(6+x = LX+(~), as required. Now if f : S(F) -+ M is a morphism in G!lrl, then we define #(f)(p) as the composition F(P) -%

@F(q) a

lf M -+-+

T(M)(p).

36

MITCHELL

Using the fact that f commutes to Y E qp, Q), #(f)(P)

ww(Y)

with

[+?I-operators,

= %Jfuw)(Y)

we compute

relative

= %fYh

= UDYfh = F(Y) %fh = F(Y) #(f )W

Hence #(f ) is a natural transformation. It is now easy to verify that # and # are inverses of each other, that C$ is natural in F and M, and that the adjunction 1 --f ST is an isomorphism. Therefore S is a full and faithful left adjoint for T. If 1%?1 is finite, then one uses the fact that the 1, form a complete set of orthogonal idempotents to show that the adjunction TS + 1 is an isomorphism. When Q?is abelian, it is clear that T is exact, and if the 1V I-coproducts in GZare exact, then S is exact by definition. This completes the proof. Remark. In the case GY= Ab, the adjunction E : TS -+ 1 evaluated, at M is just the morphism

induced by the inclusions Ml, + M. It follows from the fact that the 1, are orthogonal idempotents that E,+,is a monomorphism. It is an epimorphism if and only if the Ml, generate M as an abelian group. When M = [W], [‘Fjl, is the abelian group of matrices with nonzero entries only in the p-th column. Hence when %?has an infinite number of nonzero objects, [q is not generated by the [V]l, , and so l ,@] is not an epimorphism. The dual of Theorem 7.1 reads: THEOREM 7.1*. Let V be a small additive category, and let G?be an, amenable category with products indexed by 197 /. Then there are functorsi

such that R is a full and faithful right adjoint fey T. If GYis abelian and the 1%?j-products in GZare exact, then R is exact.

RINGS

WITH

The functors T of Theorems R is defined by

SEVERAL

37

OBJECTS

7.1 and 7.1* are the same. The functor

R(F) =

x &I). rl+g/

Remark. Theorem 7.1 and its proof remain valid if we take as [%?I the ring of row finite matrices (or any subring of the ring of row finite matrices containing the identity and all single entry matrices). Likewise Theorem 7.1* is valid if [%?‘Iis the ring of column finite matrices. In this case if z- is a small category and R is a ring, then [Rn] is the “convolution ring of a small category” defined in [31] by Lawvere, who proved Theorem 7.1* for CZ = Ab. For contravariant functors we obtain an adjoint pair

by replacing %?by %?* in Theorem 7.1 and using the ring isomorphism [%T*] = pq*. Now suppose that a is cocomplete and additive. Then we have, relative to F E @, G E Ab’*, and A E 6Y, G&S(F) Qel

S*(G), A) = AU”J*(S*(G),

@(S(F), A))

= AI[~I*(S*(G),

R*(CPI(F, A)))

= Abv*( T*@*(G)),

Q!((F, A))

= Abe*(G, CZ(F, A)) = U(F Q& G, A). By Yoneda, this composite natural equivalence

of natural

S(F) Qel S*(G)

equivalences = F &

must come from

a

G.

Now S*, being an exact functor with an exact right adjoint, preserves projective resolutions. From this we obtain when GZ is abelian, TorLW1(S(F), S*(G))

= Tor,‘(F,

G).

(1)

In the case V = Rrr where R is a ring and rr is an infinite category equivalent to 1, this isomorphism was used by MacLane [35] in order

38

MITCHELL

to reduce the construction of Tor to the case where every finitely generated left ideal over the ring is principal. (See Example 3 of this section). If Gl?is complete and additive, we obtain dually for symbolic horn, homwl(S(G), for F E 0P and G E Ab’.

R(F))

= hom(G F)

Hence when

Ex&(S(G),

R(F))

6Z is abelian, this gives = Ext$(G,

F).

(2)

Finally, we compare the Jacobson radical Jg of the small additive category % with the Jacobson radical J[‘%?] of the ring [%?I. Suppose that ideal, this means that l,al, = aPn cy E J[Vj. Since J[V] is a two-sided is in J[%‘] for all p, q. Hence 1 - cyPp/3, has a right inverse in [U] for all &, E V(q, p), where 1 is the identity of [%?I. From this it follows that 1, - c&?pp has a right inverse in %?for all pq,, , so that mP9E JV. Now suppose that / %? / is finite, and that 01 is a matrix such that aPQE J%? for all p, q. We wish to show that 01E /[VI. Since / V 1 is finite and I[%‘] is an abelian group, it suffices to show that aPn E J[%‘] for all p, q. Consider an arbitrary matrix ,k Let x be a right inverse for 1, - olP,#pP in V. Then a right inverse for 1 - olPp/3in [V] is the matrix 1 + y where yps = 0 for Y # p, yPs = x~~~& for s # p, and =x1,. YPP 8.

FUNCTOR

CATEGORIES

WHICH

ARE

MODULE

CATEGORIES

Let I;pI be equivalent to a module category AbR for some ring R, and let n be a nonempty small category with a finite cover n’ (Section 1). Then @ is equivalent to a module category, for we have from Section 1 and Theorem 7.1, ar m (AbR)n w It is somewhat

surprising

(&R)n’

z

/jbR”’

that the converse

m &[Rm’l.

is true.

Namely:

THEOREM 8.1. Let rr be a nonempty small category and let GZ be any’ category. Then 6P is equivalent to a module category if and only if 6??is equivalent to a module category and rr has a jinite cover.

I am indebted to Bill Lawvere who helped find R, and who suggested using the following theorem of Isbell. The proof can be found in [23, Theorem 3.11.

RINGS WITH THEOREM

8.2.

SEVERAL OBJECTS

39

A retract of a complete category is complete.

COROLLARY 8.3. If an is complete and cocomplete abelian and 7~ is nonempty, then a is complete and cocompleteabelian.

Proof. Consider the diagonal A : 6Y ---f 6P and the evaluation TI, : CYr---f 01, p E ) x 1. Then AT, = In, and so by Theorem 8.2 and its dual, 0l is complete and cocomplete. Hence Tp has a left adjoint S,, and a right adjoint R, , and A has a left adjoint h and a right adjoint lb. It follows that T, and A are both limit and colimit preserving, and in particular T,(O) is a zero object for 67. If p is a monomorphism in Q!, then A(p) is a monomorphism in 6P, and so A(p) is a kernel. Applying T, we seethat p is a kernel. Dually every epimorphism in 0! is a cokernel, and so C!?is abelian. Proof of Theorem 8.1. Suppose that GP is equivalent to a module category. By the corollary, Q! is complete and cocomplete abelian. Let G be a faithful, small projective in Q?. The functor A is faithful, coproduct preserving, and exact. It follows from the adjoint relation Ol(lin~G, A) = OP(G, AA) that lim G is a faithful, small projective for a. Hence by Corollary 3.2, 6Z is equivalent to a module category. Now in 0P we have the epimorphism 0 %(G(P)) -L PEInl

G

whose coordinates are induced via adjointness by the identity morphisms on G(p). Since G is projective, E is a retraction, and so since G is small we obtain an epimorphism

where 1rr’ / is finite. Let F be any nonzero object of QP. Using the epimorphism E’ and the fact that G is faithful we find 0 f @(S,(W))>F)

= WW),WN

for somepEJ7r). Hence F(p) # 0 for some p E ) n’ 1. To complete the proof it suffices now to show that if there were an object q in Vwhich is not a retract of one in rr’, then there would be a functor F such

MITCHELL

40

that F(p) # 0 but P(p) = 0 for all p E 17~’ 1. The following proved for us by Michel Andre expressly for this purpose.

lemma was

LEMMA 8.4. Let 6I? be a category with a zero object, a nonxero object, and products, and let q be an object of r. Then there is a functor F : r -+ GZ such that F(q) # 0, whereas F(p) = 0 f or every p which does not contain q as a retract. Proof. Let A be a nonzero object. the set of retractions y : p --f q. Define F(P) =

If p E / 7r /, let r(p, q) denote

x A r(D,d

and let h, denote the product projections. Note that F(q) # 0 since 1, E r(q, q), and that F(p) = 0 f or all p which do not contain q as a retract. If x E ~(p’, p), define F(x) by F(x) 4l = b, =o

if v E O’, otherwise.

One sees that F is a functor by observing then x’xy is not a retraction.

9. HOMOLOGICAL

4)

that if xy is not a retraction,

DIMENSION

Throughout this section Q? and 93 will denote abelian categories. Associated with an abelian category QY we have the graded category Ext, whose objects are those of Q! and whose morphisms of degree n from A to C are the members of Ext$(A, C). Thus a morphism of, degree zero is just a morphism of C!?, whereas a morphism of degree 12 > 0 is represented by an exact sequence E in Q! of the form

E:OcAtB,t...cB,_,cCcO. We shall not bother to distinguish notationally between the sequence E and the element of Ext,“(A, C) which it represents. When the, abelian category in question is a functor category GP, we shall write Ext,n in place of Ext”,, . If T : GZ -+ SY is an exact functor, then we can apply T to the exact sequence E to obtain an exact sequence

T(E):Oc

T(A)c

T(B,)c "'t

T(B,Jt

T(C)+0

RINGS

WITH

in L&J;.It is easy to see that abetian groups Ext,“(A,

SEVERAL

OBJECTS

T thereby

induces

C) --t Ext$(

T(A),

41 a homomorphism

of

T(C))

and that these homomorphisms respect compositions in the graded categories. Thus T : GZ+ B extends to an additive functor T : Exta 4 Ext, which preserves degrees. If To, T, : GZ-+ 98 are exact functors and I’ : To -j T, is a natural transformation, then we have the commutative diagram in .G?

which shows that

as compositions in the category Ext, . In other words, r is still a natural transformation of the extended functors To , Tl : Exta ---t Ext, . Suppose that S : GZ--t g is a left adjoint for the functor T : .B + eC. Then we have the natural isomorphism @(A, WV) = ~‘(S(4, B)

(1)

and the adjunctions -5: TS+

19,

9: In-f

ST

(2)

satisfying

(T V)(E. T) = 1,)

(q * S)(S * c) = 1,.

(3)

Now if S and T are exact, then the transformations (2) and the equalities (3) are still valid for the extended functors. This shows that T : Ext, --t Ext, has S : Ext, + ExtS as its left adjoint. Reexpressing adjointness in terms of the definition (I), this yields a natural isomorphism Ext&A,

T(B))

of abelian group valued bifunctors.

= Ext,“(S(A),

B)

(4)

42

MITCHELL

The isomorphism (4) is Lemma 1.1 of [38]. The above slick way of obtaining it is due to Brinkmann [8], who has used this and another trick to simplify substantially the exposition of the Yoneda Ext functor. Let T : 8? + GZ be an exact functor with a left adjoint 5’ : GZ + g. Then from (1) it follows that S preserves projectives. Therefore if S is also exact, then S preserves projective resolutions. Hence when GZ and 98 have enough projectives, the isomorphism (4) follows from (I). Now even when S is not exact, it may be true that a particular object A has a projective resolution which S preserves. In this case we still have the isomorphism (4) valid for that particular A and natural in B. Recall that the homological dimension of an object A E 16Z 1 is defined by h.d. A = sup{n 1Ext”(A,

) # O}.

In particular, this means that h.d. 0 = -GO. (In additivity formulas involving homological dimension, it is useful to add the convention --co + co = -co to the usual conventions regarding the addition of the extended numbers.) When there may be confusion of categories, we shall write h.d.g A, or h.d., when the abelian category in question is a functor category GP. The global dimension of Q! is defined by gl dim GZ = sup{n 1Ext” # 0} = sup{h.d. A / A E 1Ol I} Because of the duality Ext&(C, A) = Ext,“(A, C), gl dim rZ* = gl dim 02. If 59 is a small additive category,

we see that we define

r.gl dim %?= gl dim AbQ, l.gl dim GF?= gl dim Abq*. As remarked in Section 4, the (left or right) global dimension of 9 is zero if and only if 9? is semisimple. A short exact sequence 0 ---f C -+ B + A ---f 0 in G! induces a long exact sequence ... + Ext”-‘(C,

X) -

Ext”(A, X) ---f Extn(B, X) -+ Extn(C, X)

+ Extn+l(A, X) + ..’ of abelian groups.

This

gives:

RINGS WITH LEMMA

SEVERAL OBJECTS

Suppose that 0 + C + B t

9.1.

43

A ---f 0 is exact.

(a) If h.d. B = h.d. C, then h.d. A < 1 + h.d. C, (b)

If h.d. B < h.d. C, t/Zen h.d. A = 1 + h.d. C,

(c)

If h.d. B > h.d. C, then h.d. A = h.d. B.

COROLLARY

9.2.

Suppose that

is exact (n >, 0). If h.d. Bi < h.d. C for h.d. A = n + h.d. C. If h.d. Bi < h.d. C for h.d. A < n + h.d. C.

0 < i < n - 1, then 0 < i < n - 1, then

Remark. If h.d. Bi < h.d. C for 0 < i < n - 1 and if A = C, then the first statement of the corollary implies that h.d. A = 00. LEMMA

9.3.

The following are equivalent (0 < n < GO):

(a)

h.d. A < n,

(b)

ExP+r(A,

(c)

Ext”(A,

) = 0, ) preserves epimorphisms.

Proof (a) * (b) by definition. (b) * (c) by the long exact sequence. (c) =‘ (b) by th e 1ong exact sequence, in view of the fact that a sequence of length n + I is the splice of a sequence of length n and a short exact sequence. (b) + (a) since any exact sequence of length >n + I is the splice of one of length n + 1 with some other. Taking n = 0 in the lemma, we see that P is projective if and only if h.d. P < 0. Suppose that O+C+P,_,+..

.+P,-,P,-,A+O

(5)

is exact with Pi projective for 0 < i < n - 1. If C is not projective, then by Corollary 9.2, h.d. A > n. On the other hand, if C is projective, then again by Corollary 9.2, h.d. A < n. Hence if LX has enough projectives, then the homological dimension of a nonzero object A can be defined as the least integer n (or CO) for which there is an exact sequence (5) with C and the Pi projective. If OThas enough projectives, and if (5) is exact with Pi projective for 0 < i < n - 1, then C will be called the n-th kernel in a projective resolution for A. By Schanuel’s

44

MITCHELL

lemma it follows that if C and C’ are both n-th kernels in projective resolutions for A, then there exist projective objects P and P’ such that COP % COP’. LEMMA 9.4. If S : U --+ 93’ is an exact functor T : g -+ GY, then

left adjoint for

the exact

h.dg AS < h.d., A [email protected],f UTth er, every object in LX is a retract of one of the form BT, then we have equality, and consequently gl dim G!Y< gl dim 9’. Proof. The first inequality is an immediate consequence of (4)i Now suppose that C is a retract of BT. Then Ext&A, C) is a retract to Extgfi(AS, B). of Ext&A, BT), and by (4) the latter is isomorphic This shows h.d. AS > h.d. A, as required. Remark 1. M. Barr has observed that the assumption that C be a retract of BT for some B is equivalent to the assumption that the For if qc is a coretraction, adjunction vc : C -+ CST be a coretraction. then we may take B = CS. On the other hand, suppose C -5 BT % C is the identity. Then we have a commutative diagram C %1 csT-E+

which

shows

--“-+BT

B+C

1 BTST

B

VBT t

T

BT

that rlc is a coretraction.

Remark 2. In the case where a and 9 have enough projective-s, if T is exact and A has a projective resolution which S preserves, then (4) is still valid, and so the conclusions of the lemma are valid for thatt particular A. Let $2 be a small additive category, and let [%I be the associated rin of row and column finite matrices. Then by Theorem 7.1 we have functors Abrv’l -L ‘s

Ab”,

RINGS

WITH

SEVERAL

OBJECTS

45

where S is a full and faithful left adjoint for T. Since r) in this case is a natural equivalence, by Lemma 10.4 we obtain r.gl dim[YTJ 3 r.gl dim %?.

(6)

When 1%?1 is finite, T is an equivalence, and so in this case (6) is an equality. On the other hand, when 9 has an infinite number of nonzero objects, [%‘I then contains an infinite direct product of nonzero subrings, and so by a theorem of Osofsky [42], [Vj is never hereditary. Thus if V is a semisimple category with an infinite number of nonzero objects, then the left side of (6) exceeds the right side by at least two. Consider a family {cpli j i E 1) of abelian categories. Then the product category x1 6?, is abelian, and we have an obvious surjection

That it is also an injection follows from a theorem of Yoneda, which says that it is always possible to get from one representation of an element of Ext”(A, C) to another using only one intermediate sequence. A short proof of this has been given by Brinkmann [9]. Actually it suffices to know that the number of intermediate sequences necessary is bounded. From the isomorphism (7) it follows that h.d.(AJ = supI h.d. Ai , and consequently gl dim )( OZj= sup gl dim G&. I

I

From the equality (3) of Section 2, it follows, in particular, that r.gl dim u Vi = sup r.gl dim Vl I

I

relative to a family of small additive categories V< .

Remark.

If (I, and (1, are rings, then r.gl dim /1,

x

fl, = max(r.gl dim fl, , r.gl dim /1,)

This formula is not valid for general additive categories. For example, if K is a field and 2 is the totally ordered set of two elements, then as we shall see presently, r.gl dim K2 = 1,

46

MITCHELL

whereas r.gl dim(K2)

(K2) = r.gl dim(K

x

x

K)(2

x

2) = 2.

This phenomenon is perhaps explained by the fact that the obvious ring isomorphism A, x A, = [Al u A,] makes no sense if A, and A, are not rings. If 12?~= Q! for all i E I, then we denote the product category by a’. This is consistent with our notation for functor categories, since a set P can be considered as a discrete category, or in other words a disjoint union of copies of 1. LEMMA

9.5.

If a has exact I-fold

coproducts,

then there is a natural

isomorphism Ext” (@ Ai, Cj = )( Ext”(A< , C). I I Consequently, h.d. @ Ai = sup h.d. Ai . I I Proof. The diagonal functor A : r2! + 62” has as its left adjoint functor @I : 6!!’ + a which is assumed exact. The isomorphism in this case becomes Ext”((A,), Composing

this

with

d(C))

= Ext” (0

the isomorphism

I

thq (41

Ai 7 C).

(7) yields

the required

one.

LEMMA 9.6. Let L : 12 + 23 and T : 33 -+ OT be exact functors and suppose that there is a natural transformation 4 : la + LT which is a coretraction. If E E Ext$(A, C) and E # 0, then ELT # 0, and so EL # 0. Consequently, h.d. A < h.d. AL, and so gl dim M < gl dim g’.

Proof.

Let # be a retraction

for 4. Then

E = W,#, and so if E # 0, then ELT Remark. to assume

= +,ELTslr, ,

# 0.

Naturality of I,/Jis not used. In other words, that 4 is pointwise a coretraction.

it would

suffice

RINGS WITH

10. PROJECTIVE

SEVERAL OBJECTS AND FLAT

47

FUNCTORS

Throughout this section, Q? will denote a small additive category. By a direct (inverse) limit we shall mean a colimit over (the dual of) a directed set. An AB4(AB4*) category is an abelian category with exact coproducts (products). An AB5(AB5*) category is an abelian category with exact direct (inverse) limits. It was remarked by Grothendieck that a category which is at once AB5 and AB5* is trivial tsee [37, p. 861). PROPOSITION

10.1.

G is projective in Ab’* if and only if G is a retract

9f a free. COROLLARY 10.2. If G E Ab’*, then & G : ot’ -+ 6’1is exact relative to all AB4 categoriesQ?if and only if G is projective. Dually, ;f G E AbV, then hom,(G, ) : G!?’ + Q! is exact relative to all AB4* categories 0Z if and only if G is projective.

Proof. We prove the dual. If G is of the form Vr, , then hom,(G, F) = F(p), and so it is clear that horn&G, ) is exact in this case. Hence by AB4*, the same is true when G is free. Since a retract of an exact sequence is exact, the result now follows for general projective G in view of Proposition 10. I. Conversely if homV(G, ) is exact relative to all AB4* categories, then it is exact in particular for 0L = Ab. But in this case hom,(G, ) is the same as Ab’(G, ), and so G is projective. PROPOSITION 10.3. If GT is an abelian category with coproducts and if 0 -+ G’ ---f G + G” -+ 0 is exact in AbW* with G” projective, then

is exact in 0’ for any F E 0”. Proof. This follows since if G” is projective, then 0 -+ G’ -+ G + G” -+ 0 splits. PROPOSITION 10.4. If 0 --+ G’ - G --t G” - 0 is exact in AbV” (OYin any abelian category) and if G” is projective, then G’ is projective if and only if G is.

48

MITCHELL COROLLARY 10.5.

Let

be an exact sequence of projectives in Ab’*, abelian category with coproducts. Then

and let F E 6Y’ where GY is an

is exact in GZ. We define a functor G E Ab’* to be jlat if Qv G : Ab’ ---f Ab is: exact. We observe that projectives in Ab’* are flat by Corollary 10.2, and that direct limits of flats are flat in view of the fact that & commutes with direct limits. Each of the above propositions on projectives has its analogue for flats. However, the proofs are not quite so immediate.: THEOREM 10.1 b (D. Lazard-V. Govorov). A functor if and only if it is a direct limit of jinitely generated frees. Proof. Lazard’s module that of Govorov [20].

theoretic

proof

in Ab’*

[32] generalizes,

is jlat

as does

COROLLARY 10.2b. If GE Ab”*, then @& G : GZV ---f GZ is exact relative to all AB5 categories GZ if and only if G ispat. Dually, if G E Ab”,: then homv(G, ) : GZ” -+ C? is exact relative to all ABS* categories a! if and only if G is flat. Proof. If G is flat, then @& G is exact GZ by Theorem 10.1 b and Corollary 10.2, commutes with direct limits. Conversely, all AB5 categories 67, then in particular definition that G is flat.

relative to all AB5 categories: in view of the fact that 0% if & G is exact relative to taking (rl = Ab, we see by

PROPOSITION 10.3b. Let F E a’ where Q? is AB5. If 0 -+ G’ --f G -+ G” ---f 0 is exact in Ab’* with G” $at, then

is exact in GZ.

RINGS WITH

49

SEVERAL OBJECTS

Proof. By Theorem 10.1b, we can write G” as a direct limit of free G; . Then G = ld Gi where Gi is defined by the pullback diagram

O-G’+

!I 1 1

G 4 G”+O.

By Proposition 10.3, the sequences 0 -+ G’ + Gi + G; -+ 0 remain exact when tensored with F. Since F & commutes with direct limits, the conclusion follows from AB5 for GZ. 10.4b. If 0 + G’ -+ G -+ G” -+ 0 is exact in Ab’* znd if G” is flat, then G’ is jlat if and only ;f G is. COROLLARY

Proof. Let F’ + F be a monomorphism in Abg. Form the commutative diagram

The rows are exact by Corollary 10.3b, and the right vertical arrow is a monomorphism since G” is flat. From this it follows easily that the middle vertical arrow is a monomorphism if and only if the left one is. In other words, G is flat if and only if G’ is. COROLLARY

10.5b. Let . ..-x.+x,-,+

. ..--f Xl+

X,,+

G-+0

be an exact sequenceof fiat functors in Ab’*, and let FE Olg where 12 is AB5. Then ...-FO.X,-tFO,X,_,-t...-tFO,X,-tFO,G-tO is exact in LT. We now discuss briefly the homological dimension of a flat functor. It is well known (see, for example, Lazard [32, Corollary 1.41) that if ME Ab” is flat and finitely presented, then M is projective. In [26] Jensen proved that if M is flat and countably presented, then h.d. M < 1. Osofsky [43] extended these results by showing that if M is flat and 607/S/2-4

MITCHELL

50

X,-presented, then h.d. M < n + 1. Her proof was based on that of Jensen. A slightly erroneous, but simpler proof for the N, case was given by Lazard in [32]. Lazard has recently shown us how his proof can be corrected without too much extra effort. His proof also works for the N, case. Explicitly, what he shows is that if M is flat and N,presented, then the directed set of Theorem 10.1 b can be taken to have N, elements. In view of another theorem of Osofsky on the homological dimension of a direct limit (Theorem 16.1 of the present paper), this? implies immediately that h.d. M < n + 1. We shall not give details1 here, but shall only mention that in view of the examples of Section 36,, this is the best possible result. Examples of this nature for the case where % is actually a ring can be found in [43]. Thus in some sense a flat module can be as far from projective as one wishes.

11. K-CATEGORIES Let G! be an additive category, and let C(a) denote the class of endomorphisms of the identity functor 1n . Then C(a) is a commutative ring with identity (neglecting the fact that it may not be a set) which we call the center of GY.If/l is a ring, then considered as an additive category, its center is the subring of all elements c such that ch = hc for all h E fl. If %?is any small additive category, then C(A6’) is isomorphic to C(Y).’ Let K be a commutative ring. A K-category is an additive category; 577’together with a ring homomorphism K -j C(g). Equivalently, a K-category is a category a equipped with a K-module structure on each horn set in such a way that composition induces K-module homomorphisms q&4’)

&

6q(A’, A”) + @(A, A”).

Thus, in the language of closed categories, a K-category is a category which is enriched by the closed category of K-modules. A K-category with precisely one object is an associative K-algebra. A Z-category is just an additive category. Let %?be a small additive category, and let 0! be a K-category vid 4 : K + C(a). Then @V becomes a K-category using the ring homomorphism 4 : K + C(@!“) defined by 6(W)

= F W+

On the other hand, if V is a small K-category via $ : K -+ C(e) and 08

RINGS

is an additive homomorphism

WITH

SEVERAL

51

OBJECTS

category, then G!? becomes a K-category 4 : K + C(fl’) defined by

using the ring

If Q! and V are both K-categories, then these two K-category structures on 02% are not necessarily the same. For this reason we define a K-functor F : %’ + @ to be an additive functor satisfying F .4(k) for all k E K. It is equivalent

= t&z) .F

to require that the induced

maps

VC, C’) + @WC), FCC’)) be all K-module homomorphisms. When %?and 0? are both K-categories, we shall take @’ to be the full subcategory of the additive functor category consisting of all K-functors. With this convention, the two K-category structures on Q?’ defined above coincide. The limit of a diagram of K-functors is again a K-functor. In particular, if a is complete, then so is @. Likewise for colimits. Many of the definitions and constructions of the preceding sections carry through with additive categories replaced by K-categories. In particular, if 9?r and %?‘?zare K-categories, then we can form the Kcategory Vi OK Vz in the obvious way. The identities and discussion of Section 2 are valid relative to this tensor product. Of course the trivial additive category Z must be replaced here by the trivial K-category K, and the free additive category Zx is replaced by the free K-category Kvr, If V1 is a K-category and Vz is an L-category, then %i oz V2 is a K oz L category. The matrix ring [V] of a small K-category is an associative K-algebra, and Theorem 7.1 is valid relative to categories of K-functors. If n is a finite category with n morphisms, then [KT] is an n-dimensional K-algebra. If 91 is an additive category, then OTK is a K-category as described above, and if % is a small K-category, then

In particular, the category of K-functors from V to K-modules is isomorphic to the category of additive functors from %’ to abelian groups. Let G E Abv* and F E @ where 9? and Q? are K-categories using ring homomorphisms # and 4, respectively. If a is cocomplete, then we can

52

MITCHELL

form the tensor product F Q. G by regarding functor. If k E K, then we have

F as an ordinary

additive

+(k)(F 00 G)= (F.W4@aG = (W).F)00:G = FOvVW. G). The first and third equalities are clear when G is representable, and follow for general G by Corollary 6.2. The second equality follows from the fact that F is a K-functor. The third equality shows that QQ is defined on the category GY’ & Ab’*, and the first shows that it is a K-functor. A similar remark applies to symbolic horn. The natural equivalences of Section 6 are all valid relative to a K-category Q!, providing the tensor product over Z is replaced by the tensor product over K. In verifying that the associative law (5) of Section 6 is still valid, one must know that

relative to F E @@K” and G E Ab’*@@*. This is a consequence of the following easily established principle: If V + ‘3’ is a full, additive functor which is a bijection on objects, then for FE @ and G E AbQ*, we have F & G = F &’ G. If U : $9 + g is a K-functor and if a is a K-category, then since the composition of K-functors is again one, we get an induced functor

When

OT is cocomplete,

the formula

WW = FOvg(u,d for the left adjoint

S, of g” %&ml)

is still valid, When

+? = K, this becomes

= A OK WJY 9),

where S, is the left adjoint for the evaluation functor Dually, the right adjoints of GP and T, are given by &(F)(q) and

= homW(q,

U),F)

T, : Q?’ + OT.

RINGS WITH

53

SEVERAL OBJECTS

Let G E A’@@. We shall say that G is a-projective (9-f&) if G(p, ) is projective (flat) in Ab” for all p E / %?(. We say that G is K-projective (K-flat) if G(p, q) is a projective (flat) K-module for all p E 1%?1, q E / 3 I. In particular, a K-category V will be called K-projective (K-flat) if V(p, q) is a projective (flat) K-module for all p, q E / %?/. If U : %Y-+ 9 is a K-functor, then U can be regarded as a natural transformation from the two-variable functor %?to the two-variable functor g( U, U), or in other words a morphism in Ab’*@K’. As such, one can impose the condition that it be a coretraction. This is much weaker than the condition that U be a coretraction as a K-functor, i.e., that there exist a K-functor V : J?/ + V such that UP’ is the identity. For example, if rr is a group and 7 is a subgroup, then KT -+ Kr is always a coretraction of T* x T-modules, but it is seldom a coretraction of K-algebras. PROPOSITION 11.1. Let U : 9 --t 9 be a K-functor. Then the fun&or So is exact relative to all AB4 (AB5) K-categories GZ;f and only if 9( U, ) is %‘*-projective (flat). In this case we have

h.d.S S,(F) < h.d+ F for all F E @. If, f ur th er, U is a coretraction consideredas a morphism in Abv* @Kw, then equality holds, and consequently gl dim 02%< gl dim 6P. Proof. The first statement follows from and the other two follow from Lemma 9.4.

Corollary

10.2 (10.2b),

COROLLARY 11.2. Let 9 be a K-category, and let p E j 9 /. Then the functor S, is exact relative to all AB4 (AB5) K-categories 91 zf and only zf 9(p, q) is K-projective (flat) f or all q E ) B 1. In this casewe have

h.d.y S,(A) < h.d., A for all AEjGPII. If,f ur th er, 9(p, p) contains K as a K-module retract, then equality holds, and consequently gl dim cpl< gl dim 0P. PROPOSITION

11.3. Let U : $7 + 9 be a K-functor.

Then the functor

54

MITCHELL

R, is exact relative to all AB4* K-categories is Y-projective. In this case we have

Q? if and only if 9(

, U)

h.d.% UG < h.d.B G for all GE U9. COROLLARY 11.4. Let 9 be a K-category, and let p E 129 (. Then the functor R* is exact relative to all AB4* K-categories 6X if and only if 9(q, p) is K-projective for all q E j 9 1. In this case we have h.d., G(p) < h.d.9 G for all GE rZ9. Remark. The strict duals of 11.1 and 11.2 would contain statements “injective” dimension, and would also contain statement about concerning AB5* categories. We define a K-category %? to be supplemented if there is given a K-functor 9 --t K. Note that if r is any category, then the unique functor rr -+ 1 extends uniquely to a K-functor Kn + K, thereby giving Kn the structure of a supplemented K-category. If $7 is a supplemented K-category, then any object in V gives rise to a K-functor K + V, and the composition is necessarily the identity K + %?-+ K. Thus we see that the condition that V be a nonempty supplemented K-category is equivalent to the condition that +? contain K as a Kcategory retract. PROPOSITION 11.5. Let $7 -% 9 2 % be K-functors such that UV contains lQ as a retract. If G? is any abelian K-category and F E a’, then h.d.@F < h.d.B VF,

(1)

gl dim dv < gl dim a9.

(2)

and so

In particular,

if 9 is any nonempty supplemented K-category, gl dim fl < gl dim C@‘.

then (3)

Proof. Since UV contains lV as a retract, it is an easy exercise in the Godement rules to verify that OZvOP contains the identity on GZV as a retract. Consequently, the proposition is a special case of Lemma 9.6.

RINGS WITH

SEVERAL OBJECTS

55

Remark 1. Inequality (3) is not valid without some hypothesis on 9. For example, if 9 is a field, then gl dim AbQ = 0, whereas gl dim Ab = 1. Remark 2. The hypothesis that UV contain 1%as a retract implies that U is a coretraction considered as a natural transformation. For if 1$+ uv% 1%is the identity, then the composition

is the identity, where the last map is %‘(+, , I/J,). We now consider some propositions of Cartan and Eilenberg [l 1, Chap. IX, Section 21, which will be needed in the sequel. PROPOSITION 11.6. Consider F E 61W and G E AbQ*@Ko, where 01 is an abelian K-category with coproducts. If either (i) F has projective values and G is projective, or (ii) F is projective, G is 9-projective, and 0Z is AB4*, then F & G is projective in 0!“.

Proof. Both statements are immediate consequences of the natural isomorphisms 6T9(F &

G, H) = Ab ‘*‘K9(G, 6T(F,H)) = 6f(F, homp(G,

H)).

Taking 9? = K, we obtain from part (i) of the proposition: COROLLARY 11.7. If @ is an abelian K-category with coproducts, and if A is projective in 0? and G is projective in Abe’, then A ok. G is projective in Q!‘. PROPOSITION 11.8. Let V and 2 be K-projective K-categories, and let M be an A%4 and AB4* K-category. Let X be a projective resolution for F in @“, and let Y be a projective resolution for G in Ab’* @KG. If TornY;‘(F, G) = 0 f or all n > 0, then X 0% Y is a projective resolution for F 0% G in Qt”.

Proof. Since V is K-projective, it follows from Corollary 11.4 that Y is g-projective. Therefore by part (ii) of Proposition 11.6, X 0% Y is projective in QZz. It remains to be shown that X & Y is an acyclic left complex over F & G. But now since 9 is K-projective, we see again from Corollary 11.4 that Y is V-projective. Therefore since X is an acyclic left complex over F and a is AB4, we find E&(X

@g Y) = Tornv(F,

G) = 0,

for

n > 0.

MITCHELL

56 That H,,(X & tensor product.

Y) = F &

COROLLARY 11.9. Let be an AB4 and AB4* for A in 0J and let Y TornK(A, G) = 0 f or all for A OK G in G19.

G follows

from

right

exactness

53 be a K-projective K-category, and let GI? K-category. Let X be a projective resolution be a projective resolution for G in Ab9. If n > 0, then X OK Y is a projective resolution

Analogous to Proposition for flat functors.

11.6, we have the following

proposition

PROPOSITION 11.6b. Let F E AbV@Kd* and G E AbV*@K9. $9 QK &* flat and G is Sjlat, then F & G is b* OK g-flat. Proof.

of the

This is an immediate

consequence

12. DIMENSION

If

F

is

of the natural isomorphism

OF K-CATEGORIES

If %? is a small K-category, then the K-category %‘* OK ‘3 will be called the enveloping category of %?, and will be denoted by %F. We can then regard %?as an object of AbVP”. If F is another object of Ab”“, then we define the n-th (Hochschild) cohomology group of 5? with coeflcients in F as H”(V, F) = Ext;,(%, F). The cohomological

dimension (or simply

dimension)

dim, C = sup{n 1H”(%,

of V is defined

as

) # 0}

= h.d.,, %Y. When there is no confusion as to the ground ring we shall simply write dim %. We have an obvious isomorphism Ve = (Q?*)“, and under the induced isomorphism

the

object

V

corresponds

to

the

object

SF*.

Consequently,

the

RINGS

cohomology groups dim %?* = dim %. If n is any small dim, n. Thus dim, as an object in the

WITH

SEVERAL

57

OBJECTS

of %‘* are the same as those of V, and we

have

category, then we shall denote dim, Kn- simply by 71 is the homological dimension of KTT considered category

When K = Z we shall write simply dim 7~ in place of dim, V. Let % -+ 9 be a full and faithful K-functor which induces an equivalence GP = G?? for all abelian K-categories O!. Then in particular we get an equivalence Ab”” w Ab%“, and because V + 9 is full and faithful, we see that under this equivalence the object 9 goes to the object %?. It follows that ‘2? and B have isomorphic cohomology groups, and so dim %? = dim 9. In particular, this will be true if %? is equivalent to 9 (via a K-functor) or if 9 is an idempotent completion for %‘. It is also true if %? = Kn and 9 = K+ where ii is an idempotent completion for V. If %? and 23 are small K-categories, then there is a K-algebra homomorphism given by

If % and 9 are finite, then 4 as a K-module the identification

homomorphism

is simply

Hence in this case $ is a K-algebra isomorphism. Consequently, if 02 is an amenable category, then using Theorem 7.1 we obtain an equivalence ~MloKr91

In particular,

~

this yields an equivalence

@a,9

58

MITCHELL

and under this equivalence the two sided [Wj-module [U] corresponds to the two variable functor %?. Consequently, [%?I and 9? have isomorphic cohomology groups, and so dim[%?] = dim %‘. This is, of course, under the assumption that 1%F( is finite. In particular, let 7~be any finite category equivalent to 1, and let A be a K-algebra. Then we have a K-category equivalence A w AT, and so we obtain dim A = dim A?r = dim[Ax]. But [An] is just the K-algebra M,(A) of n x n matrices with entries in A, where n = / rr 1. This proves the well-known equality (see [14]) dim, A = dim, A&(A),

n 2 1.

In particular, when A = K, we see that dim, M,(K) Let V = lJ, %?t, a disjoint union of K-categories.

= 0. Then

%?~=%?*&~=(Jwf*&~?j, IXI

and so AbY;”

=

)(

Abwt*@Kq;’

.

1x1

Moreover, under this identification, %? goes to the I x I-tuple in position (i, i) and 0 at other positions. It follows that

with

pi

dim V = sup dim 5~7~. I In particular,

if A, and A, are K-algebras,

then

and, consequently, dim, A, x A, = max(dim, This formula is not valid Suppose that U : %?+ there exists a K-functor to conclude from Lemma

A, , dim, A,).

for more general K-categories. 9 is a coretraction of K-categories, so that V : 9 -+ %? with UV = lV . One is tempted 9.6, using the induced coretraction Ab”*@K”:

Abe”

+

Ab”“.

RINGS

WITH

SEVERAL

59

OBJECTS

that dim V < dim 9. However it is not true that under this coretraction the object V is taken to the object Q unless V is full and faithful, and hence an equivalence. Nevertheless we shall show in the following section that if further 9 is K-flat, then dim 99 < dim 9. On the other hand, if we assume the weaker condition that U be a coretraction as a natural transformation, and add the conditions that a( U, ) and 9( , U) be V* and V-projective, respectively, then we obtain the stronger conclusion of the following proposition. 12.1. Let U : % + 2 be a K-functor which is a coretraction considered as a morphism in Abv6, and suppose that QJ( U, ) and S( , U) are %T* and %-projective, respectively. If O! is any AB4* K-category with coproducts, then we have PROPOSITION

for all A E j 67 1. In particular, Proof.

Let V be the K-functor c?‘“(

and by replacing

dim 55 < dim Y.

, V)

=

U* @JK U : Se + 9;‘. Then qu,

)&q

1 U),

Corollary 11.7, the right side is Ve-projective. I: by V in Proposition 11.3, we obtain h.d.,,A

&9(U,

U) < h.d.,,A

@&9.

But also, because U is a coretraction in Ab”“, retract of A OK 23( U, U) in Ab”“. Hence

Therefore

(1)

we see that A OK %? is a

h.d.,, A OK g < h.d.,, A OK 9( U, U).

(2)

The result now follows from (1) and (2). If %? is a small additive category and FE Ab%‘, we define the weak dimension of F to be w.dimF

= sup{n 1Tor,(F,

) # 0}

If V is a K-category and G is an object of Ab”“, group of %? with coefficients in G is defined as H,(%?, G) = Torr(%?, G).

then the n-th homology

60

MITCHELL

The weak, or homological defined as

dimension

of the K-category

w.dim % = sup(n ( H,(%,

$? can then be

) # 01.

However, in order to keep our list of propositions from becoming oppressively long, we shall say nothing about weak dimension in this paper. The interested reader can extend well known facts concerning weak dimension of modules to facts about weak dimension of functors.

13. LEMMA

(K-j7at),

THE

SUBADDITIVITY

THEOREM

13.1. If P is projective ( jlat) in Ab’* @@ and 9 is K-projective then P is V*-prqjective (V*-Jat).

Proof. The assertion on projectivity follows from Corollary 11.4 by taking O! to be Ab Q*. It also follows from part (ii) of Proposition 11.6 by replacing a, %?, 9, and G in that proposition by AbW*, 9, K, and gQ* for each q E 1 9 I. Similarly, the assertion on flatness follows from proposition 11.6b. LEMMA

K-projective

13.2. Suppose that P is projective in AbW*@K”. If 9 i$ (K-jlat) and 02 is AB4 (AB5), then for any F in 02” we have h.d.g F @a P < ‘;g h.d.@ F(p).

Proof. Since a has exact coproducts and since a projective is a retract of a free, it follows from Lemma 9.5 that it suffices to conside the case where P is of the form ep* OK sr . But in this case we hav 1

F 0~ P = F 0~ g,* OK 97 = F(P) OK gr = %(F(P)), where S, is the left adjoint of the evaluation functor T, : Gla -+ GE Under either of the hypothesis on 9 and 6Y the functors S, are exact. The conclusion then follows from Lemma 9.4. THEOREM 13.3 (Subadditivity V-projective (V-flat) in Ab’*@‘@ !f Ol is an AB4 (AB5) K-category,

h.daF

&

Suppose that G i: theorem). and that 9 is K-projective (K-flat)i then for any F in a’, we have

G < h.d.rr*oKBG + 7;~ h.d.,F(p).

RINGS

Proof.

WITH

SEVERAL

61

OBJECTS

Let O+P,-+.

..+Pl+Po+G+O

(1)

be a projective resolution for G in &‘*@K*. then fixing the covariant variable throughout 13.1 an exact sequence of projectives (flats) tensor (1) over V with F, then by Corollary exact sequence in QP

If G is V-projective (flat), (1), we obtain by Lemma in Ab’*. Therefore, if we 10.5 (10.5b) we obtain an

O~FO,P,~...-,FO~~PP,~FO~PP,~FO,G~O. In view of Lemma 13.2, the theorem

(2)

now follows

by iterating

COROLLARY 13.4. Let V be K-projective (K-$at), AB4 (AB5) K-category. Then for FE OT’ we have

Lemma 9.1.

and let GZ be an

h.d.vF < dim,‘& + s-17 h.d.,F(p).

(3)

gl dim 02% < dim,% + gl dim a.

(4)

Consequently,

If G is K-projective

(K-jlat)

in Ab’,

h.d.vA & for all A in 1 GZ 1. In particular, h.d.vsnKv A &

then

G < h.d.vG + h.d.@A

(5)

we have %T< dim, V + h.d., A.

(6)

Consequently, dim,%? OK9 for all small K-categories Proof. Inequality subadditivity theorem, follows by taking 9 and (6) follows from V* OK V. Inequality in (6).

< dim,%

+ dim,9

(7)

9, valid under the assumption that 59 is K-jclat.

(3) follows by taking &8 = %’ and G = V in the and (4) is immediate from (3). Inequality (5) = V and Y = K in the subadditivity theorem, (5) since if ‘%?is K-projective (K-flat), then so is (7) follows by taking Q? = Ab9*@@ and A = 9

Acknowledgement. The inequality where V is a K-projective K-algebra

(3) was shown by R. Swan.

to us in the case

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Remark 1. The upper bound (3) is the best possible in the case where Q? = Ab’*. For considering Y as an object in (Ab”)‘, its values are projective in Ab’* since they are just the contravariant horn functors. Hence if d is an integer such that h.d. F < d + sup,%, h.d. F(p) for all F in (Ab’*)V, then taking F = %?we see that dim %?< d. Remark 2. In Cartan and Eilenberg [l 1, p. 1771, it is shown that if K is a field and %?and 9 are finitely generated K-algebras, then (7) is an equality. In Section 34 we shall exhibit examples where V and LB are finitely generated Z-free Z-algebras and the inequality is strict. Since we shall be primarily interested in K-categories of the form Kr., we shall restate Corollary 13.4 for this case. COROLLARY 13.4’. Let Q! be an AB4 (AB5) K-category and let v be a small category. Then for F in ali we have

h.d.,F < dim, T + SLIPh.d.nF(p).

(3’)

Consequently, gl dim 6P < dim, n + gl dim OZ.

If G is K-projective (K-flat)

in AbK”, then

h.d., A & for

(4')

G < h.d., G + h.d., A

(5’)

all A E j 6Y 1. In particular, h.d. n*XnAT < dim, n + h.d., A

forallA&

(6’)

16!1, andso dim, 9?~ < dim, QT+ dim, V

(7')

for all small K-categories V. COROLLARY 13.5. Let V 5 9 4; %? be K-functors such that contains the identity functor on %?as a retract. If 9 is K-pat, then

UP

dim V < dim B.

In particular, if 7 + r -+ r contains the identity on 02 as a retract, then dim, T < dim, rr.

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63

Proof. Let Ab’* and %?play the roles of OTand F, respectively, in Proposition 11.5. Then inequality (1) of that proposition yields dim %?< h.d. Gf?(, V). But the values of %?( , V) : 2 ---f Ab’* are representable, hence projective. Therefore the result follows from the inequality (3) of Corollary 13.4. Many of the theorems of the remainder of the paper will have to do with when the inequalities of corollary 13.4’ are equalities. To this end, the following lemma will be useful. LEMMA 13.6. Suppose that % is K-projective (K-flat), and let G be K-projective (K-flat) in Ab ‘. Suppose that D is the (n - l)-st kernel in a projective resolution for G, and let A be an object in an AB4 (AB5) category lZ?such that

h.d.%A &D

> h.d.a A.

(8)

Then h.d.w A OK G > n + h.d., A. Proof.

(9)

Consider an exact sequence in Ab’ O-tD-tP,_,j...jP,-tP,-tG-tO,

where Pi is projective for 0 < i < n - 2. Since % and G are Kprojective (flat) we obtain an exact sequence in 91’

By Lemma 13.2 we have h.d., A OK Pi < h.d. A. Hence the result follows from Corollary 9.2.

14. CHANGE

OF BASE

Let %?be a small K-category, and let L be an associative K-algebra. The functor U : V -+ V?OK L defined by U(y) = y @ 1 induces the functor

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whose left adjoint A’,] is easily seen to be given by S,(F) = F &L using the general form of S, derived in Section 6. If L is K-flat, then S is exact. On the other hand if % is K-flat and F is K-flat, then every term in a projective resolution for F will be K-flat. From corollary 10.5b’ it follows that S preserves this projective resolution. Therefore by Lemma 9.4 and the remarks following it we obtain: LEMMA 14.1. Let 9? be a small K-category and L an associative K-algebra. If F E Ab’ and G E Ab ‘@KL and if either L is K-flat or F and 9 are K-flat, then we have a natural isomorphism

Ext$,,,(F

&L,

G) = ExtVn(F, G).

Consequently h.d.,,, F &L < h.d., F, where L contains K as a K-module retract.

with

equality

in the case

Now consider the case where L is a commutative K-algebra. %?OK L can be regarded as an L-category, and we have

Then

Therefore we can replace @ by %‘* OK %?and F by %?in Lemma 14.1 to obtain THEOREM 14.2 (Change of base theorem). Let %? be a K-category! and L a commutative K-algebra. Then if either L or %9is K-flat, we have a natural isomorphism

H,“(V &L,

G) = HKB(%‘,G)

where G E [email protected], dim, V mKL < dim,%, with equality in the casewhere L contains K as a K-module retract. COROLLARY 14.3. category r we have

If

L is a commutative K-algebra, then for any small dim, x < dim, 5~,

RINGS WITH

with equality

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when L contains K as a K-module

retract. In particular,

dim, < dim n

for all commutative

rings K.

Remark. Theorem 14.2 generalizes Proposition 7.1 on p. 177 of Cartan and Eilenberg [ 1I].

15. COHOMOLOGICAL

DIMENSION

OF GROUPS

Let 7~be a small category, and let O! be a cocomplete K-category. The functor

induces a functor

as explained in Section 2. We also have the functor Qnxn + OT-induced by the diagonal 7~-+ rr x rr. Composing, we obtain the functor

satisfying

If A : AbK + AbKR is the diagonal (or “trivial we have

D(F, AK) = F. When G is a horn functor KT(P, F, . Thus Fr, is given by

action”)

functor,

then

(1)

), we shall denote D(F, G) simply by

F,(d = FM 4~ 4 where the right side denotes the coproduct of n(p, q) copies of F(q). Denoting the x-th coproduct injection by V(X) for x E ~(p, q), the morphism F,( y) for y E r(q, q’) is given by +)F,(Y) 607/8/I-j

= F(Y) ~xY).

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On the other hand, if SD : a -+ GP is the left adjoint evaluation functor, then S,(F(p)) is given by S,(W))(2) 44 WTPMY)

for the p-th

= F(P) 4P7 41, = U(XY).

For each 4 E 17~ I, let us define a morphism

Then it is easily verified that 4 is a natural transformation from S,,(F( p)) to F1, . When r is a groupoid, 4 has an inverse # given by V(X) #, = F(x-1) U(X). It follows

from Lemma

9.4 that in this case we have h.d.,F9

if G! is an AB4 category.

= h.d.,F(p)

Since D preserves

coproducts,

h.d., D(F, P) < SIP h.d.,F(p) for any projective

we deduce that (2)

P in AbKff. Now let O-P,+

. ..+P.+P,,+AK-+O

be a projective resolution for AK in AbKr. Applying (l), we obtain an exact sequence

D(F, ) and using

0 + D(F, Pd) + . ..-+D(F.P,)+D(F,P,)+F+O in @. In view

of (2), this proves h.d.,F

< h.d., AK + s,up h.d.@F(p)

for all FE GP, valid under the assumption that OT is AB4 and 7~ is a groupoid. In particular, this implies (see Remark 1 to Corollary 13.4) dim,n

< h.d., AK,

RINGS WITH SEVERAL OBJECTS

67

when 7~ is a groupoid. Since the reverse inequality holds without assumption on 7~ ((3’) of C orollary 13.4’), we have proved: PROPOSITION 15.1.

If

7~ is a groupoid, dimKr

any

then

= h.d., AK.

This proposition was proved for groups in Cartan and Eilenberg [ll, p. 1951, as an application of the general “inverse process”. Of course it suffices to prove it for groups, since a groupoid is equivalent as a category to a disjoint union of groups. The proposition is not true for general categories 7~. For example if rr is a category with an initial object s, then AK is just the representable functor K~T(s, ) and consequently h.d. AK = 0. On the other hand, it is not difficult to see (Proposition 33.1) that a poset has cohomological dimension zero if and only if it is discrete. Nevertheless, for a certain class of posets we shall be able to reduce the computation of the cohomological dimension to a one sided problem (Theorem 23.9).

16. DERIVED FUNCTORS OF THE LIMIT

FUNCTORS

Let n be a small category and let K be a commutative ring. We shall denote the constant diagram at K in AbKv by AK and in AbKn* by A*K. If Q! is a cocomplete K-category, then using the adjointness of A* and b,, , we obtain for F E OTTand A EM, (F gKv A*K, A) = AbKr*(A*K,

CY(F, A))

= AbK(K, lim @F, A)) = lim cx(F,>) = G&F, By Yoneda, this composite natural equivalence 1bF I In the case where

natural

A).

equivalence

must

come from

=F&,A*K.

Q! = AbKv* and F = KIT E (AbKKn*)a, this gives limK?r=Kv&,A*K=A*K.

a (1)

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If %? is a small K-category and I% is Ab”, then we have, using the associativity of the tensor product,

F&,A*K

=(F@,2T)&,A*K =F@vm(Gf@O,A*K) = F &,, AV.

Hence we see that are given by

the left

derived

functors

: Abvq -+ Abv

of l&

Observe that l&, : GP --f GZ is exact relative to all AB5 categories C!? if and only if A*2 is flat, in which case by Theorem 10.1 b, A*2 will be a direct limit of finite frees. In particular, this is true when the components of 7r are directed categories. It is an open question (see Oberst [40] and Isbell [24]) as to whether the exactness of b, : A67 + Ab implies that the components of 7~ are directed categories. The dual of (1) is lim F = homK,(AK, F),

77

where F E CYV and OT is a complete h(n) m

K-category.

F = Ext&,(Ag,

The dual of (2) is

F)

P*)

u,

functors of b,, : Abe= --f Aby. Thus we see that : Abvv + Abe is exact if and only if A%“, is projective in Abvs

for

all p E 1%’ 1. More

for the right derived

generally

it

follows

from

Abvr + Ab’ is zero for all K > n if and only if h.d.,, p E [ V 1. From inequality providing

(5’) of Corollary

(2*)

that

&A”

13.4’ we see that this is true

h.d.,, AK < n. It has been shown by Goblot [19] that if number it, , then lim’“’ = 0 for K > n + as an easy consequence of the following For completeness we shall let K-r denote

:

05~2~ < n for all

7~* is a directed set of cardinal 1. We shall obtain this result theorem of Barbara Osofsky. any finite cardinal number.

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69

THEOREM 16.1 (Osofsky). Let n be a directed set of cardinal number N n > - 1, and let V be a small additive category. Then for any F in A:‘s we have

h.d.e l&F s

< n + 1 + ,“~‘tz,h.d.q F(p).

Proof. Osofsky’s proof for modules [43], and those of the lemmas of Auslander [2] and Berstein [6] on which it is based, all generalize. COROLLARY 16.2. If 7~* is a directed set of cardinal number N, , n > - 1, then h.d., AZ < n + 1. Consequently,

]im’k’: Aben + Abv vanishesfor k > n + 1 and all additive categories 9?. Proof.

We have lim Z7r = AZ, 7

and so since Zr(p, ) is projective in Abn for all p E 1n / the result follows immediately from the theorem. Remark. It will be shown in Section 35 that when r* is the ordered set of all ordinals of cardinal number less than K,~, then h.d., AZ = n + 1. We now generalize Theorem 16.1 to arbitrary AB5 categories. Explicitly, COROLLARY 16.3. If (2 is an AB5 category and rr is a directed set of cardinal number N, , then

h.d.,l&F for all F E &. Proof.

< n + 1 + m;, h.d.,F(p)

n

By (1) we have hF ?i

=F&,A*Z.

By Corollary 16.2, we know that h.d.,, A*Z < n + 1, and furthermore since 7r is directed, A*Z is flat. Therefore the result follows by taking %?= Zr and 9 = Z in the subadditivity Theorem 13.3.

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Remark. Corollary 16.3 is also valid relative to directed categories . . 7r, providing that R, is interpreted as the number of morphisms in n. For n > 0, this follows from a lemma, due independently to Swan and Deligne, to the effect that any directed category rr admits a cofinal functor Z-’ -+ 7~with 7~’ a directed set. Moreover, the number of elements of z-’ is no greater than the number of morphisms of 7r, providing the latter is infinite. If n is finite, then this argument breaks down. However, it is easy to see that a finite (or even finitely generated) category ,r is directed if and only if it contains an object 4 and an endomorphism 0 E +I, q), such that ~(p, q) # 4 f orallpE1rI,andsuchthatxB=yB whenever the equation makes sense. In this case the colimit of a functor FE 09 is given by the image of F(B), which is a retract of F(q), and consequently has homological dimension less than or equal to that of F(9)It follows that Corollary 16.2 is also valid in the case where VT* is a directed category with K, morphisms.

17. THE

STANDARD

COMPLEX

Let 59 be a small K-category. We consider the complex s(V) in Ab”” whose n-th term is zero for n < -1, and for n >, - 1 is given by &W> =

0

%( >A) OK~h

,A?) OK ... OKWn

,Pn+J oK+eJ,+l>

1,

(P1....,P,+I)

where the coproduct ranges over all n + l-fold in 9. In particular, ,.!-,(%‘) = %. The boundary

sequences of objects

is given for n 3 0 by

Clearly d, is natural in V* and in V, and it is easily verified that d, d,-, = 0. We also define s, : s,(V) -P S%+i(%?)for n > - 1 by Sn(Ol) = 1 @ CL

Then s, is natural in %?(but not in U*), and we have d,s,+,

+ snd,+l

= identity,

n > -1.

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WITH

SEVERAL

71

OBJECTS

Consequently, if we let S(%‘) be the complex obtained from s(V) by replacing 8-i(%) by 0, then S(V) is an acyclic left complex over % in /lb”. Moreover, if %?is K-projective, then

is a projective K-module for all sequences p, ,..., plLfl . It follows that S,(V) is projective in Ab Se for all n > 0, and so S(V) in this case is a projective resolution for 9?. We call S(%‘T) the standard resolution for $2’. If V has just one object, then S(V) is the usual standard resolution of the associative K-algebra given in Cartan and Eilenberg [ll, p. 1741. Now let 9 be another K-category, and consider G E Ab’*@@. Since for fixed values of the contravariant variable the complex S(V) is split in Abv, it follows that S(V) ga G is an acyclic left complex over V@&G=GinAb ‘*@KB . Furthermore if %? is K-projective and G is %projective, we see from Proposition 11.6 that S,(V) & G is projective in Abv*@KB for all n > 0. Hence in this case S(Y) & G is a projective resolution for G. In particular, this is the case if 9 is K-projective and G is of the form B( U, ) where U : 9?’-+ 9 is a K-functor. Now suppose that %?is K-projective, and G E Ab’*@@’ is K-projective. Then S,(V) & G is %*-projective, and so S(e) au G is a %‘*projective resolution for G. Therefore if FE MV: where OT is an abelian category with coproducts, then we find H,(F ov: S(V) go G) = TornV(F, G). Similarly

if F E (I!‘*, Hn(hom&S(%)

BV:G, 8’)) = Ext&(G,

F).

(3)

In particular, these formulas are valid when G is of the form 9( 72, ) where U : V + 9 is a K-functor between K-projective K-categories. As a special case, let 7r ---f 5 be the inclusion of a small subcategory, and let U : Zrr --f Z+ be the induced additive functor. If F E G!? and p E j 7j 1, then the n-th term of the complex

is given by @ F(dom 4, <%.....%+&

(4)

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where the coproduct ranges over all composable sequences (01~ ,..., a,+i> with 01~,..,, 01, in r and cod 01,+i = q. Denoting the coproduct injections of (4) by u(oli ,..., 01%+i), the boundary formula is given by 4% ,..., s+J

dn = J’(4

4~2 >..., an+l?

+ g (-1y

4011 ,..., ‘y&Q+1 ,.**, %+J.

This is the complex whose n-th homology was defined by Andre in [1]1 as the n-th derived functor of F with respect to the inclusion r + ii.! Formulas (2) and (3) then bear out the observation of Ulmer [51] andi Oberst [40] that Andre (co) homology can be computed as the derived functors of the Kan extension functors. If on the left side of the boundary formula (1) we have a:i = lPi for some i satisfying 1 ,< i < n, then the same is true of all except possibly! two of the terms on the right side, and these two terms cancel. Thus, let us set @(PY P’) = U(PY P’)

if

p # p’,

@(p, p) = Cokernel (K + V( p, p)). Then we can define for n > 0,

and by the remark above, the boundaries d, induce boundaries on N(g). Furthermore, it is clear that the s,~ pass to the quotients to show that N(V) is an acyclic left complex over %‘. In the case where %?(p, p’) is K-projective for all p, p’ E 1V I, we see as before that N(V) is a projective resolution. In particular, this is the case when V is of the form KT. The complex N(V) is called the normalized standard resolution, of the K-category %?. When V has one object, N(V) is the normalized complex defined in Cartan and Eilenberg [l 1, p. 1761. The entire discussion for S(Y) above is also valid for N(V). The advantage which the normalized resolution N(V) has over the unnormalized resolution S(V) can be illustrated as follows. Let n be a small category with the property that there exists an integer m for which there is no sequence of m + 1 composable nonidentity morphisms. Such an integer will exist, for example, if 7~is a finite poset. Now S,(Kr) is a free functor in Abg(+xn) on as many generators as there are is similarly described, composable sequences (01~ ,..., an) in T. N,(Kn-)

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SEVERAL OBJECTS

73

except that here none of the oli can be identities. Hence by the assumption on V, N,(Kn) is zero for n > m. Thus KT has a projective resolution of length m in AbK(n*Xn), showing that dim, 7~< m.

18. HOMOLOGY

OF SIMPLICIAL

COMPLEXES

In the preceding section we saw that if G E Ab’*@f@ is g-projective and 59is K-projective, then N(g) & G is a projective resolution for G. In particular, let 58 = K, 97 = K rr, and G = A*K, where 7~is any small category. Then we see that N(Kn) ox,, A*K is a projective resolution for A*K in AbKn*. The n-th term of this complex, evaluated at p E 17~* i, is the free K-module on the set of all composable sequences (“0 70117..‘, a,) in 7~ such that dom 01~= p, and such that none of q ,.“, 01%are identities. The boundary formula is given by n-1 dn
,...,

a,) = 1 (-l)i

(010,..., cy&+r )...) (u,) + (- 1)‘”(ayg)..., Ol&,

i=O

where if c~~l~+r= 1 for any i > 0, then the corresponding term on the right side is considered as zero. In particular, let 7r be a poset, and let us tensor the above complex N(Kx) @;h’nA*K over Kn- with AK. We obtain the complex of Kmodules whose n-th term is the free K-module on the strictly increasing sequences (p, ,..., pTl+i) of elements of n, where the boundary formula is given by n+1

40,

>..., P,+I? = -& (-lY1

“., $5 I..., P,il>.

(1)

With regard to this complex, the following observation has been made independently by Y. C. Chen, 0. Laudal [30], and C. Watts [52]. Let X be a simplicial complex, and let m be the poset obtained by ordering the simplicies of X by inclusion. Then from the boundary formula (I), we recognize AK &,, N(Kr) BKW A*K as the complex of oriented chains of the first barycentric subdivision of X with coefficients in K. Therefore, if H,(X, A) denotes the n-th homology group of X with coefficients in the K-module A, we have

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MITCHELL

where the last equality cohomology, we obtain

is just Eq. (2) of Section 16. Similarly

Hn(X, A) = fP(hom,(dK

&,

N(Kn) mKli d *K, A))

= lP(hom,,,(N(K~) = Ext&,,(d*K,

for

&,, A *K, horn&AK, A))

d*A) = b’“’ d*A. ?7

In view of Eqs. (1) and (2) of Section 7, we could also express the homology (cohomology) of X as an appropriate Tor (Ext) over the ring [Kz-1.

19. THE

Z-CATEGORY

OF A SET OF RELATIONS

We begin with some notational remarks. If x and x’ denote finite sequences (xi ,..., x,) and (xi’,..., x~‘), respectively, then the sequence I”1 ,‘.., x, , Xl”...,. xm’) will be denoted by x i x’. The sequence X n ,***>xa , xi) will be denoted by X. If xi ,..., x, are morphisms in a category, and if they all have a common domainp, then for any morphism c with codomain p the sequence (cxi , cxa ,..., cx,) will be denoted by cx. We define xd similarly in the case where the xi all have a common codomain. By a set of relations in a category 7~,we shall mean simply a set R of ordered pairs (a, b) of morphisms of 7~satisfying dom a = dom b and cod a = cod b. If A and B are morphisms of 7r, we shall write A = B mod R if there is a system of equations in V, A = wlyl

,

xlblyl = xza2y2 p X2b2Y2

=

X3a3Y3

7

(1)

xn-lb,-ly,-l = wwn y x,b,y, = B, where for each i, 1 < i < n, we have either (ai , bi) E R or (b, , ai) E R. The quadruple (x, a, b, y) of sequences is then called a path of length n

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WITH

SEVERAL

75

OBJECTS

from A to B with respect to R. If ( x, a, b, y) is a path from A to B and (x’, a’, b’, y’) is a path from B to C, then we define (x, a, b,y) I (x’, a’,

b’,y’)

= (x i .r’, a I a’, b 1.

b’,y

LY’).

This is a path from A to C. If (x, a, 6, y) is a path from A to B, then its yeverse (5, 6, 8, p) is a path from B to A. Also taking n = 0, we can consider there to be a path from A to A. Thus = is an equivalence relation on the morphism of r. Furthermore, if (x, a, b, y) is a path from A to B, then (cx, a, b, y) is a path from CA to cB. Similarly, (x, a, b, yd) is a path from Ad to Bd. We form a category T/R, called the quotient category of n with respect to R. The objects of T/R are those of 7~. A morphism from p to q in n-/R is the equivalence class under = of a morphism from p to q in 7~. It follows from the preceding paragraph that if we compose equivalence classes by composing representatives in n, then composition is well defined. There is a natural functor 3’: n ---f X/R which satisfies F(a) = F(b) for all (a, b) R, and if G: TT-+ r’ IS a functor with the same property, then there is a unique functor N: r/R + V’ such that G = FH. This property characterizes n/R up to isomorphism. Observe that F is full and is a bijection on objects. If p, q E 1n 1, we shall denote the set of all paths between morphisms 1 defined above from p to q by Q,(R)(p, q). Th e vertical composition e structure of a category. In fact, Q,(R)(p, q) is gives Q,(R)(p, q) th simply the free category (Section 20) generated by the paths of length one between morphisms in n(p, q). The objects of this category are the members of +p, q), and its components are in one-to-one correspondence with the morphisms from p to q in n-/R. Now suppose that (x, a, b, y) is a path from A to B in SZ,(R)(p, q), and (u, c, d, 8) is a path from C to D in Q,(R)(q, Y). Then we define the horizontal composition of these two paths as (2, a, b, y) . (u, c, 4 4 = (x, a, b, ye)

1. (%

c, 4 4.

This is a path from AC to BD. It is easy to check that * gives l&(R) the structure of a category. In fact, Q,(R) comes very close to being a 2-category, as defined in Section 1. The property that is missing is Eq. (1) of Section 1. We rectify the situation as follows: First, it will be convenient notationally to assume that if (a, b) E R, then (b, a) $ R. Clearly one can always alter a set of relations so that this is the case, without changing the quotient category n-/R. Henceforth we

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MITCHELL

shall always make this assumption on R. This means, in particular, that for no a do we have (a, LZ) E R. If A = B in the system (l), then the path (x, a, b, y) is called closed. The closed path is degenerate if the indices (1, 2,..., n} can be partitioned into two element subsets {i, j} (which means that n must be even) such that: 1. 2.

(Ui , bi) = (bj 7 q), xi = xj and

yi s yi mod R.

Otherwise the closed path is nondegenerate. Observe that in the case where n-/R is a poset, condition 2 is redundant. If (x, a, b, y) and (x’, a’, b’, y’) are both paths from A to B, then they will be called equivalent if the vertical composition of one with the reverse of the other is degenerate. This is easily seen to define an equivalence relation on the elements of L?,(R) which respects both horizontal and vertical composition. Therefore if L?(R) d enotes the set of equivalence classes of members of Q,,(R), then O(R) inherits both compositions, and now one can check that L?(R) is actually a 2-category. Moreover for each pair p, q E 1T /, the category Q(R)(p, q) is a groupoid, since the reverse of (x, a, b, y): becomes an inverse for it when one passes to equivalence classes. The* 2-category L?(R) will be called trivial if R admits of no nondegenerate closed paths, or equivalently, if O(R)(p, 4) has no nonidentity endomorphisms for all p, q. A set R of relations is minimal if no proper subset of it gives rise to the same congruence relation =. If R is not minimal, then there is an ordered pair (A, B) E R such that A = B mod R - ((A, B)). This gives rise to a system of Eqs. (1) where for no i do we have (ui , bi) = (A, B). Thus if R is not minimal, then Q(R) cannot be trivial. If r is a group and R is a set of relations, then each relation of the form (a, b) can be replaced by one of the form (Y, l), where Y = ub-l, without changing rr/R. Consequently we can take R to be a subset of T in this case, instead of a set of ordered pairs. Given a path (1) from A to B, we can solve all but one of the equations for the yi’s to obtain A = fi x,r,x,‘B,

(2)

i=l

where either ri E R or ril E R. Conversely, an equation such as (2) gives rise to a path from A to B. Thus the quotient category is just the quotient group n-/N(R) w h ere N(R) is the least normal subgroup containing R.

RINGS WITH

SEVERAL OBJECTS

77

A closed path in this case is just a pair (x, Y) of sequences of the same length n such that 1 = fi xirix; i=l

)

where either ri or ril E R. It is degenerate if the indices can be paired off ioj such that rj = r;’ and xi ESxi mod N(R). One can consult [13] and [34] in this regard.

20. GENERATORS

AND RELATIONS

An orientedgraph is a nonempty set V of vertices and a set M (possibly empty) of arrows, together with two functions dom, cod: M---f V. We make G into a category G, by taking V to be the set of objects, and defining a morphism from p to 4 to be a word m,m, ..’ m, ,

m,EM,

t 3 1,

where dom ml = p, dom rni+r = cod mi for 1 < i < t - I, and cod mt = q. Composition is defined by juxtaposition. We also add one morphism ep from p to p for each p E V, and we define the composition of eP with another morphism to be the other. We shall think of eP as the empty word from p to p. A functor F: G, --t 75is completely determined by assigning objects F(p) E ) 7~1 to vertices p and morphisms F(m) E x(F(dom m), F(cod m)) to arrows m. We call G, the free category generated by the graph G. A set of generators for a category n is a set M of morphisms such that every nonidentity morphism can be written as a composition of morphisms in M. If M is any set of morphisms in rr, let G be the graph whose vertices are the objects of 7~and whose arrows are the members of M, with dom and cod as they are in r. Then there is a natural functor F: G, -+ rr which is a bijection on objects, and is full if and only if M is a set of generators for n. In this case ?Tis isomorphic to GO/R, where R is the set of ordered pairs (a, b) such that F(a) =F(b). Of course one zan usually choose a much more economical set R. We shall now illustrate the notions of this and the preceding section with some examples.

78

MITCHELL

1. If G is a graph with only one vertex, then G, is the free monoid on M generators. If R is the set of ordered pairs of the form (xy, yx) in G, with x, y E M, then ?T= G,/R is the free Abelian monoid on M generators. In this caseif/l is a ring, then /lo is the ring of polynomials in M commuting variables with coefficients in /l. If M has at least three distinct elements X, y, z, then a nondegenerate closed path with respect to R is given by the equations (XY) 2 = 4Y47 4XY)

= G4YP

(4Y

= @Y),

4Y4

= (ZY) X?

(Y4 x = Yc4,

YW 2.

= (YX) z*

Let G be the graph

where n 2 2 and all arrows are directed downwards. If we let R consist of the 2n relations of the form (XiYi 3 %+1Yi’h

(Yi’Xi , Yi+1%+1),

where the subscripts are mod n, then GO/R is a poset which we call the suspended

n-crown.

RINGS

4 nondegenerate

WITH

SEVERAL

closed path with Xl(YPl)

k2Yl’)

21

‘dY24

X,(Y,%) (~lY?L’)

3.

6,

OBJECTS

to R is given by

respect =

(XlYl)

=

dYl’%),

=

79

21

7

(“ZYZ)

x2

1

L

(%lYJ

%I

1

=

x1( yn'zn).

Let G be the graph W

X

Y

Z

W’

X’

Y’

Z’

8

where arrows are directed downwards. Let R consist of the two relations (wx, w’s’),

(YX, Y’G

Then G,/R is a poset containing the suspended 2-crown as a full subcategory. There are no nondegenerate closed paths with respect to R. h degenerate one is given by

W(Y4 =

(4(Yd

(wx)(y’q = (wx)(y’4, (w’x’)( y’z’) = (w’x’)( y’z’), (w’x’)( yz) = (w’x’)( yz). 4. Let n be a nonnegative integer. Let G be the graph whose vertices are the nonnegative integers, where there are n arrows x1k ,..., x,nk from k to k + 1 for each vertex k. Let R be the set of relations 3f the form

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MITCHELL

We call 7~~= Go/R the graded polynomial category on n letters. If A is a ring, then AbAn* is just the category of graded modules over the polynomial ring /l[x, ,..., xn] with homomorphisms of degree 0 as morphisms. Although it has nothing to do with the sequel, we shall indicate a slight generalization of some remarks made by Buchsbaum [lo] concerning the Hilbert-Samuel polynomial function of a graded module. Let OTbe an abelian category, and let G be an abelian group. Suppose that I: 1GYj -+ G is a function such that Z(A) = Z(N) + Z(A”) for each short exact sequence 0 -+ A’ + A + A” + 0 in CZ. (In other words, 1 factors through the Grothendieck group of Q!.) Suppose that D is a noetherian object in GPn. Imitating the classical method (that is, using induction on n), one can show that Z(D(k)) is a polynomial function of degree < n for sufficiently large k. 21. LENGTH If x is a nonidentity morphism in a category, then the length of x is defined to be the sup of the integers n such that x can be written as a composition of n nonidentity morphisms. The length of x is denoted by Z(x). If Z(X) is finite for all nonidentities X, then we call n an Z-category. An l-category can have no nonidentity idempotents, since x = x2 implies x = xn for all n. Nor can it have any nonidentity retractions, since I = xy implies x = (xy)%x for all n. In particular, this means that 7~is skeletal, or in other words that isomorphic objects are equal. If rr is an I-category, we define Z(1) = 0 for identity morphisms 1. Clearly the length function satisfies @Y) 3 44 + Z(Y).

If equality always holds, then rr satisfies the saturated chain condition (s.c.c.). However we shall not be using this condition. The following are examples of Z-categories. 1. A free category G, generated by a graph is an Z-category with the S.C.C. 2. A free abelian monoid on any number of generators is ani Z-category with the S.C.C. 3. Any subcategory of an l-category is an l-category. However, a subcategory of an I-category with thes.c.c. does not necessarily satisfy the S.C.C.

RINGS

4. 5.

WITH

SEVERAL

A finite product of Z-categories A finite poset is an l-category.

OBJECTS

81

is an l-category.

Remark. If a quotient of an l-category were always an l-category, then by Example 1, every category would be an Z-category. The main feature of Z-categories is that they admit of very manageable sets of generators and relations. In the first place it is clear that any nonidentity morphism can be written as a composition of morphisms of length one. Furthermore, since a morphism of length one cannot be written as a composition of two morphisms neither of which is an identity, it follows that the set of morphisms of length one is a set of generators contained in every other set of generators. Let G be the graph of morphisms of length one, and let F: G, --f 7~ be the natural, full fun&or. Then F is a bijection on objects and on morphisms of length one. Let x E n, and for a, b E F-l(x), write a N b if there are morphisms a’, b’ such that F(a’) = F(b’) an d a = fa’g and b = fb’g, where f and g are not both identities. Then - generates an equivalence relation on F-l(x), and so let {ci 1i E I> denote representatives of the set I of equivalence classes. Choose any i,, E I, and let R, = {(ci , ci,) 1i E I, i # i,}. Note that if I consists R = USm R.r .

of a single

element,

then R, is empty.

Let

LEMMA 21 .l. Let x be an l-category, and let F: G, -+ 7~ and R be as dejined above. Then F(a) = F(b) if and only if a E b mod R. Furthermore, R is a minimal set of relations in GO .

Proof. It is clear from the definition of R that if a = b, then F(a) = F(b). W e sh ow by induction on the length of x = F(a) that if F(a) = F(b), then a G b. This is trivial if Z(x) = 0, since then F-l(x) consists of only one element. Now suppose that Z(x) = n > 0. If a N b, then we can write a = fa’g and b = fb’g, where F(a’) = F(b’) and f and g are not both identities. Since f and g are not both identities, l(F(a’)) < n. Hence a’ 3 b’ by induction, and consequently a = b. It follows that a = b whenever a and b are in the same component of the equivalence relation generated by -. On the other hand, if a and b are in different components represented by ci and cj say, then by what we have already proved, a = ci and b = ci . Since ci = ci, and cj 3 ci, , this shows a = b.

82

MITCHELL

Now suppose that (ci , ti,) E R, is a consequence in R. Then we have a system of equations in G, ci

=

x1a,y,

9

xlbol

=

w2y2

,

xnbnm

=

ci,, 3

of the other relations

where (ak , &) or (b, , u,J E R - {(ci , ci,)} for each K. For some k we must have xx: and yIs both identities, since otherwise ci and ci, would be, in the same component determined by -. Assume k is the first such. Then ci and ak are in the same component determined by -, and soj ci = a, . But this is a contradiction, since (ci , ci,) is the only orderedi pair in R with ci as an entry. The following rather technical lemma will be important in characterizing weak deltas of cohomological dimension two (Theorem 33.2). LEMMA 21.2. Let rr = G,IR be an l-category where G, is the graph! of morphisms of length one. (Here R need not be minimal.) Suppose that there is a nondegenerate closed path (x, a, 6, y) in G, with respect to R. Then there is one such that x1 = 1, and such that for no k 2 2 do we have (uIi , b,) equal to either (b, , aI) OY (al , b,) with x,,. = 1 (and hence X~ = 1) and yk = y1 mod R.

Proof. We prove it by induction on the length 1 of F(x,a,y,), where F: G, -+ 7r is the natural functor. Since there are no paths corresponding to 1 = 0 or 1, the inductions gets off the ground owing to the fact that it never was on the ground. Assuming 1 > 1, we now proceed by induction on the number of i for which xi = 1. If there are no such i, then since G, is free we can write x = cx’ for some generator c of G, , and then apply the previous induction to the nondegenerate closed path! (x’, a, b, y). Thus suppose that xi = 1 for some i. By a rotation of the. subscripts, we may assume i = 1. Suppose that for some k > 1, we have. (ak., bk) = (b, , al) where yk s y1 mod R and X~ = 1. Since yk E yi , we have a system of equations in G,

RINGS

WITH

SEVERAL

OBJECTS

83

where (ai , Pi) or (Pi , CXJE R for 1 < i < t. Thus we obtain two closed paths corresponding

to the systems

b&&t

of equations:

= lb,

;

and

&LhLYn = lQlY1.

At least one of these closed paths must be nondegenerate since the original one (x, a, b, y) was, and since the pairs (ai , &) of the first closed path appear in reverse order in the second one flanked by the same terms mod R. Since each of these closed paths contains fewer terms with xi = 1 than the original, we can now apply the second induction. The argument for the case where (ak , bk) = (ai , b,) is entirely similar, and will be omitted.

22.

DELTAS

Recall that a category is skeletal if its only isomorphisms are automorphisms. If rr is any category and 7~’is a full subcategory consisting of one object from each isomorphism class, then n’ is skeletal, and the inclusion 7~’-+ 7~is an equivalence. Thus for many purposes there is no loss in generality in assuming that 7~is skeletal. A skeletal category in which the only endomorphisms are identities is called a delta (as in the Nile. The word has been suggested by Rota.) If r is a delta and r( p, q) is nonempty, then we write p < q. This defines a partial order on / 7~1, for (YE ~(p, q) and /I E ?r(q, p) implies c@ = 1, and ,!?a= l,, and so p = q since rr is skeletal. Posets are deltas, as are

84

MITCHELL

the graded polynomial categories of Section 20, Example 4. Duals, products, quotients, and subcategories of deltas are deltas, and any delta is the quotient of a free delta. A finite delta is one with only a finite number of morphisms. If n is a delta, then an initial subcategory of r is a full subcategory n’ such that if p E 1 7~’ 1 and p < q, then p E 17~’ I. Terminal subcategories are defined dually. LEMMA 22.1. Let 7~ be a delta, and let r’ be an initial M is an abelian category and D E GZr, then

subcategory.

If

h.d. D > h.d.,, D 1rr’. The restriction functor 6Zsl,+ @’ is an exact left adjoint for Proof. the exact functor GY -+ GF which extends a diagram by adding zeros. Therefore the result follows from Lemma 9.4. LEMMA 22.2. Let rr be a delta, and let r’ be a terminal subcategory. Let GZ be an abelian category, and let D E GZr be the extension of a diagram D’ E GF obtained by adding zeros. Then

h.d., D = h.d.,, D’. Proof. The extension adjoint for the restriction Hence the lemma follows

by zeros functor S: QF -+ Q? is an exact left functor T: GZn-+ 0F and ST is the identity. again from Lemma 9.4.

Remark. The lemmas are actually trivial cases of Propositions 11 .I and 11.3. If p < q in a delta 7r, then we define the following full subcategories of 77:

Note that nq is initial and pi is terminal. A subcategory of the form prp is called a muscle. Thus when 7r is a totally ordered set, a muscle is just, a closed interval. n is weak if all of its muscles are finite. Duals, finite products, quotients, and subcategories of weak deltas are weak. Clearly a weak delta is an Z-category. However, a delta which is an I-category is not necessarily weak. If 7r satisfies the stronger condition that 7re is finite for all q E / r I,

RINGS WITH

SEVERAL OBJECTS

8.5

then 7~is called initially Jinite. If r is any delta and q E / 7~1, then the height of q is the sup of the lengths of morphisms with codomain q. If 7~ is initially finite, then all of its objects have finite height (but not conversely). Let 7r be a delta, and let K be a commutative ring. By definition, dim,r is the homological dimension of K?r in AFS*Xn). Now KIT has nonzero values only on the terminal subcategory 7(n) of n* x rr consisting of all pairs (p, q) such that p < p in n. Hence by Lemma 22.2 we have dimKn

= h.d. KT / T(T).

(1)

Clearly $7~) is finite if and only if n is finite. Hence it follows from the relation +37) = +ku?) that 7~ is weak if and only if T(T) is initially finite. The minimal objects of T(T) are those of the form (p, p) with p E 17~/. The maximal objects are of the form (p, q) where p is minimal and q is maximal in n. The following table of finite posets serves to illustrate. (Here and elsewhere, posets are directed downwards.)

23. PROJECTIVES OVER DELTAS Throughout this section O?will denote an abelian category. We now employ a technique of Eilenberg and Moore [15] to characterize the projectives in a functor category O? where n is an initially

86

MITCHELL

finite delta. Let S, denote the left adjoint for the evaluation functor T,: GP + G?!.If r is initially finite, then any coproduct of the form

exists in P, even though GZ may not have infinite coproducts. because the value of the functor (1) at an object 4 in 7~ is @

codz=q

This is

Adomz

which is a finite coproduct since 7r is initially finite. Let us call an object D in 6Z relatively p rejective if it is a retract of an object of the form (1). Clearly the restriction of any relatively projective object to an initial subcategory of 7~ is also relatively projective. We shall say that D is initially relatively projective if its restriction to 3rqis relatively projective for all q E 17~I. It will turn out that such an object D is actually of the form (1). For each q E j 7~1, we define a functor R,: 0’” -+ @ by exactness of the sequence of natural transformations @

codx=q Xfl,

Tdome --

T, -k

R, -

0,

(2)

where Tdomr + T, is induced by x. Note that the coproduct is defined since ,r is initially finite and GZis abelian. Since each nonidentity x with cod x = q factors through such an x of length one, it suffices to restrict the indices of the coproduct to those x of length one. Since the TP’s have right adjoints, they preserve colimits, and consequently from (2) we see that Rq preserves colimits. An object D in CP will be called split at q if d,(D) is a coretraction. If this is true for all q, then D is spkt. Coproducts and retracts of objects which are split at q are also split at q. Observe that S,R, = 0 for q # p, whereas S,Rq = la. In fact, the morphism +,&S,(A)) can be taken to be the identity on A, and it follows that S,(A) is split. Hence any object of the form (1) is split, and so any retract of such an object is split. Consequently any initially relatively projective object is split. LEMMA

23.1.

Let n be an initially jinite delta. Consider a morphism

RINGS

WITH

SEVERAL

87

OBJECTS

where D is split at q and A, = R,(D). Suppose further that the qth coordinate of y is induced by a morphism CL: R,(D) + D(q) such that &&D) = 1. Then R&y) = 1. Proof. Since R, commutes with coproducts q # p, it suffices to consider the case where Y: WWN

-

Rut in this case we have the commutative

and

S,,R, = 0 for

D. diagram

J%(D) __- “=I + R,(D) P’=Yq i D(q) =+ which

shows

that R,(y)

1 R,(D)

= 1.

THEOREM 23.2. Let n be an initially finite delta, and let y: E---t D be a morphism in GI? such that R&y) is an isomorphism for all q E 1r 1. If E and D are initially relatively projective, then y is an isomorphism.

Proof. We must show that T,(y) is an isomorphism in Q! for all q E 1n I. We do it by induction on the height of q. If this height is zero, then R, = T, , and so T,(y) is an isomorphism by hypothesis. Thus suppose that the height of q is greater than zero, and let m, , m2 ,..., m, denote the morphisms of length one with codomain q. For each k, 0 < k < t, we define a functor R,“: Orn --f M by exactness of the sequence of natural transformations

We show by induction on k that Rqk(y) is an isomorphism. Since R,” = TQ , this will prove the theorem. For k = 0 we have R,O = R, , and so R,O(y) is an isomorphism by hypothesis. If k > 0, then we have a sequence of natural transformations @ Tciomz --

Tdomm, a

R,’ -

R;--l -+

0,

(3)

where the coproduct is indexed by all morphisms x such that xmk = x’mj for some j > k. This situation is represented in the following diagram, which we have drawn as if the domains of the mi were all distinct.

88

MITCHELL

The sequence (3) is of order two, and in fact is exact on the portion not involving the coproduct. It is not exact in general, as one seesby evaluating it at a diagram which has a nonzero object at dom mk and zero elsewhere. However it is exact at any object of the form S,(A), p E / r /, as one easily verifies. Therefore it is exact at any coproduct of such objects, hence at any retract of such, hence at any initially relatively projective object, and in particular at D and E. Thus if we evaluate the sequence at y, we obtain a commutative diagram with exact rows

@ T&ml.(E) +

T,,,,k(E) -----, R,“(E) --

Rt-w

-+

0

Using both the induction assumptions, we see that all vertical arrows except possibly the middle one are isomorphisms. By the Slemma, this one is too. This completes the proof. Combining the theorem with Lemma 1.1, we obtain: COROLLARY 23.3. Let D be initially relatively projective in GP where z is an initially Jinite delta. For each q E j x 1, let pa: R,(D) + D(q) be such that p,&,(D) = 1. Then the morphism

induced

by the morphisms

pn is an isomorphism.

RINGS WITH

SEVERAL OBJECTS

89

COROLLARY 23.4. Let 7~ be an initially finite delta, and let GZbe any abelian category (not necessarily with enoughprojectives). Then the projectives in gn are precisely the objects of the form (I) with A, projective in O! for allp E 17~I.

Proof. Since S, has an exact right adjoint, it follows that S,, preserves projectives. Hence any object of the form (1) with A, projective for all p is projective. Conversely, suppose that D is projective. Since T, has an exact right adjoint, it follows that T,(D) = D(p) is projective in 0’ for all p. Consider the epimorphism

whose pth coordinate is induced by the identity on D(p). Since D is projective, 01is a retraction, and so D is relatively projective. The result now follows from Corollary 23.3, since R,(D) is a retract of D(p), hence is projective, for all p. COROLLARY 23.5. Let V be a small additive category with the property that projectives in Ab’ are free. Then the sameis true of Vrr = %?@JZn-, where z is an initially finite delta.

Proof. Let D be projective in Ab”“. a coproduct of objects of the form

Then by Corollary 23.4, D is

S,(F) = F 0.z WA

1,

where F is projective, hence free, in Ab’. Since @ commutes with coproducts, it follows that D is a coproduct of objects of the form g(y, > 0 WP, 1. But such coproducts are, by definition, the frees in AbVW. Remark. One can define an additive category V to be (left) local if VP has a unique maximal subfunctor for all (nonzero) p E / %?I. As in the case of rings, one seesthat a category is left local if and only if it is right local, and that the nonisomorphisms in 9? form a two-sided ideal containing all left and right ideals. One can then imitate Kaplansky’s proof [28] to see that if V is local, then projectives in AbV are free. (Another proof has been given by Bernard and Michele Wedenfeldt in their theses de troisieme cycle.) If R is a local ring and n is a delta, then Rr is local. More generally, if % is an additive category such that

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MITCHELL

U(p, p) is a local ring for all p E ( 9 (, and such that %?(p, q) # 0 with p # q implies ‘S(q, p) = 0, then %?is local. In particular, if R = &>,, R, is a graded ring converted into an additive category 9 as explained in Section 1, and if R, is local, then Q?is local. COROLLARY 23.6. Let rr be an initially jinite delta, and let GI be an abelian category with enoughprojectives. Then

h.d. D = sup h.d. D 17rg 9.EInI

for D E GZn.

Proof. Let the right side be n. By Lemma 22.1 we see that h.d.D > n. Now restriction to TV, being an exact functor with an exact right adjoint, preserves projective resolutions. Hence if K, is the n-th kernel in a projective resolution for D, then K, 17~~is projective for all q. Therefore K, is initially relatively projective with projective values, and so by Corollaries 23.3 and 23.4, K, is projective. COROLLARY

23.7.

Let r be a weak delta, and let K be a commutative

ring. Then dim, x = sup dim&,n,). P<9 Proof. Since Z- is weak, T(V) is initially finite. Therefore using Corollary 23.6 and Eqs. (1) and (2) of the preceding section, we have dim, T = h.d. Kz- 17(r) = sup h.d. KT 1T(z-)(,,,) P&9 = sup h.d. KCT1+r,,) D
(4:

RINGS

WITH

LEMMA 23.8. If 7~ is a jinite then for D in @ we have

SEVERAL

91

OBJECTS

delta and 0% is an abelian K-category,

h.d. D < dKrr + s,up h.d. D(p). Consequently

dim,r

< dK .

Proof. We prove it by induction on the number of objects p for which D(p) # 0. If this number is zero, there is nothing to prove. Otherwise let q be minimal such that D(q) # 0. Then we have the obvious exact sequence in Cln 0 + D’ - D + L,(D(q)) where

D’(q)

= 0 and D’(p)

= D(p)

---f 0,

(5)

for p # q. By induction

we have

h.d. D’ < d,n + SLIPh.d. D’(p). and so by inequality

Also -W(q)) = D(q) 0 t&(K), lary 13.4’ we have h.d.L,(D(q)) The result now follows THEOREM

23.9.

(5’)

of Corol-

< d,n + h.d. D(q).

from the exact sequence (5).

If T is a weak delta, then dim, = dK .

Proof. The inequality dim,7 3 d,+r follows from the inequality (3’) of Corollary 13.4’ in view of the fact that K has homological dimension 0 in Ab”. To prove the other inequality, we have using (4), Corollary 23.7, and Lemma 23.8, dim, r = sup dim&a,) P40

< sup d&n,J 9
< dKr.

Remark. In Section 35 we shall see that the weakness assumption is necessary. In Section 17 we saw that if the maximum height of an object in a delta r is m, then dim,n < m. Thus n must have at least m + 1 objects n order that dim,m = m. On the other hand, consider the delta + whose objects are the nonnegative integers, where +(p, p + k) has one mor-

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MITCHELL

phism for all p > 0, K 2 2, and ~(p, p + 1) has one or two morphisms depending on whether p is even or odd. If we let rr be 7~truncated at the integer m, then it is easy to write down a projective resolution for I$(K: (one never has more than 2 copies of K at any vertex), and the i-th kernel is not projective for i < m. Hence h.d.,L,(K) = m, and consequently by Theorem 23.9, dim,rr = m. Note that n has m + 1 objects. However, it takes 2m objects to produce a poset of dimension m. Diagram (6) of Section 34 provides the example.

24.

RIGID

CATEGORIES

Let G be an oriented graph, and let G,, be its associated free category. Morphisms of Go will be considered as words W in the letters (arrows: morphism w’ will be said to be imbedded in s x, Y, z... . A nonidentity morphism W in G, if we can write W = W, W, W, for some morphisms W, and W, . Let ,Y#‘”be a set of nonidentity endomorphisms of G, . We are interested in the quotient category GO/R, where R is the set of relai tions of the form (IV, 1) with l+’ E YJV. This quotient category will ba denoted simply by G,,/YY. A morphism in G, is reduced with respect tc W^ if it has no member of PV imbedded in it. In particular, identities are reduced. We say that YP- itself is reduced if no member has anothei member (properly) imbedded in it. It is always possible to change -W so as to have it reduced, without changing G,/-W. For consider ths natural functor G, -+ G,,/YK, and let w’ be the set of nonidentity endo. morphisms of G, which go to identities in G,/w, but don’t have any other such endomorphism imbedded in them. It is then easy to see tha* G,,/%@” is isomorphic to Go/w’. The set PY will be called symmetric i: W,W and WW, both in ?V with W # 1 implies that W, = IV,. THEOREM 24.1. Let IV be a reduced set of nonidentity endomorphism: of a free category G,, . Then w is symmetric zf and only zf each member oJ G&Y has a unique reduced representative in G, .

Proof. Suppose that reduced representatives are unique. Let W,H and WW, be members of ,w with W # 1. Then W, and W, are reduceo since W += 1 and #‘” is reduced. Since WI and W, become left and righ. inverses, respectively, for W in G,/ W, they are equal in G,/V’, and so since reduced representatives are unique, they must be equal in G, Hence w is symmetric.

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Conversely, suppose that $/‘^ is symmetric. We make the reduced morphisms in G, into a category n as follows. The objects of n are the (reduced objects of G, , and the domains and codomains of morphisms words) in n are the same as they are in G, . We define the composition (X, of two that

reduced

(4

words

X,) * ( Y, . . . I’,)

by induction

(1)

on n, and simultaneously

..’ X,) * (Yl ..’ Y,) = (X, .‘. x,-,)

* (X, ... X,Y,

we check

.‘. Y,,)

(2)

in the case where X, ... X,Y, .*. Y,& is reduced. For n = 0 the composition (1) is just Y, .a* Y,, . For n > 0 the composition is . . . x,y, . . . Y,,, provided that this is reduced. If it is not reduced, Xl then Xi ... X,Y, **a Yi E w for some i and j, and so we define (1) to be (X, ... S(-1) * (Yj+l “’ Y,,). If x,

... xny,

(3)

... Y, E 96’” also, we must show that (3) is the same as (Xl ... A-,-,) * (Y,,, ... Y,).

If k = i, then j = 1 since w is reduced. If k < i, then 1 < j again since r;Y- is reduced, and so since %Q-is symmetric we must have xlc ... xpl

= Y,,, "' Y, .

Therefore we must show that (Xl ‘.. X&,X,

‘.’ X&l) * (Yj+l ‘.. Ym)

= (X, “’ X,_,) * (X, ... x,&lYj+l

“’ Y,).

But this is true by induction for (2). Now we must verify that (2) holds for length n. First observe that both sides of (2) are just Xl ... -Ly, ... Y,, provided that the latter is reduced. Otherwise there is an integer i < k - 1 such that Xi **aX,Y, *a* Yj E w for some j. Then both sides of (2) are (Xl “’ Xi-,) * (Yj+l ‘.. Y,). We now verify the associative law by induction on the length of the

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middle word. If this length one, we must show

is zero, associativity

is trivial.

For length

(WI ... X,) * 2) * (Yl ... Y,) = (X, .‘. X,) * (Z * (Yl ... Y,)).

(4:

There are five cases. Case 1. just this. Case 2.

Xi -*a X,ZY, X, -** X,2

-*- Y, is reduced. and ZY,

a-- Y,,, are reduced,

xi “’ X,ZY, for some i andj.

Then both sides of (4) are but

... Yj E 7Y

Then both sides of (4) are (Xi a.0 X,-l)

* (Yj+r

Case 3. X, *** X,2 is reduced, whereas sides of (4) are (X, ... X,) * ( Yi+r -*- Y,,).

ZY,

Case 4. Xi a-* XJ dual to Case 3.

... Y,,, is reduced.

E PV, whereas

ZY,

... Yj E w.

--- Y,).

Then

both

This

is

*a* Yj E VP-. Since -We is symmetric, Case 5. Xi *a* X,ZE$k’-andZY, this means Xi ... X, = Yi ... Yj . The left side of (4) is

whereas

the right side is (Xl ... X,) * (Yj+1 ... Y,).

These are equal by (2). Now we must show that

(U* V)* w= U*(V*

W)

where the length of V is greater than one. Write V = V, J V, , where V, and I’, both have length less than that of I’. Then by induction we can write (u*lq*w=(u*(vl*v2))*w=((u*v~)*v2)*w =(U*

V1)*(V2*

= U*((V,*

V,)*

W)=

U*(V,*(V,*

W) = U*(V*

W)) W).

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95

It is now easy to define two functors G,/W z? n and show that they are inverses of each other. This completes the proof. Remark. The above proof resembles a proof of Kurosh, and independently of Eilenberg, for the word problem connected with a free product of monoids. In view of the theorem we shall call a category rigid if it is isomorphic to one of the form G&V where ~9” is reduced and symmetric. COROLLARY

Proof.

24.2.

Rigid categories are idempotent complete.

Suppose that (Xl

... Xn) * (Xl

.‘. X,)

= x1 ..’ x7, (

where X, **aX, is reduced in G, . Reducing the left side as much as possible, we see that for some i and j we must have (Xi

... X,)

* (Xl

... Xj)

= 1,

and x1 ... Xi&,Xj,,

..’ x, = x1 .‘. A-, .

From the latter equation we see that i = j + 1, and so from the former we see that X, **aX, splits in G,/-tlr.

25. CATEGORIES

OF ITERATED

FRACTIONS

Let S be any set of morphisms in a small category 7r. The category of fractions obtained from 7~by inverting the members of S is a category ns together with a functor F: T + rs which takes members of S to isomorphisms, such that if G: n + n’ is any other functor taking members of S to isomorphisms, then there is a unique functor H: 7~~-+ VT’ such that FH = G. Clearly 7~~is determined up to isomorphism by these properties. If S’ C S and if every members of S is a composition of members of S’, then 7~~= 7~~’. Observe that x + nTTs is an epimorphism in the category of small categories. To see that such a category ns exists, let M denote any generating set of morphisms of n (for example, the set of all morphisms of r), and let G be the graph whose arrows are the members of &I. Then n is isomorphic to G,/R for some set of relations R. For each s E S, add one arrow s-l from cod s to dom s in the graph G, and call the resulting graph G.

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Then the free category G, contains G, as a subcategory. union of the set R with the set of relations of the form

(s-W, 11,

Let R be the

(+) s-l, I),

where T(S) is some representative for s in G, . Then it is clear that Go/R has the required universal property. Let us start with a category ~a , and let r1 be the category of fractions obtained by inverting a set S, of morphisms of 7s . Then let 7ra be the category of fractions obtained by inverting a set S, in x1 . Then 7~~ is not necessarily a category of fractions of 7~~. More precisely, it is not necessarily true that the composition n,, + rr + ~a of natural functors is the natural functor corresponding to some category of fractions of 7~s . EXAMPLE. Let nc, be the free monoid on two generators X and Y, and let 7rr be the category of fractions obtained by introducing an element 2 to invert XY. Then let ~a be obtained from rrl by introducing an element T to invert XZY. Then ~a is generated by X, Y, 2, and T, subject to the relations XYZ

=

1,

ZXY=

1,

XZYT=

1,

TXZY=l.

The left sides form a reduced, symmetric set n in the free monoid on four generators. Hence the members of ~a are uniquely representable in reduced form, and using this one deduces that the only words in X and Y alone which have two sided inverses in 7~~are of the form (XY)“, n > 0. Thus if r,, + 7rI + 7~~ were the natural functor from 7~~to one of its categories of fractions, then 7~~would necessarily be the category obtained by inverting XY. Hence ~a = rrr , a contradiction since T $ n1 . Let us now build a sequence of categories and natural functors ~o’Tl+7j-2+

“‘+?-r+

“’ )

(1)

where r8+r is a category of fractions obtained by inverting a collection S,,, of morphisms of m8 , and where n8 = b, i8 7~,,if 6 is a limit ordinal. We shall call any category n8 which can be obtained by the above procedure a category of iterated fractions of the original category z-s . Using transfinite induction we see that v,, + 7~~is an epimorphism in Cat for all 6. Remark. A direct limit in the category Cat of small categories can be formed by taking the disjoint union of the categories involved, and

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OBJECTS

identifying morphisms 01 and 01’ under the equivalence relation “there exists an index corresponding to which 01 and 01’ go to the same morphism.” Two equivalence classes are composable if they contain composable representatives, and an equivalence class is an identity if it contains an identity. The direct limits involved in the sequence (1) are particularly simple, since all the functors involved are bijections on objects. Let 7r,, + n be any functor, where 7~,,is a small category. We define a sequence of categories and functors indexed by the nonnegative integers 7To= 77(o)- 5-r(,) + T(%)-

... - 7r(n) + ‘.’

(2)

together with functors z-en) + r compatible with the functors of the sequence as follows. The functor nt,,) -+ 7~ is the given one. Assuming that we have defined ntn) + n, we let n(,+r) be the category of fractions obtained from ‘rr(,) by inverting all morphisms which go to isomorphisms in n, and we let rtn+r) + r be the induced functor. We call (2) the fraction tower of the given functor nO + rr. Let rrtw) denote the direct limit of the n(,) . Then the functors x(~) + 7~ induce a functor ~(~1 + n which reflects isomorphisms: that is, if a morphism in m(W) goes to an isomorphism in 7~, then it is already an isomorphism in r(W) . We call ?T(~) the isomorphism saturation of 7r0 with respect to the given functor 7r0 + 77. LEMMA 25.1. Let rg be any category of iterated fractions of the small category 7r0 . Then r8 is the isomorphism saturation of rO with respect to the natural functor 7~,,4 r6 .

Proof. transfinite

Consider induction

the sequence (1) used in constructing rr6 . Using we can construct a commutative diagram

where 7~o) = n(W) for X > w, and where the vertical compositions X~ + no) + nTTs are the natural morphisms nh -+ x6 of (1). The composi607/S/2-7

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98

tion YT~ -+ 7-r~~) + xd is therefore the identity functor. Now nTTs + r(a) is an epimorphism since n,, + 7~~+ n(S) = n(a) + n(8) is an epimorphism. Therefore rrs+ CT(~) , being at once a coretraction and an epimorphism, is an isomorphism. If 7~~is a category of iterated fractions of no , then in view of the lemma, we may always assume that 6 = w. For simplicity we shall always make this assumption in the sequel.

26. THE

WORD

PROBLEM

FOR G(,)

In this section we shall show that any category of iterated fractions of a free category is rigid, and hence is idempotent complete. This will not be essential for the sequel, but it will lend plausibility to a conjecture which we shall make in Section 28 concerning categories of cohomological dimension less than or equal to one. THEOREM

26.1.

Let G, = G(o)- G(l) - G(z) -

... - Go,,

be the fraction tower of a functor G, -+ VI=,where G, is the free category generated by a graph G. Then the functors of the tower are inclusions, and Gt,) is rigid for all n < w. Proof.

We construct a sequence of free categories G, C G, C G, C ... C G, C ...

together with functors G, + 7~compatible with the inclusions, and sets of morphisms Sn+l C Gn >

S;;, C Gn+1 - G, ,

O
in direction reversing one to one correspondence st) s-l (that is, dom s = cod s-i and cod s = dom s-l) such that: (a) G, is freely generated by it4 set of arrows of the original graph G; (b)

u

(JIGiGn

SF’,

where M is the

The set *w, of endomorphisms of the form s-9, ss-1

where s E u S, l
is a reduced, symmetric set in G, which go to identities in 7r; and

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WITH

(c) All morphisms of G,-, become isomorphisms in G,/#‘, .

SEVERAL

which

99

OBJECTS

become

isomorphisms

in 7~

It follows from the above properties (a)-(c) that if we can make such a construction, then G,,/W, is the category Gc,, of the fraction tower, and the functors of the tower will be inclusions because of the uniqueness of reduced representatives of the morphisms of G,,/%<, (Theorem 24.1). The direct limit G~,J of the Gt,, will then be given by G,/$iI,, , where G, is the union of the G, and yfL, is the union of the %‘,, . Since the +@i are reduced and symmetric, the same is true of their nested union 79; . Consequently Gc,, is rigid. We assume that the construction is carried out for n. We define s n+l to consist of all morphisms

which are reduced with respect to %<, , which go to isomorphisms in rr but not in G,/%i , and which are such that X, ... Xi (and hence X i+l *.* X,) does not go to an isomorphism in n for 1 < i < K. Let S;il be any set in one-to-one correspondence with Sn+i , and let G,,+i be as described in condition (a). Let 7Jrn+r be ?J?:‘;~ together with the words of the form s-lx;

. . . Xk ,

x1 ‘.’ xks-l,

where s = X1 a.0 X, E S,,+i . Condition (c) follows from the definition of Sla+i and %‘n+i . It remains to be shown that %cL+i is reduced and symmetric. 1. %L+l is reduced. This follows since %‘;, is reduced, and the fact that if X, ... X,c E SIL+l , then it is reduced with respect to %“;, . Since it does not involve any letters in S;:, , it is also reduced with respect to W%+, . 2. Wn+i is symmetric. Since “w;, is symmetric, we need only check that ifs = X, ... X, E Sn+i , then no word of the form Y, ... Y,,,Xi .** Xi is in 7?;, and dually no word of the form Xi+- **a X,,.Y, a** Y,,, is in “w;, . Observe first that if Y, a.* Y,?,X, *** Xi E “w;, , then i = K, since then x1 .** Xi goes to an isomorphism in n. Now if Y1 ..* Y,,X, .** Xk E “w;, , then X,< can’t be in ST’ for any j < n, since otherwise it goes to an isomorphism in G,/Wm . Hence Yr E ST’ and Yz *a* Y,,,X, *a* X, E G,+l for some j < n. But then since Xi e.1 X, goes to an isomorphism in r and X1 .‘. X,< E Gjel, it follows from induction on (c) that Xi *.* X1<

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100

goes to an isomorphism in G,/%$ , hence in G,/?V’~ , contradicting the definition of SR+l . This completes the proof. Combining the theorem with Lemma 15.1, we obtain: COROLLARY 26.2. Any category qf iterated fractions of a free category is rigid, and henceidempotent complete.

27. DERIVATIONS Let %7be a small K-category, and let H be an object of Abve. A derivation for H is a family of K-module homomorphisms v = *a : @(A 4) - WP, Q1,

PtCZ4~l

such that if x E ‘%?(p,q) and y E V(q, r), then GY)

= 44Y

+ XV(Y).

(1)

By induction on t, it follows more generally that 4%%

... Xt) = f: x* “’ xi-lv(xi)

Xi+1 ... Xt .

(2)

i=l

Also taking x = 1 = y in (1) we see that V( 1) = 0. Consequently if s is an isomorphism in V, then 0 = V(1) = D(X1) = V(S)s-1 + S?+i), so that v(s-1) = -s-l?+) s-l.

(3)

Let H be an object of Abv”, and let z, be a derivation for H. If I’: 9 + V? is a K-functor, then for each p, QE 153 j let wPa denote the composition =%J, 4) --

WTP),

w7)) -

fv’(P),

%))

of K-module homomorphisms. Then w is a derivation for H(V, V), called the restriction of v to 23. We shall also say in this case that w extends to v. Let n be a small category, and let H E AbK(n*Xn). Then taking V = KT in the above, it is clear that a derivation for H is completely determined by a family of functions

RINGS

satisfying (1). this form.

WITH

Henceforth

SEVERAL

the only

101

OBJECTS

derivations

considered

will

be of

LEMMA 27.1. Let G be an oriented graph with arrows M, and let H E AbGo*XCo. Suppose that for each element m E M we have an element v(m) E H(dom m, cod m). Then v extends uniquely to a derivation for H.

Proof.

This is immediate

from the formula

(2).

Now consider a category r and a set R of relations in r. Let rr = r/R. Then an object H E Ab n*Xa can be considered as an object of Abr*xr using the natural functor F + rr. In the following lemma we shall not distinguish notationally between a morphism in r and its image in rr. LEMMA 27.2. In the situation just described, let v be a derivation for H considered as an object of Abr*Xr, and let (y, f, g, z) be a path of length n from A to B in r with respect to R. Then

gYiwi)

- 4&N zi = 44

- v(B).

Hence if the path is closed, then the left side is zero. Proof.

We have using (2) +rdi%J

and similarly

i

i=l

Yiwd

=

V(Yi)fi%

+

3 +

YP(fJ

YifiW

with f replaced by g. Hence

- ski)) 3 =

[V(Yi.fGi)

- g1 MY&%)

-

V(Yi)fi%

- ri”hw1

- 4Yi)Wi

= v(A) - v(B) since fi

gi in rr, and since in r we have the equations A = Ylfl% >

- Y&441

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COROLLARY 27.3. In the situation of the lemma, if v(a) = v(b) for all (a, b) E R, then v is a derivation for H consideredas an object of AbsiXrr. LEMMA 27.4. Let 7~~be the category of fractions of a small category r with respect to somesubsetS of its morphisms,and let H E AbnS*X?rS.Then any derivation v of H regarded as an object of Abn*Xn via the natural fun&or rr -+ 7~~can be extended uniquely to a derivation of H regarded as an object of Abss*Xns.

Proof. The category rrs can be constructed by taking the graph G whose vertices are the objects of TI and whose arrows are the morphisms of rr together with a set S-r in direction reversing one to one correspondence with S, and dividing G, by the set of relations R of the types (XY, x oY),

(e, , l,),

(s-ls, I),

(ss-1,1).

Here xy denotes a morphism of length two in G, , whereas x 0y is the composition in rr, and eP is the empty word at p in G, whereas 1, is the identity morphism of p in 7~.Now regarding H as an object of AbGa*XGo, we have v defined on those arrows which are morphisms in rr by what is given, and we can define v on those arrows of the form s-l, s E S, using (3). Then v extends uniquely to all morphisms of G, by Lemma 27.1, and from the way we have defined it, we have v(a) = v(b) for all (a, b) E R. Hence by Corollary 27.3, v is defined on GO/R = 7~s. COROLLARY 27.5. Let G, be a category of iterated fractions of a free category G, , and let M be the set of arrows of the underlying graph G. Suppose that H E AbCw*XGw*,and for each m E M, let v(m) be an element of H(dom m, cod m). Then v extends uniquely to a derivation of H.

Proof. Let G, = b,>,, G, where for each n, G,n+l is a category of fractions of G, . By Lemma 27.1, v extends uniquely to G, . Assuming that it extends uniquely to G, , by Lemma 27.4 it extends uniquely to G 12+1. Therefore it extends uniquely to G, for all n, and consequently it extends uniquely to the direct limit G, . Let M be an additive category, and let D, E E GP. Then H = @(D, E) is an object of Aba*Xn. We shall refer to a derivation for H in this case as a deviation from D to E. Thus a deviation from D to E is given by a family of morphisms in OZ v(x): D(dom X) - E(cod x),

XC??7

SEVERAL OBJECTS

103

= 49 E(Y) + Wx) V(Y)*

(1’)

RINGS WITH

satisfying GY)

Formulas (2) and (3) in this case become ~(+%

... Xt) = t. D(r,

“’ Xigl) v(xJ E(x,+l

“. xt)

(2’)

i=l

and v(s-1) = --D(s-1) v(s) E(s-1).

(3’)

We also restate Lemma 27.2 and Corollary 27.5 in this context since both will be applied in the following section. LEMMA 27.2’. Let rr = r/R, and let D, E E OP. Suppose that v is a deviation from D to E considered as objects of W, and let (y, f, g, z) be a path of length n from A to B in r with respect to R. Then

Hence if the path (y, f, g, z ) is closed, then the left side is zero. COROLLARY 27.5’. Let G, be a category of iterated fractions of a free category G, , and let M be the set of arrows of G. Suppose that D, E E 6P, and let

_. v(m): D(dom m) + E(cod m) for each m E M.

Then v extends uniquely to a deviation from D to E.

28. THE

FOUR TERM

RESOLUTION

Throughout this section the following notation will be in force. G will denote an oriented graph with vertices V and arrows M, and GOwill denote the associated free category. For each nonnegative integer n, G n+1 will denote a category of fractions of G, , and G, will denote the direct limit h,>, G, . We shall let r = GJR where R is some set of relations in G, . Q?will denote an abelian category with coproducts, and if p E 1r 1 = V, then the left adjoint for the evaluation functor

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Tp: GP -+ 6Y will be denoted by S,: GY4 09. Thus S, is given explicitly bY

We are going to construct a certain four term resolution X for 257 in terms of the presentation 7r = G,/R. This will provide us with a method for dealing with problems concerning cohomological dimension less than or equal to two relative to which the standard resolution is too crude. Actually it will be convenient for future reference to construct directly the tensored complex D & X where D E 6P. The complex X can then be retrieved by taking Q! = Abn* and D = Zrr. The sequence to be constructed is an exact sequence in Q9 of the form (I), where the last coproduct is indexed by all closed paths ( y, f, g, z) in G, with respect to R. We make the convention that if a is a morphism of G, , then its image in GJR = 7~ will also be denoted by a. The context will always determine the category in which a is being considered.

Wdomy)

D(dom m)

D(P)

[21

Evaluated at a vertex 4, the sequence (1) must have the form (2), where the vertical arrows are the coproduct injections, and the dotted arrows represent a contracting homotopy which is to be constructed. We recall how the morphisms of 7~act on the terms of (1). Consider, for example, the term @Jnle,,,, S codnL(D(domm)), which we shall abbreviate to E. If x’ E ~(4, q’), then E(d): E(q) + E(q’) is given by +?z, x) E(d) = v(m, xx’). The action of x’ on the other terms is similar. Observe in particular that ~(11z,x) = v(m, 1) E(x). Now consider the family v(m) = v(m, 1): D(dom m) + E(cod m),

?llEivl.

RINGS

By Corollary

WITH

SEVERAL

27.5’, this extends uniquely

105

OBJECTS

to a deviation

v(a) = ~(a, 1): D(dom a) + E(cod a),

aEG,.

We then define v(fz, x) = a(a, 1) E(x) whenever cod a = dom x. We now define the morphisms

01,,L3,y, and 6 as follows:

(3)

In order to have the right side of the last equation make sense, we must define w(b, a, X) = -w(a, b, x) for (a, b) E R. This is unambiguous because of our convention that not both of (a, b) and (b, a) can be in R. It is immediate that 01,/3, y, 6 are morphisms in 6P, or in other words, natural transformations. In fact they are the unique natural transformations satisfying the Eqs. (3) with x replaced by 1, throughout. 28.1.

THEOREM

Proof.

First

The sequence (1) is exact.

we show that it is of order two.

We have

v(m, x)&cd, = (u(mx) - D(m) #(X)) a, = D(m) Consequently,

- D(m) D(x) = 0.

@ = 0. To see that y/3 = 0, we observe v(a, x) p, = u(ax) - D(a) u(x)

first that (4)

for all a E G, . For a E M, this is just the definition of /3, . Assuming that it has been established for all a E G, , one uses Eqs. (2’) and (3’) of Section 27 and the fact that every morphism in Gntl can be written as a composition of morphisms in G, and inverses of such morphisms to establish it for G,,, . Since every morphism in G, comes from one in G, for some n, (4) is valid for all a in G, .

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Then we have

4% b, 4 Y&J&= (+G 4 - q4 4) B, = u(ax) - D(a) u(x) - u(bx) + D(b) U(X) = 0 since a and b are the same in 7~. Hence y/3 = 0. Next we compute

and the last expression is zero by Lemma 27.2’. Hence 6y = 0. Now we construct the contracting homotopy. First we have u(L) 014= mJ and so 01is an epimorphism.

where

= lo(a) >

Then we compute,

T(X) is any representative

using (4)

of x back in G, . Hence

if we define

8, bY then we have

This establishes exactness across the first point in the sequence. Observe that the definition of 0 involves a definite choice function r which we must remain committed to. Next we compute

where at the last step we are just using the fact that v is a deviation. Now m-r(x) and I are the same when considered in r. Hence in G, we can choose a path (y, f, g, x) from mu to T(??ZX) with respect to R.

RINGS

Then using Lemma

Therefore

WITH

SEVERAL

OBJECTS

107

27.2’ again, (5) becomes

if we define A, by

we obtain

This establishes exactness at the second point of the sequence. Before establishing exactness at the remaining position, we take a closer look at A. LEMMA

28.2.

For any a E G, we have +,

4 42 = i

WY?) 4fi

, gi >3)

i=l

for somepath (y, f, g, z) from aT(x) to T(ax).

Proof. For a E M this is true by definition of A, . Suppose that it is true for a in G, . If a E G, and has an inverse in G,+r , then we can write using Eq. (3’) of Section 27, v(a-1, x) A, = --D(a-1) u(a, a-lx) A,

where aT(a-?x-)= YlfiZl yngn,zn = T(aa-?x)

= T(X).

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If we compose each of these last equations on the left with a-l, reverse the order of the equations, and interchange the left and right sides, we obtain the conclusion of the lemma for a-l. Now if a E Gn+l , then we can write a = ala2 e.0 at where each ai is in G, or is the inverse of some morphism in G, . Then we have, using what we have already proved and the fact that z, is a deviation, v(a, x) A, = 2 @a, ... U,_I) v(a, , ai+ ... a,x) h, i=l

.‘. a,&lzji) w(yji, =il g D(al where

fji, gj”),

for each i, 1 < i < t, we have

y;,gf$;,

= T(U, .‘. a,x).

If we compose the last system of equations on the left with a, e-e aiwl , and then write the t systems one under the other starting with the t-th, we obtain the conclusion of the lemma for a. Returning to the proof of the theorem, we compute w(a, b, x)(1 - Y&J = w(a, b, x) - [fJ(a, x) - v(b, 41 A, . Using Lemma 28.2 and the fact that I we find that the right side of (6) is just

= I

(6)

(since a = b in ‘rr),

f D(ri> w(fi 9gi 74

(7)

i=l

relative

to an appropriate

closed path (y, f, g, z). Thus

if we define

then we obtain

This completes

the proof of the theorem.

Remark 1. It suffices to use only the nondegenerate closed paths (y, f, g, z) in the sequence (1). For it is immediate from the definition

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WITH

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109

OBJECTS

(3) of 6 and the definition of degenerate closed path (Section 19) that the terms of the coproduct corresponding to degenerate closed paths are taken to 0 by 6. Other reductions in the number of closed paths are possible. For example, if two such paths are equivalent under the equivalence relation on paths defined in Section 19, then they map to the same thing under 6. Then using the fact that every closed path is a vertical composition of “simple” closed paths (i.e., yifixi # YjfiZj for i # j), we see that it suffices to take one path in each equivalence class of simple closed paths. Moreover, if one closed path is obtained from another by a rotation of the subscripts, then they map to the same thing under 6, hence we don’t need both of them. Remark 2. Let G, be a free group on ik? generators, and let R be a subset of G, . Then R can be identified with the set of relations of the form (Y, 1) with Y E R. Let Q? be the category Ab, and let D be the &-module Z with rr acting trivially (i.e., D = OZ). Then the exact sequence (1) reduces to a sequence of right &-modules of the form @Z?TL+

@Zr

~@ZTr-%Z74+

R

z-40,

(8)

M

where the last coproduct ranges over all pairs of sequences (x, r) satisfying fl2, xirixil = I, with either ri or rtl E R. This exact sequence first appeared in Lyndon’s paper [34]. 28.3.

COROLLARY

If OT is AB4, then for any D in GWJ we have h.d. D < 1 + sup h.d. D(p). WV

Hence gl

dim 0!‘~ < 1 +

gl

dim 6Y

and dim G,, <

1.

Proof. As usual the second and third inequalities are consequences of the first. Now since R is empty in this case, the sequence (1) becomes 0 -

@ IItEM

hod

,,,P(dom

m))

-

@ %(D(P)) FV

-

D -

0

Let n = sup h.d., D(p). S ince 02 has exact coproducts, the functors S, are exact, and so by Lemmas 9.4 and 9.5, the left and middle terms in the sequence have homological dimension < n. Hence h.d.D ,< n + 1.

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The work of Stallings [48] and Swan [50] shows that if 7~ is a group and dim z- < 1, then 7~ is free as a group, and hence is of the form G, , Also in Section 33 we shall show that if n is a weak delta and dim 7~ < 1: then 7~is of the form G, . Now there are categories r satisfying dim 7~ < 1 which are not of the form G, . Consider, for example, the monoid 71 with one nonidentity element e satisfying ee = e. Then it is easily seen that 2~ is a retract of Z(n* x rr) in A!F*~~, and so dim rr = 0. On the other hand, 7~ is not of the form G, since it is not idempotent complete. However, the idempotent completion of 7~ is the category generated by the graph X

0

subject to the relation xy = 1. This category category generated by the graph

A

in turn is equivalent

to the

Z

subject to the relations xyz = 1 and xxy = 1. This last category is oj the form G, . In fact we know of no category satisfying dim r < 1 whose idempotent completion is not equivalent to a category of the form G, . We recall at this point that dimension is preserved under idempotent completion (Section 12), and that G, is idempotent complete (Corollary 26.2). In the same way that we proved Corollary 28.3, we obtain (using Remark 1 to the theorem): COROLLARY 28.4. Let T be a category of the form G,/R where Q(R: is trivial (i.e., R admits of no nondegenerate closedpaths). If Q! is AB4 and D E Q?, then h.d. D < 2 + sup h.d. D(p). WV

Hence gl dim

6P

<

2 +

gl dim

G?

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111

znd

Remark. By duality, one can replace AB4 by AB4* in Corollaries 28.3 md 28.4. Again in Section 33 we shall show that if 7~ is a weak delta and lim n < 2, then 7~ is of the form G,/R where Q(R) is trivial. One is :empted to conjecture that a group of dimension < 2 has a presentation If the form G,/R where Q(R) is trivial. This latter condition is equivalent :o the condition that the morphism y in the exact sequence (8) be a nonomorphism. This is what happens, for example, in the case where r s a torsion free abelian group of rank one (see Balcerzyk [4, Lemma 31). However, at this point the conjecture seems prohibitively difficult to zither prove or disprove.

29.

THE

DIMENSION

RAISING

LEMMA

Throughout this section, GZ and g will denote abelian categories. The following important lemma is essentially Lemma 1.7 of [38, 3art I], somewhat cleaned up following a suggestion of Lawvere. LEMMA 29.1 (The !ors T,,, T,: 93-a fransformation q5: 1a natural transformation

dimension raising lemma). Consider exact funcandL: a + 39, and suppose that there is a natural --f LT, which is pointwise a coretraction, and a r: T, -+ Tl such that Lr = 0. Let

5e an exact sequence in 99, and suppose that there is a morphism r in 02 such !hat the diagram

/i’ i” riJ T,(L(A;) ;s commutative.

Then h.d.,B’

-“(“)

T,(B)

3 1 + h.d.nA.

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112

Proof. It suffices to prove that if E E Ext$(A, C) and E # 0, then EL(E) # 0. If EL(E) = 0, then the long exact sequence induced by E yields F E Ext,“(B, L(C)) such that L(E) = pF. Then since LI’ = 0: we have

which contradicts Lemma 9.6. EXAMPLE 1. Let 2 be the category 0 5 1. Then a2 is the category of morphisms in CPIwith commutative squares as morphisms. There are two evaluation functors To , T, : GY2-+ &‘, and x induces a natural transformation r: T,, + T1 . If A E ( CY1, then the diagram

o-+0 +A====A---to 1 1 /I -1 1 O+A=====A---to-+0

(1:

can be considered as an exact sequence

0 -+

L(A) -+

S,(A) 4

M(A) -

0

in 0Z2. Clearly Lr = 0, and LT, = I n . Hence if we take 7 = 1, , we find that all of the conditions of the dimension raising lemma are satisfied. Therefore h.d. M(A) 3 1 + h.d. A, and consequently gl dim cpG2 > 1 + gl dim a. By Corollary 9.7, the same is true for any category containing 2 as a retract. But the condition that 7r contain 2 as a retract is equivalent to the existence of two objects p, 4 E 1x 1 such that n(p, 4) # 0 whereas ~‘(9, p) = 0 (see Lemma 1.2). In particular, if v is a delta, then gl dim @ = gl dim 91 if and only if n is discrete (assuming gl dim a < co). EXAMPLE 2. Let n be the linearly ordered set of n elements, and let ii be obtained from n by inverting the morphism from the first to the last

RINGS WITH

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113

element. Then the objects of 12’”are the sequencesA, -+ A, --t *a*-+ A,-, in GZsuch that the composition is an isomorphism. If n > 3, then we may add a row of zeros to the top of the diagram (1) and n - 3 rows of zeros to the bottom to obtain a short exact sequence in 6Yz. Then precisely as in Example 1 we obtain gl dim G@> 1 + gl dim 0J. If n < 3, then the argument breaks down. In fact in the previous section we saw that dim 3 = 0, with the result that gl dim C@= gl dim /X EXAMPLE 3. Let N denote the free monoid on a single generator x. Then GP is the category of endomorphisms in GZ.Let To = Tl : GZN+ Ol be the evaluation (forgetful) functor, and let L: Q?-+ GZNassign to an object its identity endomorphism. Then LT = la, and if we let r = 1 - X: T,, + T, , then Lr = 0. If A E 1OT/, then the exact sequence in OT

O-+AUA@AL-4)

(2)

can be considered as an exact sequence from L(A) to L(A) in aN if we assign to A @ A the endomorphism (-: y). Also we have the commutative diagram in 6!! (but not in a”)

7=(0.1) /I

(;

;,=1-x

AA’LA@A

and so the dimension raising lemma yields h.d.L(A) > 1 + h.d. A. Consequently, gl dim 6YN> I + gl dim a. EXAMPLE 4. Let Z denote the free group (written multiplicatively) on a single generator x. Then az is the category of automorphisms in 0’. Now in the sequence (2) of the preceding example, the endomorphisms 607/8/2-S

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assigned to each term are automorphisms. to the present situation to give h.d.L(A)

>

1 +

Hence that example

h.d.

applies

A,

and gl dim QZz >, 1 + gl dim 02. Remark. Since the categories 2 and N are of the form G, , and ii and Z are of the form G, , it follows from Corollary 28.3 that the inequalities of Examples 14 are actually equalities when U is AB4. EXAMPLE 5. Let 7r denote the cyclic group of order n generated by an element x. Then 0P is the category of automorphisms 01: A -+ A in 6Y satisfying Q = 1, . Let A be an object of a such that nA = 0 where n, is the sum of 1, with itself n times. Then relative to the endomorphism (-: y) of A @ A we have

(-:

Y)” = (_l

;, = (:,

;,.

Consequently, the sequence (2) of Example 3 may be considered sequence in GP, and so that example shows once again that h.d.L(A)

3 1 + h.d. A.

as a

(3)

But we also have the exact sequence in GY!

where A and V are the diagonal and codiagonal morphisms corresponding to the coproduct AT. This sequence may be considered as an exact sequence from L(A) to L(A) in @. Since h.d.,Ar = h.d.A, it follows from (3) and (4) and the remark to Corollary 9.2 that h.d.L(A) = co providing A # 0. This can be used to prove the second half of the following generalization of Maschke’s theorem: If rr is a finite group of order m, then gl

dim@

=

gl

dim Q!

or

co

depending on whether or not m is invertible in C(a). Details may be found in [38, part I, Section 31.

RINGS

LEMMA

29.2.

WITH

SEVERAL

115

OBJECTS

If C!?is AB4 and T is one of 2, N, OY Z, then h.d .,,*xr An 3 1 + h.d., A

(5)

foranyA~jOr1. Proof. Let 0 denote the initial element of 2* x 2. Then we have the exact sequence in GF!2*x2 0 + An --t S,(A) + L,(A) + 0, where L,(A) is the diagram with A at 0 and zero elsewhere, is the left adjoint for the evaluation functor T, . Applying twice, we see that h.d.L,(A)

and S,,(A) Example I

> 2 + h.d. A.

Since h.d.S,(A) = h.d.A, this yields (5) in the case 7~ = 2. Now consider the case rr = N. Then we have an exact sequence Qpxn

0--An-%Arr~A-+O,

in (6)

where A is considered to have trivial action by n* x n. If x is the generator of 7~, and v,, and u, denote the coproduct injections, respectively, for the left and middle terms in (6) corresponding to xn(n > 0), then 01and ,8 are defined by l&a

=

1,

)

%B

=

%A+1

-

%I *

The sequence (6), considered as a sequence in G!?, is just a trivial case of the four-term resolution, and hence is exact. (Note, however, that /3 would not be a morphism in M n*Xn if n were a free monoid on more than one generator.) Now applying Example 3 twice, we see that h.d +xnA

3 2 + h.d.w A.

Hence from (6) we obtain (5) in the case 7r = N. The argument for Z is exactly the same as that for N, except that here n ranges over all the integers, and Example 4 replaces Example 3. We shall encounter a further application of the dimension raising lemma in Section 32. Still another having to do with algebras of the form K[X]/f((X) where f(X) is a manic polynomial can be found in [38, Part I, Section 41.

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30. B-SUBCATEGORIES Let 7~be a small category, and let T be a subcategory. Let B (for basis) be a set of morphisms of 7~satisfying: (i)

IfbEB,thendombE/rj;

(ii) If x E n and dom x E 1T 1, then x can be written uniquely as x=tbwitht~randb~B; (iii)

I, E B for all p E 1T 1.

If such a set B exists, we shall call 7 a B-subcategory of VT.We say that 7 is a B*-subcategory of T if T* is a B-subcategory of n*. For example, if rr is a group and 7 is a subgroup, then 7 is both a B and B*-subcategory of 7~.Also if 7 and VTare categories with r nonempty, then injecting 7 into T x n by putting a fixed identity morphism in the second position, we see that 7 can be considered as a B and B*-subcategory of T x n. Let 7 be a B-subcategory of n, and consider the inclusion u: 27 -+ Zn. Then conditions (i) and (ii) guarantee that Zr(U, ) is ZT*-free. In fact, for 4 E / n /, we have an isomorphism in AbT* z~(u, q) = @ ZT( , dom b), bEB,

where B, is the subset of B consisting of those morphisms whose codomain is 9. Dually if 7 is a B*-subcategory of VT,then Zz-( , U) is ZT-free. From Proposition 11.3, we therefore obtain: LEMMA 30.1. category. Then

Let

T

be a B*-subcategory of vr, and let C7!be an AB4* h.d.,D / 7 < h.d.,D

for all D E GP.

Remark. If 7 is a subgroup of 7~and D E Ab- is Z with trivial action, then Lemma 30.1 is just Shapiro’s lemma [ll, p. 1961. If 7 is a B-subcategory of YT,then using (i)-(iii) it is easy to see that if tx E 7 and t E 7, then x E T. Therefore if 7 is also a B*-subcategory of rr, then t,xt, E 7 together with t r , t, E 7 implies x E 7. From this it follows that Z T is a retract of Zrr( U, U) in A&*X7. Thus from Proposition 11.1 we obtain:

RINGS WITH LEMMA

IfS,:Q?

SEVERAL OBJECTS

117

30.2. Let T bea B-subcategory of rr, and let be an AB4 category. -r GP is the left adjoint of the restriction functor, then h.d.,S,(D)

< h.d.,D

for all D E Q?. If, further, 7 is a B*-subcategory of 7~,then equality holds,

and consequently gl dim Q? < gl dim GF Similarly, from Proposition 12.1 we obtain: 30.3. Let T be a B and B*-subcategory of 7~, and let GZbe an category with coproducts. Then

LEMMA

AB4*

h.d .7iXr* AT < h.d.,rX, Arr for

all AE 1GZ/.

3 I. BRIDGE CATEGORIES By Corollary 28.3 we know that if G, is a category of iterated fractions of a free category, and if OLis an abelian category with exact coproducts, then gl dim OL< gl dim GYG”< 1 + gl dim GY. We do not have any general theorem which distinguishes the two cases. Example 2 following the dimension raising lemma gives an idea of the sort of difficulties involved. However, suppose that 6 = 1, and that G, is obtained from G, by inverting only morphisms of length one (that is, arrows of the original graph). We shall call such a category a bridge category. If M is the set of arrows and 5’ C M is the set of morphisms to be inverted, then every morphism of G, can be expressed uniquely in the form m’lm’”

12

mc’ t’

m,EM,

(1)

where ci = &l if mi E S, ci = 1 if mi $ S, and the combinations mm-’ and m-lrn do not appear in (1) for any m E S. The reason for the terminology is the following. Suppose that one is driving in a modern city comprised of a network of islands V and

118

MITCHELL

bridges M. One will naturally ask if it is possible to cross each bridge exactly once. The problem will be complicated somewhat by the fact that only a subset S of the bridges will be two-way, and the solution will no longer depend on the local properties of the islands. The following cities illustrate this.

n I7 city A

city

B

Having thus realized that the classical method fails, we convert the city into a category. Specifically, we give each of the two-way bridges a direction, and we consider the free category G, generated by all the bridges. Then, letting G, be the category of fractions obtained from G, by inverting the two way bridges, the problem amounts to finding a morphism (1) in which each bridge occurs exactly once, possible with an exponent of minus one. Now although we are not a bit closer to a solution of the original problem, it turns out that we are on the way to generalizing the Hilbert syzygy theorem. First let G be a graph with vertices I’ and arrows M, and let r be the bridge category obtained by inverting a set S of arrows. Also let G’ be a subgraph with vertices I” C V and arrows M’ C M, and let n’ be the bridge category obtained by inverting the set S’ = M’ n S. Then 7~’ is a B-subcategory of n, for we can take as B the set of all morphisms (1) whose domains are in I”, but m, $ M’. Dually n’ is a B*-subcategory of 77. LEMMA 31.1. discrete category, B*-subcategory.

If 7~ is a bridge category which is not equivalent to a then TI contains at least one of 2, N, or Z as a B and

Proof. If m E M - S, then by the remark preceding the lemma we see that rt contains N or 2 as a B and B*-subcategory depending on whether or not dom m = cod m. Hence we may assume M = S, so that 7~is a free groupoid. Since 7~is not equivalent to a discrete category, there is a nontrivial endomorphism m1m2 *.. mI , where mi or rnii E M for all i, and taking t to be minimal we may assume that mi + m, or

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WITH

SEVERAL

119

OBJECTS

my1 for i > 1. Let &r be the set of arrows obtained from M by replacing m, by an arrow m starting and ending at dom m, . If 3 = a, then the corresponding bridge category ii is isomorphic to rr via an isomorphism which sends m to m, *-. m, , and since we now have an arrow m such that dom m = cod m, we see as above that ii contains Z as a B and B*-subcategory. Remark. retract.

One can also show that 7~ contains

one of 2, N, or Z as a

THEOREM 31.2. Let 7~ be a bridge category which is not equivalent a discrete category. If 6Z is an AB4 category, then gl dim

If

further

GZ is AB4*,

6P

=

1 + gl dim

to

CPI.

then h.d.,.x,

for all A E 1GZ’I. In particular, dim,

AT

=

1 +

h.d.,

A

;f GZ is any small K-category, @?n =

1 +

dim,

then

V.

Proof. The inequalities in one direction follow from Corollary 28.3. The inequalities in the other direction have been proved for the cases 7~ = 2, N, and Z in Section 29. The general case now follows from Lemmas 30.2, 30.3, and 31.1. 31.3. Let rr be a product of n bridge categories, none of is equivalent to a discrete category, and let be an AB4 category.

COROLLARY

which Then

gl dim

If further GI is AB4*,

GP

for all A E 1GY I. In particular, dim,

1.

When

+ gl dim

GI

then h.d .neX,, An

Remark

n

=

=

n +

h.d.,

A

;f V is any small K-category, ‘37~ =

n +

dim,

each of the n categories

then

V.

of the corollary

is a free

MITCHELL

120

monoid on one generator and a is a module category Ab”, the first equality of the corollary yields r.gl dim fl[X,

,..., X,] = n + r.gl dim (1.

This is the version of the Hilbert syzygy theorem first proved by Eilenberg, Rosenberg, and Zelinsky [14]. L’k1 ewise taking % to be a K-algebra /l, the third equality becomes dim, JX,

,..., X,] = n + dim, /I.

This theorem also appeared in [14]. Still another theorem of these authors is had by taking x to be n in Theorem 31.2 with a = Ab” again. Since [An] is just the ring T,(A) of triangular matrices over (1, we obtain r.gl dim 3’,(A) = 1 + r.gl dim A, and when /I is a K-algebra, dim, T,(A) = 1 + dim, A.

Remark 2. The corollary is also valid if n = CO.For then for each finite n, 7~can be expressed as the product of a nonempty category with a product of n bridge categories, and so the latter can be considered as a B and B*-subcategory of r.

32. DIMENSION By Corollary

OF DELTAS

13.4’ we know that if @ is an AB4 K-category, gl dim OP < dim, QT+ gl dim 02

then (1)

and h.d .n*X,An < dim, rr + h.d.a A

(2)

for all A E ) r% I, where n is any small category. In the previous section, we saw that (1) and (2) are equalities whenever x is a finite product of bridge categories. In the present section we shall be interested in these inequalities in the case where r is a weak delta. By Theorem 23.9 we know that in this case dim,n can be computed as the sup of the homological dimensions of objects of the form L,(K), p E 1n 1, where L,(K) is the object of AbK” with K atp and zero elsewhere. Also by Lemma 22.2

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WITH

SEVERAL

121

OBJECTS

we can consider L,(K) as an object of Pn in computing its homological dimension. Note that pr is initially finite since rr is weak. Our main tool will be the following application of the dimension raising lemma. LEMMA 32.1. Let 7~ be an initially $nite delta, and let D be a K-projective object of AbKW, where K is a commutative ring over which projective modules aye free. Suppose that q E / 7~ / is such that D is split at q but D 1VT” is not projective, whereas D 1 7~,>is projective for all p < q. If 67 is an abelian k-category with coproduck, then

h.d., A OK D 3 1 + h.d. A

(3)

forallAElli(1. Proof. By Lemma 22.1, we may assume r = rq . For each p E / 7~ 1, let D,’ be the image of the morphism @

D(dom X) + D(p).

(4)

cod Z=LI X#l,

Since D is split, we can write D(p)

= 0; @ D;

(5)

for all p E / 7~ I. We then get an exact sequence O-M-4

@ S&D;)tD-0 DElfll

(6)

where by Corollary 23.3 and the fact that D / nfl is projective for p < q, we have M(p) = 0 for p < q. Furthermore, M(q) # 0, since otherwise D would be projective. If we evaluate (6) at q, we obtain an exact sequence of K-modules 0 --

M(q) 2+cogEq

D;,,

2-+

0; @ D;-

0,

(7)

where the image of pQ is in those terms of the coproduct corresponding to x f 1, . Therefore if we denote the injections and projections for the coproduct by U(X) and p(x), then we can write PP = co~zE*PqPc4 44 Z#l”

(8)

122 Now

MITCHELL

if E denotes the middle term of (6), then by definition u(x)

Therefore

=

U(ldomz)

where

‘%%

of E we have

cod x = q.

if we define +)

=

P&(X)

U(ldomz),

then from (8) we obtain Pn = ,,E=, 44 EW

(9)

X#l,

Now let A E 1OT 1, and let us tensor (6) over K with K-projective, we obtain an exact sequence in an: 0---,A@,M~A@,E In the notation

A. Since D is

---tA@,D---tO.

of the dimension 1’:

raising lemma,

@

Tdo,.

-

(10)

let

T,

cod z=q X#l,

denote the obvious natural transformation, where T, : 09 --f Q? is the p-th evaluation functor. Also denote L(A) = A OK M. Then since M(p) = 0 for p < q, we certainly have LI’ = 0. Since M(q) is a nonzero projective K-module, it contains a copy of K as a retract (since projective K-modules are free), and consequently LT, contains 1, as a retract. Therefore if we apply A OK to Eq. (9), we see that all of the conditions of the dimension raising lemma are satisfied, and so that the lemma yields the desired inequality (3). Remark. All that is needed in order to insure that LT, contains la as a retract is that M(q) contain a copy of K as a retract. This may be directly verifiable in some cases, thus eliminating the requirement that projective K-modules be free. If Z- is a weak delta, then an object p E j z 1 will be called (n, K)-strong (n > 1, K a commutative ring) if there is an object q > p such that (i) h.d.L,(K) 1rq = n; (ii) h.d.L,(K) 1ng’ < n for p < q’ < q; (iii) The (n - 1)-st k ernel in a projective splits at q.

resolution

for L,(K)

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SEVERAL OBJECTS

123

It is an easy consequence of Schanuel’s lemma that condition (iii) is independent of the projective resolution. Note that if K is a field and h.d.L,(K) > n then p is (n, K)-strong. For in this case we need only take 4 to be minimal with respect to property (i), since property (iii) is automatic. LEMMA 32.2. If r is a weak delta and if p E I n ] is (n, K)-strong, where projective K-modules are free, then

h.d. L,(A) 3 n + h.d.or A

(11)

for all A in an AB4 K-category Q!. Hence gl dim 02” 3 n + gl dim a. Proof. If h.d.A = co, then (ii) follows from the fact that the evaluation functor T, has an exact right adjoint. Otherwise, let D be the [n - l)-st kernel in a projective resolution for L,(K). Then conditions (i)-(iii) show that D satisfies the hypothesis of Lemma 32.1. Hence h.d., A OK D > h.d., A, and so the conclusion follows from Lemma 13.6. COROLLARY 32.3. Suppose that projective K-modules are free. If rr is a weak delta such that dimKrr = n < co, and ifm contains an object which is (n, K)-strong, then (1) zs . an equality for all AB4 K-categories G?. The Fameis true when dim,g = co, providing that for each positive integer N there is an object in 7r which is (n, K)-strong for somen > N.

Remark 1. is a field.

The hypothesis of the corollary is always satisfied if K

Remark 2. If dim n = n < co, then it is not difficult to show that there is always a prime field F such that gl dim AbFW= dim,z-. In other words, when K = 2, there is always an abelian category (namely, AbF) relative to which (1) is an equality. We now turn to the inequality (2). Recall that if n is a delta, then r(r) denotes the full subcategory of 7~* x r consisting of those pairs [p, q) such that p ,< q in 7. We form a delta T(T) by adding an object 0 to T(T) and taking as morphisms from 0 to (p, q) the morphisms f of ~(p, q). The composition of f with a morphism (g, h) from (p, q) to

124

MITCHELL

(p’, q’) is defined as gfh. If v is a poset, then this construction simply amounts to adding an initial element to A. Now consider the object KT of APS*Xa). It is nonzero only on the final subcategory T(~T) of n* x 7~, and hence can be regarded as a diagram overmif we place the zero module at the object 0. We then have an exact sequence in AbK T(n) 0 + KT + S,(K) -L,,(K)

--, 0,

where S, denotes the left adjoint of the evaluation of T(T). From Lemma 32.2 we now deduce:

functor

of the object 0

COROLLARY 32.4. If 0 is (n + 1, K) strong in T(X), and if projective K-modules are free, then h.d .*iv,, AT > n + h.d., A for all A in an AB4 K-category a. In particular, if dimKr = n, then (2) is an equality. Likewise if dim,n = 00 and if for each N there is an n > N such that 0 is (n, K)-strong, then (2) is an equality.

33. DELTAS OF DIMENSION

<2

We now proceed to characterize those weak deltas such that dimK7r = 0, 1, or 2. These characterizations turn out to be independent of K. In passing, we shall show that if p is any object of a weak delta, then the first and second kernels in a projective resolution for L,(K: are split. From Corollary 32.3 it follows that if dim,x < 3, then the inequality (1) of the preceding section is always an equality. Likewise from Corollary 32.4 we see that if dim,n < 2, then the inequality (2: is always an equality. On the other hand, in the following section we shall exhibit finite posets v such that dim,n = 4, and where the inequalities (1) and (2) can be strict. We don’t know if (2) can be strict when dimzrr = 3. For dimension zero, we don’t even require that 7r be weak. In fact. PROPOSITION 33.1. tative ring, then dim,n Proof.

If T is any delta and K is any nontrivial = 0 if and only if 7~is discrete.

If = is discrete,

then clearly dim,n

commu-

= 0. On the other hand:

RINGS

WITH

SEVERAL

if rr is not discrete, then 7~contains 2 as a retract, we have dim,n > 1 by Corollary 13.5. THEOREM

commutative (a)

33.2. Let rr be a weak ring. Then,

dim,n

125

OBJECTS

and so since dim,2

= I,

delta, and let K be a nontrivial

,< 1 if and only &f r is free;

(b) dimKn < 2 a. and only ;f rr has a presentation where G, is free and Q(R) is trivial. Furthermore, if p is any object of n, then the first a projective resolution for L,(K) are split.

of the form

Go/R

and second kernels in

Proof. The “if” parts follow from Corollaries 28.3 and 28.4. Now let G be the graph of morphisms of 7~ of length one, and let G,, be the associated free category. We recall from Section 21 how to construct a minimal set R of relations in G, such that the natural functor F: G, + rr induces an isomorphism GO/R w n. For x E rr, we consider the set I of equivalence classes of F-l(x), where two morphisms are in the same equivalence class if and only if one can be obtained from the other as a consequence of relations of the form (a’, b’) with F(a’) = F(b’) of length strictly less than that of x in n. We let ci , i E I be representatives of equivalence classes, and we choose any i,, E I. Then we define R, to be the set of relations of the form (ci , ci,j where i 5 i,, , and R is defined as the union of the R, over all x E rr. Now let p be any object of r, and let us apply the four term resolution of Section 28 to the object D = L,(K) in A@“. Since n is a delta, there are no inverses involved here, and furthermore D(q) = 0 if q # p. Thus the four term resolution takes the relatively simple form

-B,

S,(K) A

L,(K)

-----f 0.

(2)

For x E ~(p, q), the coproduct injections U(X) defined in the diagram (2) of Section 28 will be regarded as a basis of the free K-module at q relative to the term S,(K) of (2). Likewise, the coproduct injections v(m, x), w(a, b, x), and h(y, f, g, z, x), where cod x = q and dom m = dom a = dom y = p, will be regarded as basis elements of the free K-module at q relative to the other terms of (2).

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MITCHELL

Let D’ denote the image of p. Suppose that we have an equation of the form

where IQ , Kj’ E K, the m, are distinct morphisms of length one from p to q, the mj’ are morphisms of length one with domain p, and the xi are nonidentity morphisms in rr with codomain q. Applying the definition of /3 (see Eqs. (3) of Section 28), we obtain

Since the mi have length one and no mj’xj has length one, it follows that no basis element on the right is equal to one on the left. Hence both sides are zero, and since the mi are distinct, it follows that the ki are all zero. This shows that if Rq : AbKn ---f AbK is the functor introduced in Section 23, then R&D’) is a free K-module on as many generators as there are morphisms of length one from p to 4 in n, and furthermore R&3’) is an isomorphism, where /3’ is p considered as a morphism onto D’. Hence by Theorem 23.2, we see that if D’ is projective (which by exactness of (2) is equivalent to saying that h.d.L,(K) < l), then /I’ is an isomorphism, and so /3 is a monomorphism. Hence to prove the “only if” part of (a), it suffices to show that if ,r is not free (that is, if R is not empty), then there is a vertex p relative to which fi is not a monomorphism. Again using exactness of (2), it suffices to show that y # 0. Let (c, d) E R, and write c = ma and d = nb where m and n are morphisms of length one. If we take p = dom c in the above, then by definition of y (Eqs. (3), Section 28) we find Y&4,

4 1,)) = v(m, a) - 4% b),

where q = cod c. Hence if y = 0, then the right side is zero, which by linear independence of the v’s implies that m = n and a = b mod R. But then (c, d) would be a consequence of shorter relations, contradicting the construction of R. We now proceed in an entirely similar fashion for the “only if” part of (b). Let D’ be the image of y. Suppose that we have an equation of the form ,

ya i=l

i=l

RINGS

WITH

SEVERAL

127

OBJECTS

where (ci , di) are distinct members of R, (Q’, dj’) are members of R, dom ci = dom ci’ = p, cod ci = q, and the xj are nonidentity morphisms of n with codomain q. Write ci = miai ,

di = nib, ,

cj‘ = mj’aj’

,

dir = nj’bj’

in G,, where the m’s and n’s have length one. Applying of y, we obtain

gl k&h , 4 - 4ni , 41 = i kj’[+mj’, aj’xj) -

the definition

v(tij’, bj’xj)].

(3)

As before we shall show that the ki are zero. If we define the spine of a basis element u(m, X) to be the morphism mx in n, then since basis elements appearing in the same bracket on either side of (3) have the same spine, we may assume that all basis elements appearing in (3) have the same spine. Then (3) can be rewritten il

k,n(m, , ai) - (t

kii v(m, , a,) = i i=l

kj’[v(mj’,

aj’xj)

- n(ni,

bj’xj)],

(4)

j=l

where moue = ci, as in the definition of R. Let us suppose that k, # 0. Using the fact that the (ci , dJ are all distinct, one can see from the explicit construction of R that the basis element v(m, , a,) appears only once on the left side. We consider the equivalence relation on the basis elements which appear on the right side generated by the requirement that two elements in the same bracket be equivalent. Then the sum of the coefficients of all basis elements in a given equivalence class is zero. In particular, this is true of the class of a(m, , a,). Since the sum of the coefficients of v(m, , a,) itself is k, # 0, there must be some other v in the class of v(m, , a,) the sum of whose coefficients is not zero. Hence this v appears on the left, say v(ul , ml) where I # s. But since the xi are not identities, this would make (m,u, , m,u,) a consequence of shorter relations, contradicting the construction of R. As before, this proves that R&D”) is a free K-module on as many generators as there are elements (c, d) E R with dom c = p and cod c = q, and, furthermore, R&y”) is an isomorphism, where y” is y considered as a morphism onto D”. Hence by Theorem 23.2, we see that if D” is projective (which by exactness of (2) is equivalent to saying that h.d.L,(K) < 2), then y” is an isomorphism, and so y is a monomorphism. To complete the proof, it suffices now to show that if L?(R) is not trivial,

128

MITCHELL

then there is a vertex p relative to which y is not a monomorphism. By exactness of (2) it suffices to show that 6 # 0. Thus suppose that Q(R) is not trivial, or, in other words, that there exists a nondegenerate closed path (y, f, g, .z) in G, with respect to R. By Lemma 21.2, we may assume that yi = 1, and that for no k >, 2 do we have (fk , gh.) equal to either (fi , g,) or (g, , fi) with yh: = 1 and zli = zi mod R. Let p = dom y and q = cod Z. Then we have

where D = L,(K). Th e only nonzero terms on the right are those for which yi = 1, , and so by the choice of path we see that the term w(fi , g, , ZJ appears isolated on the right. Consequently, 6 # 0.

Remark, In Section 36 we shall remove the weakness assumption in part (a) in the case where rr is a poset. It would be interesting to know if this can be done in the rest of the theorem. Certainly the above proof uses this assumption very strongly.

34. THE ~-GEM The first example of a finite poset 7~ such that the difference gl dim Q? - gl dim GZ depends on 02 was found by William Spears [47]. Using a nontopological method, Spears exhibited several finite posets 7~ with the property that if K is a field, then r.gl dim Kr = 4 =3

if

char K = 2

if

char K # 2.

We are now going to consider a family of such examples, one for each characteristic n (not necessarily prime), which in the case n = 2 will yield the smallest of Spear’s examples. Moreover, by taking two such examples of relatively prime characteristic, we shall obtain examples of finite posets n and T’ such that dimm

x

QT’< dimr + dimrr’.

We had considered generating such examples by ordering the faces of a finite simplicial complex by inclusion. However, the number of

RINGS

WITH

SEVERAL

129

OBJECTS

simplices necessary to triangulate a given space usually makes it difficult to write down projective resolutions in the corresponding diagram category. Albrecht Dold suggested considering the following cellular decomposition of the projective plane:

cl3 b

a

a

b

There are three O-cells, six l-cells, and four 2-cells. This is not a triangulation, but it is a regular cell complex. If we order the cells by inclusion, and then add an initial element and a terminal element to the resulting poset, we come upon the example of Spears mentioned above. More generally, consider the cell complex of Fig. 1 (n 3 2). If we order the cells by inclusion and add an initial element 0 and a terminal element t, we obtain the poset of Fig. 2. We shall call this set the n-gem, and we shall denote it by g, . We are now faced with the problem of writing down a projective resolution for L,(Z) in Abgm. Rather than give the details in this particular example, we shall describe the general method for writing down projective resolutions for L,(Z) in Abff where r~ is a finite poset with initial element 0. First let us recall how to compute the kernel of homomorphism

of free abelian groups of finite rank. Such a homomorphism is given by an m x n matrix of integers, which by elementary row and column operators (i.e., changes of base for the domain and codomain off) can be reduced to a diagonal matrix (i.e., aij = 0 for i # j). The map f is an epimorphism if and only if this diagonal matrix has 1 or - 1 in each column. Let j, ,..., j, be those of the m rows which consist entirely of 0’s. Then the kernel

off 607/8/I-9

is given, using the new basis on Z”,

by the 1

x

m matrix

with

1 in

FIGURE

1

t FIGURE

2

RINGS

WITH

SEVERAL

131

OBJECTS

position (k, ik) for 1 < k < I and 0 elsewhere. To compute g relative to the old basis on Z”, suppose that there were N row operations originally performed on f, and that the i-th one consisted in subtracting c times the p-th row from the q-th row. Then we perform N column operations on g, the (N - i + I)-st one being to subtract c times the g-th column from the p-th column. Now let 7~be a finite poset with initial element 0 and maximal element t. A resolution for L,(Z) may always be started off with the short exact sequence

0 -+

D--j

S,(Z)-+&,(Z)-+

0,

where S,(Z) is the constant diagram at Z, and D is constant at 2 over 7~ - {O] and is zero at 0. Suppose that we have constructed the n-th kernel D in a projective resolution for L,(Z), n > 0. Using induction we may assume that D(p) is a free abelian group of finite rank for each p E 7~. We then look for an exact sequence

O-D'>+

@ S,(F,)"+D+O, PEP

(3)

where the F, are free abelian groups of finite rank. Of course we can always take Fp = D(p) f or all p, but this is usually wasteful. In fact, in many situations it will suffice to take F, = D(p) for those p which are minimal with respect to the property D(p) # 0, and F,, = 0 otherwise. This will certainly be true when D is the first kernel described above. In any case one must verify that 01 is actually an epimorphism at each vertex. Suppose now that we have chosen such an epimorphism 0~. If q < t, then we let D,,t denote the corresponding homomorphism D(q) -+ D(t) of abelian groups. Then (3) yields a commutative diagram of free abelian groups of finite rank

0---,D'(q)J%

@ F,LL

D(q)-+0

PET”

D’ @,t

0 -+

! D'(t+

n 1

D CI.t I

@ F, -"L D(t)4 lJsn

0.

The matrix of fll can be computed as described above. Since D,,, is a monomorphism, /3, can be computed as the kernel of the restriction of a1 to OprIP F p > and the matrix of the latter is had by taking just some 607/8/I-g*

132

MITCHELL

of the rows of the matrix of OI~. Then one must express the generators of D’(g) in terms of the generators of D’(t), considering both as elements of @,,,, Fp , to compute the matrix of Dht. If we apply this process to the n-gem, we obtain as third kernel the diagram D such that D(P) = z,

1
<2n,

= zzn,

P = t,

z 0,

otherwise.

The morphism @zr D(p) ---f D(t) is given (which we block off into four n x n matrices)

by the 2n

0

-1

..

..

0 -1 : '.

'.

..

i

-1

0

0

0

0

-1

-1

0

: : -1”

-1 0

0 0

:

-.

1

0 -

2n matrix

-

1 -1 1

x

-, . . 0

(41

:

: 11

0

0

-1

i

-

1

Here the inclined dotted lines represent -l’s, the vertical dotted lines represent 1‘s, and all positions not otherwise indicated are zero. One can easily reduce this matrix to diagonal form, and one finds that 2n - 1 of the diagonal positions contain 1 or - 1, whereas the last position contains n. Hence D is not split at t, and so, in particular, is not projective. Therefore dimg, = 4. If K is an arbitrary commutative ring, then a projective resolution for L,(K) in AbKgn is obtained by tensoring the one above for L,(Z) over Z with K. In particular, if K is the ring Z, consisting of all rational numbers whose denominators are products of prime factors of n, then we find that D @z K is projective in Ab K. On the other hand, if K = Z/(n),

RINGS WITH SEVERAL OBJECTS then D oz K is not projective, we obtain THEOREM 34.1.

133

but it is split. Thus using Corollary

32.3,

If 0l is an AB4 ZTL-category, then gl dim 6Ygn= 3 + gl dim Q?.

If Gf is an AB4 Z/(n)-category,

then

gl dim flgfi = 4 + gl dim a. Remark. To be sure, projective Z/(n)-modules are not always free. However, this is a case where we can dispense with this assumption. (See the remark following Lemma 32.1.) One can also eliminate the AB4 assumption, since the only coproducts which are involved here are finite. If OT is a K-category and rr is any small category, then Mn is a K-category. In particular, this is true when K = Z, or Z/(n). Therefore if we let g,” be the product of g, with itself m times, we can iterate theorem 34.1 to see that gl dim agF = 3m + gl dim a if M is an abelian Z,-category,

whereas

gl dim 6YgT= 4m + gl dim CPI if @ is an abelian Z/(n)-category. In other words, for each integer m > 0, there are categories rr such that gl dim GP - gl dim a varies by as much as m. We return to the case K = Z, and we complete the projective resolution for L,(Z). This can be done using the short exact sequence

0-

S,(Z)JL (6

S,(Z)) 0 St(Z) -L

D+

0.

p=1

The homomorphism all is given by the matrix (4), with an additional row

134

MITCHELL

consisting of 1 in the last column and 0 elsewhere. pt can then be computed to be the matrix [n - 1, -l,...,

-1,

The homomorphism

l)...) 1, -n],

(5)

where the first dotted line represents n - 3 positions containing - 1, and the second dotted line represents n - 2 positions containing 1. Consider now two integers n, n’ 3 2. If we tensor (over Z) the above projective resolution for L,(Z) in Abgm with that for L,(Z) in Abgn’, we obtain a projective resolution for Lt,,,,,(Z) in AbgmXQ by Corollary 11.9. The seventh and eighth terms at levels seven and eight look like the following:

where the vertical arrows in the seventh term are just coproduct injections, and Zi = Z for all i. The matrix of pt @ 1 + 1 @ /3;, is obtained by writing the matrix (5) for n’ beside that for n. A splitting y for fl @ 1 + 1 @ fl’, in order to be a morphism of AbgnXg*‘, must be such that Y(~,~‘) is 0 on all terms of the coproduct except Zt and Z,, . Then in order to have /3r = 1, we see that Y(~,~‘) must be given on Z1 and Zt, by integers X and h’ such that -nh - n’h’ = 1. In other words, ’ spiti s i f an d only if n and n’ are relatively prime. There~o~l+l@B dimg,

x g,, < 7 =8

if

(n, 72’) = 1

if

(n, n’) #

1.

That the inequality is actually an equality follows from Lemma 32.2, since 0 is (3, Z)-strong in g, . We now obtain a result for algebras by passing to the associated matrix algebra [Zg,]. This is a free Z-algebra on 28n + 22 generators. Since passage to the matrix algebra preserves tensor products and dimension (in the finite object case), we obtain: THEOREM

34.2.

For any n > 2 we have dim[Zg,]

= 4.

RINGS

WITH

SEVERAL

135

OBJECTS

Furthermore, dim[Zg,]

@ [Zg,,] = 7

if

(n, a’) = 1

z= 8

if

(n,n’)

# 1.

Finally we remark that the phenomenon which we have just seen in the third kernel of a projective resolution for L,(Z) could be reproduced in any kernel beyond the third by using the following trick. If n is a poset with an initial element 0, then we define Z;r by adding a new element 0’ to n which precedes every element except 0 and is incompatible with 0, and then adding an initial element - 1 to the result. Then in the exact sequence

0-D-t

S,(Z) @&,,(z)-

s-,(z)+L-l(z)-0

the object D is zero at - 1, 0, and 0’, and is the constant diagram at Z over the rest, hence is the same as the first kernel in a projective resolution for L,(Z). Therefore the (K + 1)-st kernel in a projective resolution for L-,(Z) can be taken to be the same as the k-th kernel in a projective resolution for Lo(Z) for all K > 0. Iterating the process m times, we see that the (k + m)-th kernel in a projective resolution for L-,(Z) over ES- can be taken to be the same as the k-th kernel in a projective resolution for L,(Z). The set JP2 is drawn below -m

-In+1

-m+l’

-m+2

62

-m+2’

-2

-2’

-1

-1’

0

0’ B

136

MITCHELL

This set was first brought to our attention by Fred Linton, who jectured correctly that its dimension is m + 1. The results of this section can now be elaborated on as follows:

dim (,Yyn) x (,XTi,) = 7 + m + m’ =8+m+m’ gl dim @%z

if

con-

(n, n’) = I if

(n,n’)

# 1,

= 3+m+gldimG?

if

G! is a Z,-category

= 4 + m + gl dim a

if

G! is a Z/(n)-category.

In closing this section, we remark that the n-gem provides an example of another phenomenon which does not otherwise appear to be easily come upon. Observe first that the category of groups is a reflective subcategory of the category Cat of all small categories (that is, the inclusion functor has a left adjoint). The group reflection of a small category n can be constructed explicitly as the free group on the morphisms of 7r divided by relations of the form xy = x 0 y where x and y are composable morphisms in rr. Of course, when n is given by generators and relations, the group reflection can be constructed more economically as the group with the same generators and relations. Now when rr is a poset, the group reflection tends often to be free. For example, if 7r is a directed set (in fact, a directed category) then its group reflection is free. More generally, if rr is a poset with the property that (in the terminology of the following section) every (unsuspended) crown which it contains has either an upper bound or a lower bound, then the group reflection of rr is free. The smallest poset which we know of which does not have a free group reflection is the 2-gem with the initial and final elements omitted. Here the group reflection can be shown to be the free product of a free group on 12 generators with a cyclic group of order 2. More generally, if one omits the initial and final elements from the n-gem, the group reflection is the free product of a free group on 4n + 4 generators with a cyclic group of order n. Some of the details of these facts involving group reflections have been carried out by E. Cooper.

RINGS

35. AN

UPPER

BOUND

WITH ON

SEVERAL THE

137

OBJECTS

DIMENSION

OF

A WEAK

POSET

Throughout this section, unless otherwise specified, 7r will denote a weak poset. In Section 17 we saw that if rr is a weak poset, then an upper bound on dim 7r is given by the sup of lengths of chains in 7~. However, this upper bound tends to be very crude. For example, if x is a linearly ordered set of n + 1 elements, then there is a chain of length n, whereas dim n < 1 since T is free. In this section, we shall obtain a much more refined upper bound S(x) which will turn out to be equal to dim r if either is less than three. On the other hand we shall exhibit a set n (Fig. 3 below) such that dim m = 3, whereas 6(n) = 4. Even without this example, we knew that dim rr cannot be equal to 6(r) in general, since one can prove (we shall not do it) that a(~-, x r2) = a(~,) + S(mJ, whereas the examples of the preceding section show that the corresponding equation does not hold for dim. This consideration indicates that it may be difficult to find a nonhomological description of dim. Note that such a description must satisfy dim rr = dim CT*, since this is a relation which holds for any small category r. We shall use the membership notation p E r to denote that p is an object of 7~, and subsets of 7~ will always be considered as full subcategories. In particular, if S is a subset of 7~, then 71- S is the full subcategory of T whose objects (elements) are not in S. As usual, we denote {v 1z, < 4) by rQ and {v / p < u < 4) by prTTq . If P~q has exactly two elements (or, equivalently, if the unique morphism from p to q has length one), then q is called a cover for p. We shall say that p is unicovered in r if it has precisely one cover in n. In particular, this means that p is not maximal. LEMMA 35.1. If p and q are distinct unicovered elementsof rr, then p is unicovered in r - (q}.

Proof. If q is the unique cover of p in 7~,then the covers of q will become the covers of p in 7~- {q}. Hence if there is only one such, then p is unicovered in n - {q}. If the unique cover of p in 7~is not q, then it will remain the unique cover of p in 7r - {q}. LEMMA 35.2. Let p, ,..., p, , p be distinct elements of rr such that p is unicovered in r, and such that for each i, 1 < i < k, pi is unicovered in r - (p, , p, ,..., p,-J. Then for each i, pi is unicovered in 7.r- {P, PI 3P, ,.“, Pi-l>.

138

MITCHELL

Proof. By Lemma 35.1 and induction on i, p is unicovered in rr - {PI 9P2 ,..., p,-r}. Since pi is also unicovered in this set, it follows again from Lemma 36.1 that pi is unicovered in r - {p, p, ,..., piwl}. Let rr be a muscle, or in other words a finite poset with an initial element s and a terminal element t. We define the measureof VT,denoted by ~(‘rr), inductively as follows. First we set p(n) = 0 if and only if n has only one element. Now assume n > 0, and that we have defined what we mean by p(n) < n - 1. We define an n-sequencein x to be a sequence p, ,..., p, such that for each i, 1 < i < k, we have (a)

pi is unicovered in 7~- {pi ,..., pipI}, and

(b) cL((n - {PI ,..apPi--ll)p,) < n - 1. We shall say that the n-sequence is successfulif p, = s. Then p(r) < n is defined to mean that there exists a successful n-sequence. By induction on n one verifies that if p(n) < n, then CL(~)< n + 1. This justifies defining p(n) as the least integer n such that ~(71.)< n. It is immediate from the definition that p(r) = 1 if and only if the initial element s is unicovered in 7~.Also it is easy to seeby induction on n that if the unique morphism from s to t has length n, then CL(~)< n. If q E 7, then we shall refer to ~(vTJ as the measureof q in r. Thus p(r) < n means that we can eliminate s by iterating the process of eliminating unicovered elements of measure < n - 1 in what is left. PROPOSITION

35.3.

If p is unicovered in rr and if p # s, t, then cL(n- {Ia G 44.

Proof. By induction on the number of elements of 7~.Letp, ,..., p, = s be a successful n-sequence in r, where p(n) = n. Then using Lemma 35.2 and the induction, we see that p, ,“.., p, is a successful n-sequence in r - {p} if p is not pi for any i, and otherwise p, ,..., p,-r , p,+r ,..., p, is a successful n-sequence in r - {p}. COROLLARY 35.4. If P(T) < n, then any n-sequence in 7~ can be extended to a successfulone.

If rr is any weak poset, we define SC4 = SUPK%)

IP

d 4b

Observe that even when r is a muscle, it is not necessarily true that S(T) = /L(T). A s remarked at the beginning of the section, one can show

RINGS

WITH

SEVERAL

OBJECTS

139

that 8(n1 x z-a) = 8(7~i) + a(~-,). H owever it is not always true that s(?T*) = S(T). c onsider, for example, the following set rr (ordered downwards):

FIGURE

Here S(r) = 4, whereas

S(r*)

3

= 3.

It is clear that 6(r) = 0 if and only if n is discrete. Furthermore, it is easy to see that 6(n) < I if and only if every muscle of r is totally ordered, or equivalently, if and only if r does not contain 2 x 2 as a full subcategory, Such a weak poset is sometimes called a tree. Alternatively, a tree is a poset which is free as a category. THEOREM

35.5.

For any weak poset n- we have dim v < 6(r).

Proof.

By Theorem

23.9 it suffices to show that h.d. k(z)

< CL(~)

(2)

for every muscle 7~, where L,(Z) is the object of A@ with 2 at the initial element s and zero elsewhere. We prove (2) by induction on the number of elements of r. If r has only one element, then (2) is clear. Now suppose that p(n) = n > 0. Then there is an element 9 which is unicovered < n. Let in 7~ such that ~(7rJ < n - 1. If n=’ = rr - {q}, then I

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F: Abn + AbS’ be the restriction functor. Let S,, : Ab + Abn be the left adjoint for the evaluation functor at p, and SJ1’: Ab + Abn’ the corresponding left adjoint for p E rr’. Then we have S,F = S,‘,

if

P f 9,

and S,F = S,‘, where r is the unique cover of 9. It follows that F preserves and since it is also exact, it preserves projective resolutions. O+D+P,-,-t

~~~-Pp,+P,+Ls(Z)-+O

projectives, Let (3)

be an exact sequence in Abn with Pi projective, 1 < i < n - 1. Since p(vq) < n - 1, we may assume by induction that D(p) = 0 for all p E 7rq. (This uses Corollary 23.3.) Since 7~- 7rq is a terminal subset of n, it follows from Lemma 22.2 that the homological dimension of D is the same as that of its restriction to 7r - nQ . Applying F to (3) and using the induction again, we find that F(D) is projective. Hence D is projective, and so the theorem is proved. Remark. To see that the inequality of the theorem can be strict we consider the set r of Fig. 2. Here we have 6(r) = 4, whereas dim T = dim 7~* < S(n*) = 3. Consider now the suspended n-crown (n 3 2)

FIGURE

4

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WITH

SEVERAL

OBJECTS

141

We shall denote this set by c, . In Fig. 4, I and J denote the two rows of n vertices. In Section 20, Example 2, we considered the unique minimal sets of generators and relations for c, , and we found that there was a nondegenerate closed path. Theorem 33.2, part (b) then shows that dim c, = 3. One can also verify this directly by writing down a projective resolution for L,(Z) in A&. If 57is any poset (not necessarily weak), we shall say that r contains a suspended crown. if it contains c, as a full subcategory for some n 2 2, with the further stipulation in case n = 2 that there be no element of z- which simultaneously follows both members of I and precedes both members of J. Note that if n > 2, then fullness rules out the possibility of there being an element of r which follows two among I and precedes two among J. LEMMA 35.6. If rr is a (not necessarily weak) poset which contains a suspended crown, then it contains it as a retract.

Proof. Since c, is a complete lattice suffices to consider n = 2 (Fig. 9, where

for n > 2, by Lemma

1.2 it

S

J 0

t FIGURE

5

by definition of containing a crown, there is no member of r which follows both members of I and precedes both members of J. We define Y: 7r + ca as follows: r(p) = s if p precedes both members

of J but does not follow a member of I = that member of I it follows if it follows one and precedes both members of J = that member of J it precedes if it precedes just one = t otherwise.

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MITCHELL

Then it is easily verified identity. THEOREM

35.7.

that Y is order preserving,

and that r 1 ca is the

Let 7~ be a weak poset. Then the following

are equiva-

lent: r contains a suspended crown, (4 (b) If GZ is an AB4 category of finite global gldim@33+gldim@, (c) dim n > 3, (4 s(n) Z 3, (e) r has no presentation Go/R with Q(R) trivial. Proof. The rem 33.2.

equivalence

of (c) and (e) was

dimension,

established

then

in Theo-

(a) 3 (b) By Lemma 35.6, 7r contains c, as a retract, and so it suffices to show that gl dim 6Z’n = 3 + gl dim 6Y. But this follows from Lemma 32.2 and Theorem 33.2 since dim c,, = 3. (b) + (c)

by inequality

(4’) of Corollary

(c) 3 (d)

by Theorem

35.5.

13.4’.

(d) + (a) We may assume that r is a muscle such that p.(r) 3 3. If we remove a maximal l-sequence in n, then since I 3 3, this cannot remove s. Thus we may assume that every element q of 7r such that 7~~is linearly ordered has at least two covers-i.e., 77is “well-covered” in the terminology of [38, Part II, Section 41. It follows now from [38, Part II, Lemma 4.51 that 7~ contains a suspended crown. Remark. The lemma of [38] w h ic h we have just used is, when taken with the page which precedes it, self-contained, and so we have chosen not to reproduce it here. It should be remarked that the definition of “crown” given in [38] is a little more general than the definition given here. However it is easy to see that r contains a suspended crown according to one definition if and only if it does according to the other. As in Section 32, let T(T) be the poset obtained from 7(n) by adding an initial element 0. In the last figure of Section 22 we have drawn T(c.J, and so 7(c2) is had by joining all the minimal elements to a common point 0. The diagram for T(CJ is similar, only wider. Using the method described in the last section, it is possible to write down a projective resolution for L,(Z) in A6rfCJ, and one finds that 0 is (4, Z)-strong.

RINGS WITH

SEVERAL OBJECTS

143

Hence, denoting n = c, , we have by Corollary 32.4, h.d.ne;rrAT = 3 + h.d., A for all A in an abelian category a. However, we don’t know if this is true for all weak posets (or deltas) of dimension 3. One might try to prove that if n is any weak poset with a vertex which is (n, Z)-strong, then 0 is (n + 1, Z)-strong in T(T), and conversely. If n is a weak delta, one can still define the notion of the measure of a muscle, and the discussion up to and including Theorem 35.5 is valid. In this case an object p is “unicovered” if, whenever m and n are morphisms of length one with domain p, an equation of the form mx = ny implies that x = y and m = n. However, there is no generalization of Theorem 35.7 in this case. Indeed, if rTTzis the graded polynomial category on two letters (Section 20, Example 4), then dim x2 = 2, whereas S(7r,) = co. 36. DIRECTED

FUNCTORS

In this section we shall imitate parts of Osofsky’s paper [44] to obtain her results on directed modules in the more general setting of a functor category. Among other things, this will enable us to show that the upper bound on the number of nonvanishing derived functors of the inverse limit functor obtained in Section 16 is the best possible. For easy comparison with Osofsky’s paper, we shall adhere to her notation and arguments as closely as possible. We shall be working in a functor category Abe, where %?is a small, additive category. The notational and terminological conventions introduced in Section 3 regarding a functor M E Abv will be in force. If x is an element of M, then the subfunctor of M generated by x is denoted by (x). Thus the value of (x) at an object Q is the subgroup of M(q) consisting of all elements of the form XY where dom r = x and cod Y = q. If M E Ab’, then M is called directed if it is generated by a set M’ of elements such that (i) (ii)

XT = 0, x E M’, YE V > Y = 0. The set M’, ordered by x < y if and only if x E (y), is directed.

The set M’ will be called a set offree generators for the directed functor M. Note that condition (i) is equivalent to the condition that the natural transformation V ,s, + (x) defined by 1 ,I, b x be an isomorphism for

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all x E M’. The inverse of this isomorphism will be denoted by x-l. Thus X-~(W) = r. Conditions (i) and (ii) imply that M is finitely generated if and only if it is representable. Example. The example of a directed functor which we have most in mind, and which will help in understanding the sequel, is the functor AR E AbRn where R is a ring and 7~* is a directed set. The set M’ of free generators can be taken to be the set of identity elements of R at the various p E 177 I. Throughout the remainder of this section, M will denote a directed functor with a set M’ of free generators. Note that while this puts a heavy restriction on the additive functor M, there is, on the other hand, no assumption on the additive category %?. We fix an upper bound function U: M’ x M’ + M’ (that is, U(X, y) >, x, y), and we say that a subset X C M’ is u-closed if u(X x X) C X. Clearly every subset X C M’ has a smallest u-closed set cl(X) containing it. Explicitly, set X,, = X, and define Xn+r = X, u u(X, x X%). Then cl(X) = lJz=s X, . If X is infinite, then X and cl(X) have the same cardinal number. If X is u-closed, then the subfunctor of M generated by X is directed. Let X be any subset of M’. For n > 0, let P,(X) be the free functor on all sequences (x0 , xi ,..., x,) with xi E X and x0 > xi 3 .a* > x,, , where the generator (x0 ,..., xn) is at / x, 1. That is, we define PnP)

=

Go , Xl ,..., %> @l?,l f

0 X$‘X

2&E1>‘..>Zo Also let P-l(X) denote the subfunctor morphisms in Abv 4: PnV)

of M generated

by X. We define

- Pn-dX)

by the rules n-1 4(x,

,...I

x,) = c (-l)i(xo

)...) ai )..., x,) + (-l)n(x,

,..., X,-i) X,‘,(Q

i=O

for n > 0, and d,,(xo) = x0 . Then it is a straightforward verify that d,d,-, = 0 for n > 0, or, in other words, complex. Now let x be any element of M’, and let

s(x) = {y EM’ 1y < x}.

calculation to that P(X) is a

RINGS

Then

WITH

SEVERAL

145

OBJECTS

define morphisms

x*: P&(x)) in Ab’

n > --I

- Pn+1(W,

by the rules x*(x0

Xn)

)...,

=

for n > 0, and for n = -1 by x*(xr) to check that E = x*M5))

(x,

X” ,...,

xn)

= (x)Y. Then it is straightforward

(1)

+ d,+l(~*(o)

for all [ E PJs(x)), n > 0. LEMMA

36.1.

Let X be a u-closed

subset of M’.

Then

P(X)

is acyclic.

Proof. In any case d,, is an epimorphism since X generates Ppl(X). Now let 5 be an element of P,(X), say

E =

i

(X(,i

)...)

x,i)

ri

.

i=l

Since X is u-closed, we can find x E X such that x 3 x,,%for 1 < i ,< t. and so if d,( 0 = 0, then by (1) we have Thus 4 E Pn(s(x)>, t = 4+,(x*(f))LEMMA

36.2.

If M’ has N, elements, n > - 1, then h.d. M < n + 1.

Proof. This follows from Theorem 16.1, since M is the directed union of the A,’ representable functors (x) where x ranges through M’. Our task now is to find a condition under which the inequality of the lemma is an equality. LEMMA

countably Proof.

verbatim.

36.3 (Kaplansky). generated

Any

projective

in Ab”

is a coproduct

of

functors.

Kaplansky’s

module theoretic

argument [28] carries over

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Now let t E P,(M’)

where

n 2 0, say

5 = f: {Xd,...,

x,f) Yi .

i=l

We shall say that x E M’ appears in f if x = xji for some i and j, 0 < j < n, 1 < i < t, ri # 0. If S is a subset of P,(M’), we define a(S) = {x ( x appearsin .$for some8 E S>. Note that if S is infinite, then since each member of S has only a finite number of members of M’ appearing in it, we have 1 a( S)l < 15’ (. LEMMA 36.4. Suppose that h.d. M < k where k > 0, and let n be a nonnegative integer such that no set of cardinal number < K, generates M. Then there is a u-closed subsetX C M’ such that:

(a) I X I = u,; (b)

No set of cardinality < N, generates P-,(X);

Cc) 4c(PdW

is a retract of d,(P,(M’)).

Proof. The situation is represented in the following diagram, where the vertical arrows are inclusions.

PkC M’) Since h.d. M < k, we know that dk(Pk(M’)) Kaplansky’s theorem, we can write

* P,-,(M’) is projective.

Hence by

RINGS

WITH

where the Qi are countably dh(Pk(M’)), set

SEVERAL

generated. N+ = @ &.I

147

OBJECTS

If N

is any subfunctor

of

Qi 3

where J is minimal such that N C oisJ Qi . Observe that if N is generated by an infinite set N’, then N+ is generated by a set of no greater cardinality since each element of N’ appears in only a finite number of the Qi and the Qi are countably generated. We now construct a u-closed subset X, C M’ for each ordinal (Y < N, . First take X,, to be the empty set. If OLis a limit ordinal, define

Note that 1X, / < K, providing ing that 1 X, j < N, , define

/ X, j < h,’ for all j? < LY. Now

assum-

where x is any element of M’ which is not in P-,(X,), and R[d,(P,(X,))]+ is a set of elements of P,(M’) in one-to-one correspondence with and mapping onto a set of generators of [d,(P,(X,))]+. Such an x can always be found since / X, / < xt,, whereas M is not generated by N, elements. Also, since ( X, / < N, , Pk(X,) is N,-generated, and so the same is true of d,(P,(X,)), hence of [d,(P,(X))]+. Hence R[dk(PI;(X))]+ may be assumed to have at most N, elements, and consequently 1X,,, j < N, . Now set x = Then (a) is satisfied.

Since P-&x)

is a strictly ascending it follows that P-i(X) Hence (b) is satisfied.

u x,. Z
=

tJ Pl(xJ Q
union of a chain of subfunctors of order type N,, , cannot be generated by fewer than ~~ elements. Finally, we have by construction

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MITCHELL

for all a. Consequently,

and so d,(P,(X))

= [&(P,(X))]+,

proving

(c).

LEMMA 36.5. Under the hypothesis of the preceding lemma, assume further that every subset of M’ of cardinality N, has an upper bound in M’. Then h.d. Pel(X) < k - I where X is the set of the conclusion of the lemma.

Proof.

First we have (2)

Now let z > x for all x E X. Such exists since 1X 1 = N, . In (2), we may subtract any element of the second summand from each free generator of the first summand and still have a coproduct. In particular, this gives

But using the relation

we see that

dk(Pk(X))c 4c[~*(~k--1ml c 4cPkW’)). Therefore d,(P,(X)) is a retract of dk[z*(Pk--l(X))], hence of Pkel(M’), hence of Pk-l(X). This proves the lemma. The following lemma is sometimes known as the dual basis lemma [ll, p. 1321. LEMMA

36.6.

Let P E AbV be any functor, and let {xi 1i 6 I} be a set

of generators for P. Then P is projective if and only if there is a family fi : P + Vl~iI such that for all x E P, fi(x) = 0 for all but a jinite number of i, and x = J& xifi(x)e

RINGS

Proof.

Consider

WITH

the natural

SEVERAL

OBJECTS

149

epimorphism

If P is projective, then we obtain f: P + & %?,szI such that f+ = 1, and we can take fi to be f composed with the i-th projection from the coproduct. Conversely, if the fi exist, then they determine a morphism f as above. For the purpose of the next lemma, we define a morphism Y in @ to be a zero divisor if its composition with some nonzero morphism yields zero. Thus in categorical language, Y is a zero divisor if it is not both an epimorphism and a monomorphism. LEMMA 36.7. Suppose that M is projective, and that whenever x < y in M’, the element r such that x = yr is not a zero divisor. Then M is representable.

Proof. Let f: M -+ VP be a nonzero morphism. Then f(x) # 0 for some x E M’, hence f(y) # 0 for all y > X. If x is any element of M’, then x < y for some y 3 x since M’ is directed. Write .s = yr, where Y is not a zero divisor by hypothesis. Then f(x) = f (y) r # 0 since f(y) # 0. Thus f is. nonzero on all elements of M’. It follows that if there is a family fi as in Lemma 36.6, then all but a finite number of them are zero. But then M is finitely generated, hence representable. Remark. Osofsky assumed simply that V have no nonzero zero divisors. However, our main application will require the slightly weaker hypothesis of the lemma. In the following theorem, x-i is to be interpreted as 1, and N, stands for K, where a: is any infinite ordinal. THEOREM 36.8. Let M be a directed functor with a totally ordered set M’ of free generators with the property that if x = yr, x, y E M’, then r is not a zero divisor. Suppose that the smallest cardinality of a generating set for M is N,, - 1 < n < CO. Then h.d. M = n + 1. Moreover, if n = - 1 or 0, M’ need not be totally ordered.

Proof. In any case we know h.d. M < n + 1 by Lemma 36.2. When n = - 1 there is nothing to prove. When n = 0, the result follows from Lemma 36.7. We proceed by induction on n. Suppose the result true for n > 0, and that M is not generated by N, elements. Then since M’ 607/8/l-10

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is totally ordered, all subsets of N, elements have upper bounds. Therefore if h.d. M < n + 1, then we can find X as in Lemma 36.5 such that the smallest generating set for P-i(X) has N, elements and h.d. P-i(X) < n. This contradicts the induction, and establishes the theorem for all finite n. The same argument can now be used in the case n = cg. COROLLARY 36.9. Let 7~be a totally ordered set, and let be any (nonzero) additive category (in particular, any nonzero ring). Then the highest nonvanishing derived functor of !&I,, : AbVF”+ Ab’ is the n + 1-st, where the minimum number of elementsin a cojinal subsetof rr* is N,, . Moreover, if n = 0 or - 1, then rr* need only be directed.

Proof. Let p be any nonzero object of V. Let M = A%“, E Ab”“. Then A%Tpis generated by the elements 1, at (p, i) where i runs through the vertices of V. These form a set of free generators which, as an ordered set, is isomorphic to n *. If i < j in n, then 1, at (p, i) is carried into 1, at (p, j) by the morphism 1, @ m in %n = $7 @ Zn, where m is the unique morphism in r from i to j. Clearly, 1, @ m is not a zero divisor in V’n. Hence by the theorem, h.d. AVp = n + 1, and so by Eq. (2*) of Section 16 there is a functor F E Abe’” such that (sm’“+“F)(p) 77

= Ext;;’

(dVD , F) # 0.

- 1 < n < co, then the corollary shows that Remark 1. Ifr=N$, the highest nonvanishing derived functor of lb is the n + I-st. (Here K, is considered as the well-ordered set of ordinals of cardinal number < N,, .) Note, however, that it, has a countable cofinal subset, and so l&ck) = 0 for k > 1 in this case. Moreover, it is not difficult to see that any totally ordered set of cardinal number K, has a (well-ordered) set of cardinal number N, cofinal in it for some n < w. Thus a totally ordered set must have cardinal number at least ~,+i in order to have l&ck) f 0 for all k. An example is provided by ~,+i itself. Remark 2. The assumption that M’ be totally ordered in the theorem is necessary. However, we shall show in a forthcoming paper that the corollary can be extended to general directed sets. COROLLARY 36.10. Let r be any delta, and suppose that 7~ or n* contains 8, + 1 as a subcategory for some n satisfying -1 < n < 00.

RINGS WITH

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151

Then dim,n- > n + 2 for any nonzero commutative ring K. Furthermore, if r or rr* contains N, as an ordered subset, then dim,rr = GO. Proof. The hypothesis for n = COimplies the hypothesis for each GO,and so it suffices to consider n finite. Also since dim z = dim n*, we may assume that n* contains N, + 1 as a subcategory. Since every nonlimit ordinal is a complete lattice, it suffices by Lemma 1.2 and Corollary 13.5 to assume that 7~* is N, + 1 itself. Let p denote the minimal element of n. Let M denote the subfunctor (right ideal) of Kr(p, ) such that M(q) = K for q > p and M(p) = 0. Then the smallest cardinality of a generating set for M is N,, , for otherwise N,, would contain a cofinal subset of cardinality < N, . Hence by the theorem, h.d. M = n $- 1, and so

n (

h.d. K+p,

)/M = n + 2.

In view of inequality (3’) of Corollary 13.4’, this gives the desired result. COROLLARY 36.11. Let rr be any delta, and supposethat dim,r = 1. Then n is generated by its morphisms of length one. Moreover, ;f rr is a poset, then z is free.

Proof. If x is any nonidentity morphism of n, then we can write x = my for some morphism m of length one, for otherwise r* would contain o + 1 as a subcategory, and so dim TX > 2 by the preceding corollary. Now suppose that we have written x = mims **. m,z, where each mi has length one. If z is not an identity, then we can write z = mktlz’ by the above, and so x = mr .** mkt.lz’. This process must terminate since otherwise rr would contain w + 1 as a subcategory. Now assumefurther that 7~is a poset, and letp < q. If there are a pair of incomparable elements in the muscle *xq , then n contains 2 x 2 as a full subcategory, hence as a retract since 2 x 2 is a complete lattice. Therefore, since dim, 2 x 2 = 2, we find that dim, n > 2, a contradiction. We have now that m is a poset all of whose muscles are totally ordered and finite. But this is equivalent to the poset being free as a category. Remark 1. A category can be generated by its morphisms of length one and still have morphisms of infinite length. Remark 2. The second statement of the corollary is probably true for all deltas, but we have not been able to prove it.

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It is a well-known theorem of Auslander [2] that the right global dimension of a category %’ is the sup of the homological dimensions of its cyclic functors. Thus if % is not semisimple, then r.gl dim% = 1 + sup{h.d. M 1M is a right ideal}.

(3)

Let us say that 5~3is a right valuation category if for each p E j 9 I, the subfunctors of VI”,, are totally ordered by inclusion. For example, if rr is a partially ordered set with the property that r,~ = (q 1Q > p> is totally ordered for all p E 7, and if K is a division ring, then KTTis a right valuation category. We also define a category %Tto be a division ringoid if all of its nonzero morphisms are isomorphisms. COROLLARY

36.12 (Osofsky [44]).

Let %?be a right valuation category.

If $? is semisimple,then it is a division ringoid. Otherwise, assumingfurther that %?has no zero divisors, we have r.gl dim %T= 2 + sup{n 1%?possesses a right ideal generatedby K, but no fewer elements).

Proof. If %?is semisimple, then since the subfunctors of Yfl are totally ordered for all p, it follows that YZ, is simple for all p. Therefore, 9? is a division ringoid. Otherwise, assuming further that V?has no zero divisors, the result follows from Eq. (3) and Theorem 36.8 since in this case every ideal will have a totally ordered set of free generators. COROLLARY 36.13. If 7~ is any well ordered set with at least two elementsand K is a division ring, then

r.gl dim Kx = 1. On the other hand if 71is N,, + 1, - 1 < n < GO,then l.gl dim KT = n + 2. Remark. such that

There exist examples due to Jategaonkar [25] of rings R r.gl dim R = 1,

l.gl dim R = n + 2.

RINGS WITH

37. DIRECT

LIMITS

153

SEVERAL OBJECTS

OF FUNCTORS

WITH

VARYING

DOMAIN

Let FE (XV and G E OZ9 where a, V’, and 9 are additive categories. If U: ‘S’+ 9 is an additive functor, then a natural transformation r): F-F UG will be called a morphism from F to G over U. We let Addfun OTdenote the category whose objects are pairs (V, F) (sometimes denoted more simply by F) w h ere ?? is a small additive category and FE Q!‘. A morphism from (%‘,F) to (9, G) is a pair (U, 7) where U: V -+ 9 is an additive functor and rl is a morphism from F to G over U. Composition is defined by

If Addcat denotes the category of all small additive categories and additive functors, then we have the functor Addfun OZ+ Addcat which assigns9 to (V, F) and U to (U, q). If U: 9?-+ B is an additive functor, and GE a9, then the identity transformation on UG can be considered as a morphism from UG to G over U. This morphism has the property that given an additive functor V: d + %?and a morphism h from H to G over VU, there is a unique morphism p from H to UG over V (in fact p = h) such that (V, Y)(U 1°C) = (W

4.

In other words Addfun GZhas the structure of a jibred category over the base category Addcat. The fibre over 9? is just the functor category a”. (The basic ideas concerning fibred categories can be found in Grothendieck [21]. However, we shall not be using any properties of fibred categories except one easily verifiable one below.) Henceforth, we shall assume that 0~’ is cocomplete and additive. In this case we know that if U: 9 -+ 9 is an additive functor, then the functor 00’: 6X” + 6YV has as its left adjoint the functor BV 9( U, ). If F E C!?‘, then the adjunction morphism

can be considered as a morphism from F to F & 9( U, ) over U. This morphism has the property that given an additive functor V: 9 -+ & and a morphism h from F to H over UV, there is a unique morphism p from F & 9( U, ) to H over V such that

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MITCHELL

In other words, Addfun has the further structure of a cofibred (and hence bifibred) category over Addcat. Let r be a small (nonadditive) category, and let U: 7~--+ Addcat be a functor. If i E 1n /, we denote U(i) by Vi (so as to remember that it is an additive category), and if m E +i, j) we denote U(m) by Urn (so as to remember that it is an additive functor). Thus Umn = UmUn whenever mn is defined in 7~, and lJ1i is the identity functor on W. A factor over U is a functor r + Addfun QZ whose composition with the projection to Addcat is U. Thus a functor over U consists of a functor Fi E Gl? for each i E 1n 1, and a natural transformation qm: Fi -+ U”Fj for each m E n(i, j), satisfying the rules mn = ,~wz(~, rl

yL),

$I = 1Fi .

In order to avoid introducing too many symbols, we shall denote the above functor 7~+ Addfun O! simply by {Fi, T”}. Let { Ui: Vi -+ 5%j i E 1~11) be the colimit of U in Addcat. Then in keeping with the general principles of bifibred categories with cocomplete fibres and cocomplete base, the colimit of the functor {Fi, 7”) in Addfun @ is given by

where the colimit on the right is taken in the fibre flV. Addfun @ can be constructed similarly when CY is complete.

Limits

in

If 0’ is complete, then dualizing the above and using the Remark. symbolic horn functor, one obtains a certain bifibred category over the dual of Addcat. Now consider a fixed functor U: n ---f Addcat. If (F”, qm] and (Gi, pm} are functors over U, then a natural transformation 4 of these functors will be called a natural transformation over U if @: Fi -+ Gi is a morphism over the identity on W’ for all i E 1n 1. In other words, a natural transformation over U is a family @: Fi + Gi where @ is an ordinary natural transformation for each i E j n /, such that for each m E ~(i, j) the diagram

RINGS

WITH

SEVERAL

155

OBJECTS

is commutative. The subcategory of (Addfun Up whose objects are functors over U and whose morphisms are natural transformations over U will be denoted by M, . If 6l! is abelian, then so is 67, . Remark. The category 67, is actually isomorphic to a functor category Wn, where U~T is the additive category described as follows. The objects of UX are ordered pairs (p, z’) with i E 1m 1 and p E 1‘S !. A morphism from (p, ;) to (4, j) is a formal sum

where

CL,,,= 0 for all but a finite number of m. Composition (%F)(B??)

= (UY%J

is defined by

&J(mfi).

When U is the constant functor at %?, Un is just the %?rr = V @ Zrr. However, we shall not adopt this approach.

category

The colimit of a functor U: rr -+ Addcat is rather complicated in general. However, we are mainly interested in the case where v is a directed set, and in this case we have the following explicit construction. First let us denote Urn by Uij where m is the unique morphism from i to j. The functors Uij: W + W induce a direct system Uij: 1Vi / -+ 1 $3 / in Ens, the category of sets, and we let {Cl’: / W 1+ 1 %? I} be the direct limit. Given two members of 12? 1, we choose an index i such that these members can be represented as Ui(p) and V(q), and we define @( Wp),

w7))

= 12 ~V’(P>,

U”‘(P))

in Ab (which setswise, of course, is the same as the direct limit For p, q, Y E / %F (, we let w WP), be the homomorphism homomorphisms %Tj(W(p),

wd)

0 w U”(s), WY)) -

of abelian groups W(q))

@ W( W(q),

in Ens).

@‘(U”(P), WY))

induced

on direct limits by the

W(Y)) 3 W( W( p), W(r))

for j > i. In this way %? becomes an additive category, and functors Ui: Vi + %?are built into the construction so as to give %?the structure of a direct limit for the W in Addcat. Now if {Fi, $j} is a functor over 72, we carry out a similar process to

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MITCHELL

construct a functor F E 02%. Given an object of 1%?1, we represent Ui(p) for some i, and we define F( U”(p)) = 2 Fj( U”Q)),

it as

(2)

the direct limit this time being in c!Z.If 01E ‘+(p, q), we let F(a): F( U”(P)) + F( Wd) be the morphism in 02 induced on direct limits by the morphisms Fq W(a)): P( W( p)) - Fj( W(q)). We leave it to the reader to check that F is a well-defined additive functor from % to OT,and that natural transformations

are present in the construction so as to give F the structure of a direct limit for {Fi, vii} in Addfun OL. If 12 is an AB5 category, then it is readily verified that

is exact. We shall now restrict ourselves to the case where GZ= Ab. We consider an auxiliary category Aux whose objects are pairs (V, X) where 9 is a small additive category and X E ,?&&I. A morphism from (%?,X) to (9, Y) is a pair (U, f) where U: %?-+ 9 is an additive functor and f is a family of functions f,: X(p) -+ Y( U(p)), p E / %?/. (In other words, f is a natural transformation from X to the composition of Y with the restriction of U to 1%?I.) We have the forgetful functor T: Addfun Ab -+ Aux which assigns to the object (%?,F) the object (V, X) where X(p) is th e underlying set of the abelian group F(p), and to the morphism (U, 7) the morphism (U, f) where f, is the underlying set map of the abelian group homomorphism qP . The functor T has as its left adjoint the functor S: Au - Addfun Ab defined on objects by

RINGS

WITH

SEVERAL

157

OBJECTS

where 1x 1 denotes the object p such that x E X(p). phism from (V, X) to (9, Y), then

If (U, f)

is a mor-

is the morphism over U which takes 1 1X,to 1 ,,(J, . Direct limits in Aux are formed similarly to the way that they are formed in Addfun Ab. Thus, if {Xi, f”j} is a functor over U: 7~+ Addcat, then the direct limit is the object (V, X) where (Ui: 99 -+ S’} is the direct limit of the ei, and X is given by

J?UYP)) = 5 WWPN 3

(3)

where the right side is the direct limit in Ens. Comparing (3) with (2) and recalling that direct limits of abelian groups are formed setwise as in Ens, we see that the functor T: Addfun Ab + Aux preserves direct limits. The functor 5’: Aux + Addfun Ab also preserves direct limits because it is a left adjoint. Thus if {P, #j} is a direct system in Addfun Ab over U: x + Addcat, then the adjunction epimorphisms

provide a natural transformation over U such that the induced formation on direct limits is the adjunction epimorphism

trans-

Taking kernels of the ci and repeating the process, we see that we can construct an acyclic left complex in Ab, which at each i yields a free resolution of Fi in Ab *, and such that the direct limit sequence is a f r ee resolution for F mVdb Q. We are now in a position to apply Berstein’s argument [6] to prove the following theorem. THEOREM 37.1. Let U: 7~+ Addcat, where r is a directed set of cardinal number x~, -1 < n < CO. If (9?, F) is the direct limit of objects (Ui, Fi) over U in Addfun Ab, then

~.cI.~F < n + 1 + sup l~d.~-gi Fi.

158

MITCHELL

Proof. Using the acyclic complex in Ab, constructed above, one immediately reduces to the case where Fi is projective in Abvi for each i. But then Fi & %‘(Ui, ) is projective in Ab’ for each i. By the alternative formula (1) for the direct limit of the Fi, the result now follows from Theorem 16.1. COROLLARY 37.2. Let %?= l&, V:’ in Addcat, where r is a directed set of cardinal number N, , - 1 < n < co. Then

r.gl dim %’ < n + 1 + sup r.gl dim V”. 1 Proof. Let FE Ab ‘. Using the explicit construction of direct limits in Addfun Ab, we see that (V, F) is the direct limit of the (Vi, U(F) in Addfun Ab. Hence the result follows from the theorem. 37.3. Let % = l&, W in the category of small K-categories, where rr is a directed set of cardinal number N, , - 1 < n < CO. Then dim, %?< n + 1 + sup dim, Vi. i COROLLARY

Proof. The direct limit in K-cat is the same as it is in Addcat. Furthermore, since 7~is directed, we have

The objects Vi E Ab (Qi)e form a direct system in Addfun Ab whose direct limit is %?.Hence the result follows again from the theorem. COROLLARY 37.4. Let 7~= b,& in the category of small categories, where I is a directed set of cardinal number ~~ , - 1 < n < CO. Then

dim, n < n + 1 + sup dim, ni. i Proof. The functor Cat + K-Cat which assigns KIT to r~ is direct limit preserving since it is a left adjoint. Hence this corollary is just a special case of the preceding one. We recall that an invariant 6(n) of a finite poset n was introduced in Section 35 such that dim r < S(n) (Theorem 35.5). If r is any pose& we can define S(n) to be the sup of the S(,‘) where V’ ranges through all

RINGS WITH

SEVERAL OBJECTS

159

full, finite subsets of 7r. In particular, 6(n) = 0 if and only if 7~is discrete, 6(r) < 1 if and only if 7~does not contain 2 x 2 as a full subcategory, and S(n) < 2 if and only if rr does not contain a suspended crown (Theorem 35.7). Since every poset is the direct limit of its full, finite subsets, we obtain from Corollary 37.4: COROLLARY

37.5.

If TI is a poset with u, elements, -1

< n < co,

then dim 71< n + 1 + 6(n). COROLLARY

37.6.

If rr is a totally ordered set with H, elements,then dimn < 71$2.

The inequality of Corollary 37.6 is the best possible in view of Corollary 36.10. In fact, these two corollaries combine to show that if 01is any ordinal number of cardinal number N, other than N, itself, then dim CY. = n + 2. For 01= N, , we can deduce only that the dimension is n + 1 or n + 2. If one could show in general that the dimension of a poset is the sup of the dimensions of its muscles (without the weakness assumption of Corollary 23.7), then it would follow that dim N, = n + 1. This is the case when n = 0, since N, is free. We have tried to show that if a totally ordered set has dimension < 2, then it has a presentation of the form G,/R where G, is a free category and Q(R) is trivial (cf. Corollary 28.4). We have been able to show that any countable totally ordered set has such a presentation. However, we don’t know if countability characterizes dimension < 2. For example, as remarked above, there is a chance that dim or = 2. We observe finally that the ordered set of rational numbers has dimension two by Corollary 36.10 and Corollary 37.6. On the other hand, we have not been able to pinpoint the dimension of the ordered set R of real numbers, even assuming the continuum hypothesis. For Corollary 37.6 then tells us that dim R < 3, whereas Corollary 36.10 tells us only that dim R > 2. More generally, if we assume that the cardinal number of R is it, (which by Cohen is consistent providing X, is not a countable limit of ordinals), then by Corollary 37.6 we have dim R < n + 2. We expect that equality holds. In fact, it may be true more generally that the dimension of a totally ordered set is n + 2 where N, is the sup of the cardinalities of closed intervals.

160

MITCHELL REFERENCES

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