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Pure-Injective Modules
CHAPTER PURE-INJECTIVE
V MODULES
A module is called pure-injective if it is a direct s u m m a n d of every module in which it is a pure submodule. A non-zero pure-injective module is not slender; in fact, any non-zero h o m o m o r p h i c image of a pure-injective module is not slender (see L e m m a 2.1 below). Thus any module which contains a non-zero homomorphic image of a pureinjective module, or contains a copy of R ~, is not slender. In C h a p t e r IX we shall prove a theorem of Nunke which is a converse of this observation for modules over a principal ideal domain. In preparation for t h a t we shall, in the first section of this chapter, investigate the structure of pure-injective modules over a p.i.d., and in the second section study the homomorphic images of pure-injectives, which are the cotorsion modules. Pure-injective modules over general rings have been studied in great d e p t h by algebraists and model-theorists (see the Notes at the end of the chapter), but we shall present here only what we need for the later chapters of this work.
Structure theory Recall t h a t the definition of a pure submodule of a module was given in IV.2.1. A h o m o m o r p h i s m p: A -~ B is called a pure embedding if it is an embedding of modules such t h a t ~[A] is a pure submodule of B. It is easy to see t h a t a module N is pure-injective if and only if it has the injective property with respect to pure embeddings, i.e., for every pure embedding c~: A --+ B and h o m o m o r p h i s m f : A ~ N, there is a h o m o m o r p h i s m g: B -+ N such t h a t g o c~ = f (cf. 1.2). Just as is the case for injectives, the class of pure-injectives is closed under direct products and direct summands. We shall make i m p o r t a n t use of another characterization of pureinjective modules.
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MODULES
1.1 D e f i n i t i o n . Let )~ be an infinite cardinal. A module N is called s compact if whenever S is a set of ~ linear equations over N (in any number, finite or infinite, of variables) such that every finite subset of S has a solution in N, then S has a solution in N. N is called algebraically compact (or, alternately, equationally compact) if it is )~+-algebraically compact for every infinite ~. 1.2 T h e o r e m . The following are equivalent, for any R-module N " (1) N is (IRI + No)+-algebraically compact; (2) N is pure-injective; (3) N is algebraically compact. PROOF. (1) =V (2)" Given a pure submodule A of B and a hom o m o r p h i s m f" A --+ N, we must show that f can be extended to g- B --+ N. Let us say that a map ~" J~ --+ N is a partial homom o r p h i s m if B is a submodule of B containing A and for every finite system of linear equations m
E
rijyj -- bi
(i - 1 , . . . , n )
j=l
where the bi belong t o / ~ , if the system has a solution in B then m
rijyj - ~(bi)
(i - 1, . . . , n)
j=l
has a solution in N. Notice that f is a partial h o m o m o r p h i s m because A is a pure submodule of B By Zorn's L e m m a there exists a m a x i m a l extension of f to a partial homomorphism ~" B --+ N. We will be done if we show that /~ - B. If not, let c C B \ / ~ , and aim to contradict the maximality of ~. Let T be the set of all finite systems of equations m
(T)
~
rijyj - vi + sic
(i - 1 , . . . , n )
j=l
(in unknowns yj, vi) such that for some choice of vi - bi in B
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V.1 S t r u c t u r e theory
m
(*)
y~rijYj
- bi + sic
(i - 1 , . . . , n )
j=l
has a solution in B. For each T in T fix ~(~-) -- ( b l , . . . , bn) such that (.) has a solution; also let {y~r). j _ 1 , . . . , m} be a new set of variable symbols. Now form a system of equations, S, with coefficients in N by putting into S for each ~- E T the equations m
(**)
E
riJy~ "r) - six = {](bi)
(i - 1 , . . . , n)
j=l
where r - ( b l , . . . , bn). Then S is finitely solvable in N because is a partial homomorphism (and because there is no "clash" of the (~) variables yj for different w). Hence S has a solution in N by the hypothesis on N. (Notice that ISI <_ IRI + R0.) If d is the "x-value" of a solution of S, then we extend ~ to h" B + R c -+ N by defining h(c) - d. (Notice that h is well-defined because if s c - b E / ? , then sx - it(b) belongs to S.) To see that we have a contradiction, it remains only to check that h is a partial homomorphism. Now if m
(* * *)
EriJyJ
-- b; + sic
(i - 1 , . . . , n )
j=l
has a solution in B (where b~,..., b~n E B), then for the corresponding ~- we have put the system (**) into S (where ~(~-) - ( b l , . . . ,bn)). Moreover, by subtracting (,) from (, 9 ,), we see that the system m -
-
(i -- 1 , . . . , n )
j=l
has a solution in B, and hence m
Z j=l
- O(b ) - O(b )
(i -- 1 , . . . , n )
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126
has a solution in N, because ~ is a partial homomorphism. From these facts, and the choice of d, one can readily deduce the existence of a solution of m
E
rijYj -- O(b~)+ sid
(i - 1 , . . . , n )
j=l
in N . (2) =~ (3)" Suppose that N is pure-injective and that $ is a system of equations over N which is finitely solvable in N. Let
M - (N @ F ) / K where F is the free module whose basis is the set of unknowns in the system S, and K is the submodule generated by the equations in S, m i.e., if ~-~.jm1 rjyj -- b belongs to S, then (-b, ~ j = l rjyj) belongs to K. Let t be the canonical embedding of N into M. It is easy to see that t is injective, and using the finite solvability of S in N, one can verify that t is a pure embedding. Let us identify N with tiN]. Since N is pure-injective, M - N | C for some C. By construction S has a solution in M; the projection of this solution onto N is then a solution in N. (3) =:v (1) is obvious. [:] 1.3 C o r o l l a r y . Every module is embeddable as a pure submodule of a pure-injective module. PROOF. Let ~ - (]R] + R0) +. Given a module M, we construct by induction a chain {My" v _~ ~} of modules such that M0 - M and for all v: (1) M~ is a pure submodule of M~+I; (2) if v is a limit, My - (.J~
It is also possible to prove 1.3 by using ultrapowers (cf. Exercise II.5). By a more careful construction, one can embed M as a pure submodule of a pure-injective module M, called a pure-injective
V.1 Structure theory
127
envelope of M, such that whenever N is a pure-injective module containing M as a pure submodule, then there is a pure embedding of M into N which is the identity on M. (See Warfield 1969.) Now we t u r n our attention to principal ideal domains, R; to begin with we even assume that R is local ~ that is, R is a discrete valuation ring, or d.v.r, with maximal ideal pR. Recall that a module is called reduced if it contains no non-zero divisible submodule. See 1.3.1 for the definition of the p-adic topology. 1.4 P r o p o s i t i o n . Let R be a d.v.r, and M a pure-injective Rmodule. Then (i) pWM is a divisible R-module; and (ii) if M is reduced, M is Hausdorff and complete in the p-adic topology. PROOF. (i) If b E pWM we need to show that for all n there exists Cn E pWM such that pncn - b. For n E w consider the system Sn of equations -
b} u
-
m
e
It is enough to show that Sn is solvable in M, for then the "x-value" of a solution will serve as cn. Since M is pure-injective, it suffices to show t h a t Sn is finitely-solvable. But any finite subset of Sn reduces to {p
x - b} u { p k v k --
for some k, and is solvable because b E pn+kM. (ii) Thus if M is reduced, M is Hausdorff. Now to see t h a t M is complete in the p-adic topology, consider a Cauchy sequence {an" n E w}. W i t h o u t loss of generality we may assume t h a t for all n > m, an - am E p m M . The system of equations
{x - an -- pnyn" n E w} is finitely solvable because for any k, x
{x-
an - p n y n ' n
-
ak
yields a solution of
~_ k}.
Hence it is solvable in M, and the "x-value" of a solution is the desired limit of the Cauchy sequence. D
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V. P U R E - I N J E C T I V E M O D U L E S
We shall say that a module M over a d.v.r. R is complete if it is Hausdorff and complete in the p-adic topology. We shall show that the reduced pure-injectives over a d.v.r, are precisely the complete modules, and we shall determine the structure of complete modules. Since the structure of divisible modules is known (cf. 1.2 for the structure of divisible groups), this will give us a complete structure theory for pure-injectives over a d.v.r. We begin with the structure of complete modules. One way of obtaining a complete module is to start with a direct sum of cyclic modules, B, and take its completion, /~ - l i m B / p k B with respect to the p-adic topology. T h e n B is an R-module whose topology is its p-adic topology; moreover B is pure in B, because its p-adic topology is induced by that on B, and B / B is divisible, because B is dense in B. (Compare 1.3.5.) This suggests the following definition. 1.5 D e f i n i t i o n . Let R be a d.v.r, and M an R-module. A submodule B of M is called a basic submodule of M if: (1) B is pure in M; (2) B is a direct sum of cyclic modules; and (3) M / B is divisible. In order to make our exposition as self-contained as possible, we shall give a sketch of the proof of the existence and uniqueness of basic submodules. (For more details, see for example Fuchs 1970 or K a p l a n s k y 1969.) 1.6 L e m m a . Let R be a d.v.r. If an R-module M is not divisible, then it contains a non-zero pure cyclic submodule. PROOF. Let T = {x E M : pnx = 0 for some n}, the torsion submodule of M. If T is divisible, then M - T | F for some torsion-free, non-divisible submodule F. If c E F such that c ~ pR, then c generates a pure cyclic submodule. If T is not divisible, then there is an element, b, of T of order p such that for some n, b C pnR \pn+IR. L e t c E R such that b - pnc. It is not hard to see t h a t (c) is pure in T, and hence in M. [::]
V.1 Structure theory
129
1.7 T h e o r e m . Let R be a d.v.r. If M is an R-module, M possesses
a basic submodule, which is unique up to isomorphism. PROOF. A subset Y of M is called pure-independent if (Y), the submodule generated by Y, is pure in M and equals ~ y e Y ( Y } , the (internal) direct sum of the cyclic submodules generated by the elements of Y; Y is called a basis of M if it is pure-independent and in addition M / ( Y ) is divisible. To prove the existence of a basic submodule of M we shall prove the stronger fact that every pure-independent subset of M (e.g., 0) is contained in a basis of M. In fact, we use Zorn's Lemma to extend the given pure-independent subset of M to a maximal one, Y. If we let B = {Y), it remains only to check that M / B is divisible. If not, then by 1.6, there exists c E M such that the cyclic submodule of M / B generated by c + B is pure in M / B . But then one may easily verify that Y U {c} is a pure-independent subset of M, contradicting the maximality of Y. This takes care of existence. For uniqueness we must show that if B - 0 n/p | n>l
is a basic submodule of M, then the cardinals an and/~ are uniquely determined by M. But, in fact,
an - dim p n - 1M ~9]/pn M ~9] the nth Ulm invariant of M, and /~ = dim M / (T + pM) where T is the torsion submodule of M. F-1 Now we can derive a structure theorem for complete modules. 1.8 C o r o l l a r y . Let R be a d.v.r, and M a complete R-module. Then M is the completion of a direct sum, B, of cyclic modules; and B is uniquely determined, up to isomorphism. PROOF. Let B be a basic submodule of M. Then property (1) of a basic submodule implies that the p-adic topology on B is induced by
130
V. P U R E - I N J E C T I V E MODULES
the p-adic topology on M. Property (3) implies that B is dense in M in the p-adic topology. Thus M is a completion of B. If M is the completion of another submodule C which is also a direct sum of cyclic modules, then C is a basic submodule of M see the remarks before 1.5 and hence isomorphic to B. [3 Finally, we can characterize the reduced pure-injective modules over a d.v.r, as the complete modules. 1.9 T h e o r e m . If R is a d.v.r, and M is a reduced R-module, then M is pure-injective if and only if M is complete. PROOF. The implication from left to right was proved above in 1.4. For the converse, suppose that M is complete. By 1.3, M is a pure submodule of a pure-injective module N. We can suppose that N is reduced, since pWN is the divisible part of N by 1.4, and p ~ N N M = {0} since M is pure in N and Hausdorff in the p-adic topology. Let Y be a basis of M, and B the basic submodule of M it generates. By the proof of 1.7, Y extends to a basis of N. Hence there is a basic s u b m o d u l e o f N of the form C = B | ~ for some B t. Since N is the completion of C and M is the completion of B, the projection of C onto B extends to a projection of N onto M; so M is a direct summand of N, and hence pure-injective. El Now we turn to an arbitrary p . i . d . R . We shall prove that every reduced pure-injective R-module is a product of pure-injective R(p)modules, where p ranges over the set, P, of primes of R. (Here R(p) is the localization of R at the prime ideal pR, i.e., R(p) is the subring of the quotient field, Q, of R consisting of all ab -1 E Q such that b~pR.) 1.10 L e m m a . Let M be an R(p)-module. Then M is pure-injective as R(p)-module if and only if it is pure-injective as R-module. PROOF. (:::~)" Suppose that M is pure-injective as R(p)-module. By 1.2 it's enough to prove that every system, $, of R-module equations which is finitely solvable in M has a solution in M. But every Rmodule equation is an R(p)-module equation, so this is true since M is algebraically compact as R(p)-module by 1.2.
V.1 S t r u c t u r e theory
131
(r Suppose t h a t M is pure-injective as R-module. Let M be a pure s u b m o d u l e of an R(p)-module N; then the inclusion of M in N is a pure embedding of R-modules as well, so M is a direct s u m m a n d of N (as R-modules and hence as R(p)-modules). [2] 1.11 T h e o r e m . Let R be a p.i.d, and M a reduced R-module. Then M is pure-injective if and only if M is isomorphic to a product [Ipep Mp, where each Mp is a pure-injective R(p)-module. PROOF. (r By 1.10, a pure-injective R(p)-module, Mp, is a pureinjective R-module. Since a p r o d u c t of pure-injectives is pure-injective, we are done. (=~)" First of all notice that, just as in the proof of 1.4, we can show t h a t f T { r M : r E R \ {0}} is a divisible submodule, a n d hence is zero. Now fix a prime p of R, and let
Mp = {u E M : rlu if gcd(r, p) = 1}. (Here, riu , read "r divides u", means t h a t u E r M . ) We claim t h a t
(t)
for every a E M there exists ap E Mp s. t. p n l a - ap for all n.
To show this, consider the following system,
Sa,p = {pnyn = a -
~a,p, of linear equations:
x: n E w} U {rzr = x: gcd(r, p) = 1}.
This system is finitely solvable in M because given n E w and r E R such t h a t gcd(r, p) = 1, there exist s, t E R such t h a t sr + tp n = 1; if we let x = sra, t h e n r(sa) = x and pn(ta) = a - x. Hence Sa,p is solvable in M; if we let ap be the "x-value" of a solution, we have the desired element. Notice t h a t for a given a and p, the element ap is unique, because M { r M : r E R \ {0}} = {0}. Now we define a function f : M --+ HpE7~ Mp by: f ( a ) = (ap)pEp, where ap is as in (t). It is easy to see t h a t f is a h o m o m o r p h i s m . Moreover, f is one-one because if f ( a ) = (ap)pE p = f(b), t h e n for all p and n,
pn divides a - b = (a - ap) - (b - ap),
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132
so a - b belongs to M { r M ' r E R \ {0}} - {0}. To see that f is surjective, let (ap)pe~, E YIpe~, Mp, and consider the system of equations
{pnynp - x - ap" p E ~, n E w}. This system is finitely solvable because for any set { P l , . . . ,Pm} of m ap~, then for all n E w, pnlx-ap~ for i -- 1 , . . . m. primes, if x -- ~-~i=1 Therefore the system has a simultaneous solution, and if a is the "xvalue" of such a solution, then f(a) = (ap)pE p. So we have proved that M is isomorphic to HpET' Mp. All that remains is to show that each Mp is a pure-injective R(p)-module. For MB to be an R(p)-module, we need unique divisibility by elements q of R \ pR; but divisibility holds because of the definition of Mp and because of the isomorphism f; uniqueness holds because {x E Mp: qx = 0} C_ M{rM: r E R \ {0}} = {0}. Finally, MB is pure-injective as R-module since it is a direct s u m m a n d of the pureinjective module M; hence by 1.10 it is pure-injective as R(p)-module. E] 1.12 S u m m a r y . Let us sum up what we know about the structure of pure-injective R-modules for the case R = Z. In this case, an abelian group A is pure-injective if and only if A is isomorphic to a group of the form 1-Ip~p Ap 9 D where for each prime p, Ap is the completion of a group of the form
0 Z(p)
9
Z(#p) (,)
new
where Z(p n) is the cyclic group of order pn (= the cyclic Z(p)-module Z(p)/pnZ(p)). We can also replace Z(p) by its completion, Jp (cf. L3.#), and say that Ap is the completion of
0
z(p )
9
new
Moreover, D is of the form
0 z(p~162 9 pEP
J(#')
V.1 Structure theory
133
(see L 2 for definitions). It follows from the proof of 1.7 and from the structure theory of divisible groups, that the cardinal numbers C~p, n, ~p, %, and 5 form a complete set of invariants for A.
1.13 T h e o r e m . If T is a torsion abelian group and C is any abelian group, then Hom(T, C) is a reduced pure-injective group. PROOF. Since T is the direct sum, T - ~ p E 7 ~ Tp of its p-primary parts, and Hom(T, C) ~- I-Ipep Hom(Tp, C), we can assume without loss of generality that T is a p-group for some prime p. Then Hom(T, C) is a Z(p)-module because for any f E Hom(T, C) and any m relatively prime to p, m fiT] = f[mT] = f[T] and the fact that f[T] is a p-group imply that there is a unique g E Hom(T, C) such that m g -- f . What we must show is that Hom(T, C) is a complete Z(p)-module. First of all, Hom(T, C) is Hausdorff in the p-adic topology: indeed, if f E pW Hom(T, C) and a E T, then pna = 0 for some n; now f = pug for some g E Hom(T, C), so f ( a ) = pUg(a) = g(pna) = O. Secondly, every Cauchy sequence in Hom(T, C) has a limit: indeed, let { f n : n E w} be a sequence in Hom(T, C) such that for all n, fn+l - fn = pngn for some gn; then define h: T -+ C by: h(a) - f l ( a ) + E ( f n + l
- fn)(a) - f l ( a ) + ~-~pngn(a);
n>l
n>l
for all a E T; this is a finite sum because a has finite order; then it is straightforward to check that h is the limit of the sequence. [-1 See Fuchs 1970 (section 46) for information about the cardinal invariants of Horn(T, C). 1.14 C o r o l l a r y . If D is a divisible abelian group and A is any abelian group, then Hom(A, D) is pure-injective. PROOF. Let T be the torsion part of A. The short exact sequence 0--+ T ~ A --+ A / T --+0
induces the Cartan-Eilenberg sequence
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134
0-+
H o m ( A / T , D) -~ Hom(A,D)--+ Hom(T, D) ~ E x t ( A / T , D) = O.
The last term is 0 because D is injective. We claim that H o m ( A / T , D) is divisible. If so, then the preceding exact sequence splits, and Hom(A, D) ~- H o m ( A / T , D ) @ Hom(T, D). Now Horn(T, D ) i s pure-injective by 1.13, so Hom(A, D) is pure-injective. Thus it remains to prove the claim. Let D ~ be a torsion-free divisible group containing A / T . We have the Cartan-Eilenberg sequence . . . - + Hom(D', D)--+ H o m ( A / T , D) --+ E x t ( D ' / ( A / T ) , D) - O. Moreover, Hom(D ~, D) is divisible since D ~ is torsion-free and divisible, so Horn(A/T, D) must be divisible. [:] Note that we have identified the divisible part, H o m ( A / T , D), of Hom(A, D) and the reduced part, Hom(T, D). 1.15 T h e o r e m . Let F be a filter on a set I such that there exist sets Xn E F such that ~new Xn - ~. Then for any abelian groups Ai(i E I), the reduced product 1-II A i / F is pure-injective. PROOF. (The definition of the reduced product I]I A i / F is given in II.3.1.) Let F and Xn be as in the statement of the theorem. Let A - 1-II A i / F . By 1.2 it suffices to show that A is Rl-algebraically compact. So let S - {Sn" n E w} be a system of equations over A which is finitely solvable in A. Thus each Sn is of the form m j=l
where m - re(n) may depend on n, rnj E Z, and an E ~ I Ai. Let Sn(i) denote the corresponding equation m -
j=l
in Ai. By hypothesis, for each k E w there is a solution in A of { S n ' n ~_ k}, so by definition of the reduced product and because
V.2 Cotorsion groups
135
F is closed under finite intersections, there exists a set Yk E F such that for all i E ]Irk, {Sn(i): n < k} has a solution in Ai. By replacing Yk with Y0 M . . . M Yk M Xk we can assume that ~kewYk -- ~ and Yn D_Yn+l. As well, we can assume Y0 = I. Now we are ready to define elements bj E I-II Ai (j E w) such t h a t yj - (bj)F is a solution to S in A. For each i, there is a unique k(i) so t h a t i E Yk(i) \ Yk(i)+l. Choose the bj(i) so t h a t they are a solution of the system {Sn(i)" n < k(i)}; this is possible since i E Yk(i). This will then give us the desired solution of S because for every n {i E I" the bj(i) solve Sn(i) in Ai} contains Yn+l and hence belongs to F. [:3 1.16 C o r o l l a r y . For any abelian group B, BW/B (W) is pure-injective. PROOF. In the theorem, let Ai = B, let F be the cofinite filter on I = w, and let Xn = w \ {n}. [:3
2
Cotorsion groups
T h r o u g h o u t this section we will be dealing with abelian groups, i.e., Z-modules, which we will refer to simply as groups. (Of course, our discussion extends routinely to a r b i t r a r y p.i.d.'s.) A group A is called cotorsion if E x t ( J , A) - 0 for all torsion-free groups J. T h a t is to say, A is cotorsion provided t h a t it is a direct s u m m a n d of every containing group B with the property t h a t B / A is torsion-free. Since B / A torsion-free implies t h a t A is pure in B, it follows t h a t every pure-injective group is cotorsion. Also, (see 2.2), a h o m o m o r p h i c image of a cotorsion group is cotorsion. Later we shall show t h a t the cotorsion groups are precisely the homomorphic images of pureinjective groups (see 2.6). This is of interest to us because of the following (which holds over a r b i t r a r y rings). 2.1 L e m m a . If M contains a non-zero homomorphic image of a pure-injective module, then M is not slender.
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V. P U R E - I N J E C T I V E M O D U L E S
PROOF. Suppose there is a homomorphism h: N is pure-injective and there exists a E N such that a homomorphism f" R (w) --+ N such that for all Since R (W) is pure in R W, f extends to g" R ~ ~ N. M shows that M is not slender. [21
-+ M such that N h(a) ~: O. There is n E w, f ( e n ) - a. Then h og. R W
2.2 L e m m a . For any short exact sequence 0 ~ A ~ B --+ C ~ O, if A and C are cotorsion then B is cotorsion. Conversely, if B is cotorsion then C is cotorsion. PROOF. Both assertions follow immediately from the exactness of the induced sequence E x t ( J , A ) --+ E x t ( J , B ) ~ Ext(J, C) --+ 0 for any (torsion-free) group J. [:3 Note that it is not true that every subgroup of a cotorsion group is cotorsion; in fact every group is a subgroup of a divisible, hence cotorsion, group. Here is an important way in which cotorsion groups arise: 2.3 P r o p o s i t i o n . For any groups A and B , Ext(A, B) is cotorsion. PROOF. Let D be a divisible group containing B. Then the short exact sequence 0--+ B ~ D ~ D / B ~ 0
induces the exact sequence Hom(A,D/B)-+
Ext(A, B) -~ Ext(A, D ) = 0.
The last term is 0 because D is injective. The first term is pureinjective (and hence cotorsion) by 1.14. Hence Ext(A, B) is the homomorphic image of a cotorsion group, and is thus cotorsion. V] E x a m p l e . Not every group which arises in this way is pure-injective. Let A be Q / Z and let B be a reduced p-group such that pWB ~ {0}. The short exact sequence 0-+ Z-+ Q-+ A-+ 0
V.2 Cotorsion groups
137
induces the exact sequence Hom(Q,B) Ext(Q, B)
--+ Hom(Z,B) -~ Ext(Z, B).
-+
Ext(A,B)
--+
The last term is 0 since Z is projective, and the first term is 0 since B is reduced. Thus B = Hom(Z, B) is isomorphic to a subgroup of Ext(A, B), and, in fact, B is the torsion part of Ext(A, B) because Ext(Q, B) is torsion-free. Therefore Ext(A, B) cannot be pureinjective since the p-torsion part, T, of a reduced pure-injective group satisfies p ' T = {0} (cf. 1.4 and 1.11). Here is a simple criterion for being cotorsion: 2.4 P r o p o s i t i o n . A is cotorsion if and only if Ext(Q, A) = 0. PROOF. Suppose that Ext(Q, A) = 0 and that J is a torsion-free group. Now J can be embedded in a divisible group, which is a direct sum of copies of Q. Thus by the Cartan-Eilenberg sequence we have an epimorphism Ext(Q (~) ,A) --+ E x t ( J , A) -+ 0 for some cardinal g. Since Ext(Q (~) , A) -~ Ext(Q,A) ~ -- O, conclude that Ext(J, A) = 0. V]
we
can
2.5 L e m m a . If A is reduced and cotorsion, then Ext(Q/Z, A) ~ A. PROOF. The short exact sequence
0-+ Z-+ Q + Q/z +0 induces the exact sequence Hom(Q, A) --+ Hom(Z, A) --+ Ext(Q/Z, A) ~ Ext(Q, A). The first group is 0 because A is reduced; the last is 0 because A is cotorsion. Therefore A ~ Hom(Z, A) -~ Ext(Q/Z, A). W1 2.6 T h e o r e m . If A is a cotorsion group, then A is the homomorphic image of a pure-injective group. Moreover, if A is torsion-free, then A is pure-injective.
V. P URE-INJECTIVE MODULES
138
PROOF. Let D be a divisible group containing A; if A is torsion-free choose D to be torsion-free. Then the short exact sequence
0--+ B ~ D --+ D / B --+0 induces the exact sequence
H o m ( A , D / B ) - ~ Ext(A, B) --+ Ext(A, D ) = 0. The last term is 0 since D is injective, and the next-to-last is isomorphic to A by 2.5. Hence A is a homomorphic image of Hom(Q/Z, D/A), which is pure-injective by 1.14. If A is torsion-free, then the first term is 0, so A is isomorphic to Horn(Q/Z, D/A). [:] 2.7 C o r o l l a r y . If A is a reduced torsion-free non-zero cotorsion group, then A contains a copy of Jp for some prime p. PROOF. This follows immediately from Theorem 2.6 and the structure theorem for pure-injectives (cf. 1.12). [:] We conclude with a brief discussion of a class of groups which arise, among other places, in the study of endomorphism rings of groups. (See section XIV.3.) 2.8 D e f i n i t i o n . A group A is called cotorsion-free if it does not contain any non-zero subgroups which are cotorsion. 2.9 T h e o r e m . For any group A the following are equivalent: (1) A is cotorsion-free; A
(2) Hom( , A)= O,
A
Z - 1-I,e, J,
(3)A does not contain a copy of Q, Z(p), or Jp for any prime p; (~)A does not contain any non-zero pure-injective subgroup; (5)A is reduced and torsion-free and does not contain a direct summand isomorphic to Jp for any prime p; (6)A is reduced and torsion-free and does not contain a subgroup isomorphic to Jp for any prime p.
V.2 Cotorsion groups
139 A
PROOF. (1) ~ (2) follows immediately from the facts that Z is pure-injective (cf. 1.9 and 1.11) and that a homomorphic image of a pure-injective group is cotorsion. (4) =v ( 3 ) i s easy, and (3) =, (4) follows from 2.7 because (3) clearly implies that A is reduced and torsion-free. (3) obviously implies (5), and (6) =~ (1) follows from 2.7. So it remains to prove (2) =v (3) and (5) =~ (6). Assume (2). If A contains a copy of Q, then there is a non-zero homomorphism from Z to A by the injectivity of Q. If A contains a copy of Z(p), we can obtain a non-zero hom omorphism of Z into A by composing the canonical surjection from Z onto Z / p Z with the projection of Z / p Z - which is a direct sum of copies of Z(p) onto one of its summands. Finally, A obviously cannot contain a copy of Jp. This proves (2) =v (3). Now assume (5), and suppose, to obtain a contradiction, that there is an embedding f : Jp --4 A. Since A is reduced, there is a maximal power, pk, of p which divides f(1). Define g" Jp -4 A by g(x) - p - k f (x) for all x e Jp. (Note that pk divides f (x) because there exists n e Z such that pkl(x -- n), so pk divides f ( x ) - n. f(1).) Clearly g is an embedding, so we will obtain a contradiction if we show that im(g) is pure in A, for then it will be a direct summand of A. But for any x E Jp there exists n E Z such that p divides ( x - n) in Jp. Since p divides ng(1) in A only if p divides n, we get that p divides g(x) in A only if p divides g(x) in im(g). V] h
As a consequence of the theorem and the results of Chapter III, we obtain immediately: 2.10 C o r o l l a r y . (i) If A has cardinality < 2 s~ then A is cotorsionfree if and only if A is reduced and torsion-free if and only if A is slender. (ii) If A is R i-free, then A is cotorsion-free. [3 By 2.1 and 2.6 we know that a necessary condition for A to be slender is that it be cotorsion-free. That this condition is not sufficient is seen by the example of Z ~, which is cotorsion-free. However, Nunke's theorem (IX.2.4) will tell us that Z W is the only additional obstacle to A being slender.
140
V. P U R E - I N J E C T I V E M O D U L E S
EXERCISES In Exercises 1-3, D is an wl-incomplete ultrafilter on a cardinal ~, M is a group, and M* is the ultrapower M ~ / D of M with respect to D. (See II.3.1; note that here M* does not mean the dual of M.) By Theorem 1.15, M* is Rl-algebraically compact, hence pure-injective. Since M* is pure-injective, it is determined by cardinal invariants Olp, n, ~p, ")'p, 5, aS in 1.12. 1. Prove" ( a ) ~ p - l i m n ~ c ~ d i m p n M * / p n + l M * ; (b)'~p - l i m n ~ dimpnM*[p]; (c)(~ - l i m n ~ rank (n!M*). [Hint" the inequalities " _< " hold in any pure-injective group; the opposite inequalities need that M* is an ultrapower with respect to an wl-incomplete ultrafilter: e.g., in (a), use that p n M * / p n + l M * ~( p n M / p n + I M ) ~ / D and that ~p --dim(M*/T+pM*), where T is the torsion subgroup of M*.] 2. Prove that for any group A, the following are equivalent: (1) A ~- A W / D for every ultrafilter D on w; (2) A ~- M ~ / D for some wl-incomplete ultrafilter D on some ~; (3) A is pure-injective, and if C~p, n,/~p, 7p and ~ are the invariants of A, as in 1.12, then l(a), (b), and (c) hold as well as: (d) if a = C~p, n, ~p, or 7p, then either a is finite, or a W = a; (e) either (f = 0 or 5W = ~. [Hint: For (2) ::v (3), use the fact that IIW/DI = IIW/DI W if I is infinite (see Exercise II.4).] 3. If A ~- I-Inew B n / D for any groups Bn and some non-principal ultrafilter D on w, then A ~- A w / u for every ultrafilter U on w. [Hint" Use Exercise II.14 and Exercise 2 above.] 4. The cardinal invariants of Z w / z (W) a r e : ~,p - O ; ~ - 2 ~~ (cf. 1.12 and 1.16).
~p, n -
0; /~p -- 2R~
In Exercises 5-7, let F be a filter on I. Let us say that F is weakly t~-complete if the intersection of < t~ elements of F is non-empty. (The definition of ~-complete is given in II.2.4.) 5. Generalize Theorem 1.15 as follows" Let F be a ~-complete filter which is not weakly + -complete. Then for any modules Mi (i C I),
V. Notes
141
the reduced product N = l-IieI Mi/F satisfies" any system of ~ linear equations over N is solvable in N whenever every subsystem of size < ~ is solvable in N.
6. (i) If F is a-complete and M is pure-injective, then N - M I / F satisfies: any system of a linear equations over N is solvable in N whenever every finite subsystem is solvable. (ii) Suppose F is weakly a+-complete and M is such t h a t there is a system of ~ linear equations over M which is not solvable in M but is such t h a t every subsystem of size < ~ is solvable. T h e n M I/F has the same property. 7. Let A be an abelian group. (i) If A is pure-injective, then for all filters F, A 1IF is pure-injective. (ii) If A is not pure-injective, then A 1IF is pure-injective if and only if F is not weakly wl-complete. [Hint: use 5 & 6.] 8. Suppose M is an R-module and a -- (IRI + b~0). Let I be the set of finite subsets of a and F the filter generated by {{X E I : a E X}" a < a}. Show t h a t the diagonal embedding 5M" M --+ M I/F is a pure embedding and every finitely solvable system of a equations whose p a r a m e t e r s are from 5M[M] is solvable in MI/F. (This can be used to give an alternate proof of 1.2 (2) ::~ (3).) 9. If A is cotorsion-free and ~: Z ~ --+ A, then for all x E A \ {0}, {n E w: ~(en) = x} is finite. [Hint: assume ~ and x form a counterexample. Let B - ~.x), and choose p so t h a t B is Hausdorff in the p-adic topology. Let B denote the p-adic closure of B in A. Choose E Zp SO t h a t ~x ~ B (and hence not in A). Show ~x E rge(p).]
NOTES The theory of pure-injective modules was developed by Kaplansky, Log, Maranda, and Fuchs, among others; see Fuchs 1970, Chapter VI, for more on this subject; see also Warfield 1969. The notion of a basic subgroup is due to Kulikov; see Fuchs 1970, Chapter VI. See Prest 1988 for an exposition of the important role of pure-injectivity in the model theory of modules. Our proofs of 1.2, 1.4 and 1.11 are inspired by model-theoretic approaches to the subject (particularly Eklof-Fisher 1972). Theorem 1.13 is due to Fuchs and
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V. P U R E - I N J E C T I V E M O D U L E S
Harrison. Theorem 1.15 is in Mycielski 1964 in more general form. (For modules over a general ring R, there are ultrafilters with respect to which arbitrary ultraproducts will be pure-injective; this is due to Keisler 1964b and Kunen 1972; see also Eklof 1977b (section 10).) Corollary 1.16 is due to Balcerzyk 1959. Cotorsion groups were discovered independently by Fuchs, Harrison and Nunke; for more on this subject see Fuchs 1970, Chapter IX. The notion of cotorsion-free comes from GSbel 1975, where it is called stout. The various parts of Theorem 2.9 are proved in GSbel 1975, GSbel-Wald 1979, and Dugas-GSbel 1982a. Exercises. 1-3:Eklof 1973; 4:Balcerzyk 1959; 5:Mycielski 1964; 7:Dugas-GSbel 1979a.
V MODULES
A module is called pure-injective if it is a direct s u m m a n d of every module in which it is a pure submodule. A non-zero pure-injective module is not slender; in fact, any non-zero h o m o m o r p h i c image of a pure-injective module is not slender (see L e m m a 2.1 below). Thus any module which contains a non-zero homomorphic image of a pureinjective module, or contains a copy of R ~, is not slender. In C h a p t e r IX we shall prove a theorem of Nunke which is a converse of this observation for modules over a principal ideal domain. In preparation for t h a t we shall, in the first section of this chapter, investigate the structure of pure-injective modules over a p.i.d., and in the second section study the homomorphic images of pure-injectives, which are the cotorsion modules. Pure-injective modules over general rings have been studied in great d e p t h by algebraists and model-theorists (see the Notes at the end of the chapter), but we shall present here only what we need for the later chapters of this work.
Structure theory Recall t h a t the definition of a pure submodule of a module was given in IV.2.1. A h o m o m o r p h i s m p: A -~ B is called a pure embedding if it is an embedding of modules such t h a t ~[A] is a pure submodule of B. It is easy to see t h a t a module N is pure-injective if and only if it has the injective property with respect to pure embeddings, i.e., for every pure embedding c~: A --+ B and h o m o m o r p h i s m f : A ~ N, there is a h o m o m o r p h i s m g: B -+ N such t h a t g o c~ = f (cf. 1.2). Just as is the case for injectives, the class of pure-injectives is closed under direct products and direct summands. We shall make i m p o r t a n t use of another characterization of pureinjective modules.
V. P U R E - I N J E C T I V E
124
MODULES
1.1 D e f i n i t i o n . Let )~ be an infinite cardinal. A module N is called s compact if whenever S is a set of ~ linear equations over N (in any number, finite or infinite, of variables) such that every finite subset of S has a solution in N, then S has a solution in N. N is called algebraically compact (or, alternately, equationally compact) if it is )~+-algebraically compact for every infinite ~. 1.2 T h e o r e m . The following are equivalent, for any R-module N " (1) N is (IRI + No)+-algebraically compact; (2) N is pure-injective; (3) N is algebraically compact. PROOF. (1) =V (2)" Given a pure submodule A of B and a hom o m o r p h i s m f" A --+ N, we must show that f can be extended to g- B --+ N. Let us say that a map ~" J~ --+ N is a partial homom o r p h i s m if B is a submodule of B containing A and for every finite system of linear equations m
E
rijyj -- bi
(i - 1 , . . . , n )
j=l
where the bi belong t o / ~ , if the system has a solution in B then m
rijyj - ~(bi)
(i - 1, . . . , n)
j=l
has a solution in N. Notice that f is a partial h o m o m o r p h i s m because A is a pure submodule of B By Zorn's L e m m a there exists a m a x i m a l extension of f to a partial homomorphism ~" B --+ N. We will be done if we show that /~ - B. If not, let c C B \ / ~ , and aim to contradict the maximality of ~. Let T be the set of all finite systems of equations m
(T)
~
rijyj - vi + sic
(i - 1 , . . . , n )
j=l
(in unknowns yj, vi) such that for some choice of vi - bi in B
125
V.1 S t r u c t u r e theory
m
(*)
y~rijYj
- bi + sic
(i - 1 , . . . , n )
j=l
has a solution in B. For each T in T fix ~(~-) -- ( b l , . . . , bn) such that (.) has a solution; also let {y~r). j _ 1 , . . . , m} be a new set of variable symbols. Now form a system of equations, S, with coefficients in N by putting into S for each ~- E T the equations m
(**)
E
riJy~ "r) - six = {](bi)
(i - 1 , . . . , n)
j=l
where r - ( b l , . . . , bn). Then S is finitely solvable in N because is a partial homomorphism (and because there is no "clash" of the (~) variables yj for different w). Hence S has a solution in N by the hypothesis on N. (Notice that ISI <_ IRI + R0.) If d is the "x-value" of a solution of S, then we extend ~ to h" B + R c -+ N by defining h(c) - d. (Notice that h is well-defined because if s c - b E / ? , then sx - it(b) belongs to S.) To see that we have a contradiction, it remains only to check that h is a partial homomorphism. Now if m
(* * *)
EriJyJ
-- b; + sic
(i - 1 , . . . , n )
j=l
has a solution in B (where b~,..., b~n E B), then for the corresponding ~- we have put the system (**) into S (where ~(~-) - ( b l , . . . ,bn)). Moreover, by subtracting (,) from (, 9 ,), we see that the system m -
-
(i -- 1 , . . . , n )
j=l
has a solution in B, and hence m
Z j=l
- O(b ) - O(b )
(i -- 1 , . . . , n )
V. P URE-INJECTIVE MODULES
126
has a solution in N, because ~ is a partial homomorphism. From these facts, and the choice of d, one can readily deduce the existence of a solution of m
E
rijYj -- O(b~)+ sid
(i - 1 , . . . , n )
j=l
in N . (2) =~ (3)" Suppose that N is pure-injective and that $ is a system of equations over N which is finitely solvable in N. Let
M - (N @ F ) / K where F is the free module whose basis is the set of unknowns in the system S, and K is the submodule generated by the equations in S, m i.e., if ~-~.jm1 rjyj -- b belongs to S, then (-b, ~ j = l rjyj) belongs to K. Let t be the canonical embedding of N into M. It is easy to see that t is injective, and using the finite solvability of S in N, one can verify that t is a pure embedding. Let us identify N with tiN]. Since N is pure-injective, M - N | C for some C. By construction S has a solution in M; the projection of this solution onto N is then a solution in N. (3) =:v (1) is obvious. [:] 1.3 C o r o l l a r y . Every module is embeddable as a pure submodule of a pure-injective module. PROOF. Let ~ - (]R] + R0) +. Given a module M, we construct by induction a chain {My" v _~ ~} of modules such that M0 - M and for all v: (1) M~ is a pure submodule of M~+I; (2) if v is a limit, My - (.J~
It is also possible to prove 1.3 by using ultrapowers (cf. Exercise II.5). By a more careful construction, one can embed M as a pure submodule of a pure-injective module M, called a pure-injective
V.1 Structure theory
127
envelope of M, such that whenever N is a pure-injective module containing M as a pure submodule, then there is a pure embedding of M into N which is the identity on M. (See Warfield 1969.) Now we t u r n our attention to principal ideal domains, R; to begin with we even assume that R is local ~ that is, R is a discrete valuation ring, or d.v.r, with maximal ideal pR. Recall that a module is called reduced if it contains no non-zero divisible submodule. See 1.3.1 for the definition of the p-adic topology. 1.4 P r o p o s i t i o n . Let R be a d.v.r, and M a pure-injective Rmodule. Then (i) pWM is a divisible R-module; and (ii) if M is reduced, M is Hausdorff and complete in the p-adic topology. PROOF. (i) If b E pWM we need to show that for all n there exists Cn E pWM such that pncn - b. For n E w consider the system Sn of equations -
b} u
-
m
e
It is enough to show that Sn is solvable in M, for then the "x-value" of a solution will serve as cn. Since M is pure-injective, it suffices to show t h a t Sn is finitely-solvable. But any finite subset of Sn reduces to {p
x - b} u { p k v k --
for some k, and is solvable because b E pn+kM. (ii) Thus if M is reduced, M is Hausdorff. Now to see t h a t M is complete in the p-adic topology, consider a Cauchy sequence {an" n E w}. W i t h o u t loss of generality we may assume t h a t for all n > m, an - am E p m M . The system of equations
{x - an -- pnyn" n E w} is finitely solvable because for any k, x
{x-
an - p n y n ' n
-
ak
yields a solution of
~_ k}.
Hence it is solvable in M, and the "x-value" of a solution is the desired limit of the Cauchy sequence. D
128
V. P U R E - I N J E C T I V E M O D U L E S
We shall say that a module M over a d.v.r. R is complete if it is Hausdorff and complete in the p-adic topology. We shall show that the reduced pure-injectives over a d.v.r, are precisely the complete modules, and we shall determine the structure of complete modules. Since the structure of divisible modules is known (cf. 1.2 for the structure of divisible groups), this will give us a complete structure theory for pure-injectives over a d.v.r. We begin with the structure of complete modules. One way of obtaining a complete module is to start with a direct sum of cyclic modules, B, and take its completion, /~ - l i m B / p k B with respect to the p-adic topology. T h e n B is an R-module whose topology is its p-adic topology; moreover B is pure in B, because its p-adic topology is induced by that on B, and B / B is divisible, because B is dense in B. (Compare 1.3.5.) This suggests the following definition. 1.5 D e f i n i t i o n . Let R be a d.v.r, and M an R-module. A submodule B of M is called a basic submodule of M if: (1) B is pure in M; (2) B is a direct sum of cyclic modules; and (3) M / B is divisible. In order to make our exposition as self-contained as possible, we shall give a sketch of the proof of the existence and uniqueness of basic submodules. (For more details, see for example Fuchs 1970 or K a p l a n s k y 1969.) 1.6 L e m m a . Let R be a d.v.r. If an R-module M is not divisible, then it contains a non-zero pure cyclic submodule. PROOF. Let T = {x E M : pnx = 0 for some n}, the torsion submodule of M. If T is divisible, then M - T | F for some torsion-free, non-divisible submodule F. If c E F such that c ~ pR, then c generates a pure cyclic submodule. If T is not divisible, then there is an element, b, of T of order p such that for some n, b C pnR \pn+IR. L e t c E R such that b - pnc. It is not hard to see t h a t (c) is pure in T, and hence in M. [::]
V.1 Structure theory
129
1.7 T h e o r e m . Let R be a d.v.r. If M is an R-module, M possesses
a basic submodule, which is unique up to isomorphism. PROOF. A subset Y of M is called pure-independent if (Y), the submodule generated by Y, is pure in M and equals ~ y e Y ( Y } , the (internal) direct sum of the cyclic submodules generated by the elements of Y; Y is called a basis of M if it is pure-independent and in addition M / ( Y ) is divisible. To prove the existence of a basic submodule of M we shall prove the stronger fact that every pure-independent subset of M (e.g., 0) is contained in a basis of M. In fact, we use Zorn's Lemma to extend the given pure-independent subset of M to a maximal one, Y. If we let B = {Y), it remains only to check that M / B is divisible. If not, then by 1.6, there exists c E M such that the cyclic submodule of M / B generated by c + B is pure in M / B . But then one may easily verify that Y U {c} is a pure-independent subset of M, contradicting the maximality of Y. This takes care of existence. For uniqueness we must show that if B - 0 n/p | n>l
is a basic submodule of M, then the cardinals an and/~ are uniquely determined by M. But, in fact,
an - dim p n - 1M ~9]/pn M ~9] the nth Ulm invariant of M, and /~ = dim M / (T + pM) where T is the torsion submodule of M. F-1 Now we can derive a structure theorem for complete modules. 1.8 C o r o l l a r y . Let R be a d.v.r, and M a complete R-module. Then M is the completion of a direct sum, B, of cyclic modules; and B is uniquely determined, up to isomorphism. PROOF. Let B be a basic submodule of M. Then property (1) of a basic submodule implies that the p-adic topology on B is induced by
130
V. P U R E - I N J E C T I V E MODULES
the p-adic topology on M. Property (3) implies that B is dense in M in the p-adic topology. Thus M is a completion of B. If M is the completion of another submodule C which is also a direct sum of cyclic modules, then C is a basic submodule of M see the remarks before 1.5 and hence isomorphic to B. [3 Finally, we can characterize the reduced pure-injective modules over a d.v.r, as the complete modules. 1.9 T h e o r e m . If R is a d.v.r, and M is a reduced R-module, then M is pure-injective if and only if M is complete. PROOF. The implication from left to right was proved above in 1.4. For the converse, suppose that M is complete. By 1.3, M is a pure submodule of a pure-injective module N. We can suppose that N is reduced, since pWN is the divisible part of N by 1.4, and p ~ N N M = {0} since M is pure in N and Hausdorff in the p-adic topology. Let Y be a basis of M, and B the basic submodule of M it generates. By the proof of 1.7, Y extends to a basis of N. Hence there is a basic s u b m o d u l e o f N of the form C = B | ~ for some B t. Since N is the completion of C and M is the completion of B, the projection of C onto B extends to a projection of N onto M; so M is a direct summand of N, and hence pure-injective. El Now we turn to an arbitrary p . i . d . R . We shall prove that every reduced pure-injective R-module is a product of pure-injective R(p)modules, where p ranges over the set, P, of primes of R. (Here R(p) is the localization of R at the prime ideal pR, i.e., R(p) is the subring of the quotient field, Q, of R consisting of all ab -1 E Q such that b~pR.) 1.10 L e m m a . Let M be an R(p)-module. Then M is pure-injective as R(p)-module if and only if it is pure-injective as R-module. PROOF. (:::~)" Suppose that M is pure-injective as R(p)-module. By 1.2 it's enough to prove that every system, $, of R-module equations which is finitely solvable in M has a solution in M. But every Rmodule equation is an R(p)-module equation, so this is true since M is algebraically compact as R(p)-module by 1.2.
V.1 S t r u c t u r e theory
131
(r Suppose t h a t M is pure-injective as R-module. Let M be a pure s u b m o d u l e of an R(p)-module N; then the inclusion of M in N is a pure embedding of R-modules as well, so M is a direct s u m m a n d of N (as R-modules and hence as R(p)-modules). [2] 1.11 T h e o r e m . Let R be a p.i.d, and M a reduced R-module. Then M is pure-injective if and only if M is isomorphic to a product [Ipep Mp, where each Mp is a pure-injective R(p)-module. PROOF. (r By 1.10, a pure-injective R(p)-module, Mp, is a pureinjective R-module. Since a p r o d u c t of pure-injectives is pure-injective, we are done. (=~)" First of all notice that, just as in the proof of 1.4, we can show t h a t f T { r M : r E R \ {0}} is a divisible submodule, a n d hence is zero. Now fix a prime p of R, and let
Mp = {u E M : rlu if gcd(r, p) = 1}. (Here, riu , read "r divides u", means t h a t u E r M . ) We claim t h a t
(t)
for every a E M there exists ap E Mp s. t. p n l a - ap for all n.
To show this, consider the following system,
Sa,p = {pnyn = a -
~a,p, of linear equations:
x: n E w} U {rzr = x: gcd(r, p) = 1}.
This system is finitely solvable in M because given n E w and r E R such t h a t gcd(r, p) = 1, there exist s, t E R such t h a t sr + tp n = 1; if we let x = sra, t h e n r(sa) = x and pn(ta) = a - x. Hence Sa,p is solvable in M; if we let ap be the "x-value" of a solution, we have the desired element. Notice t h a t for a given a and p, the element ap is unique, because M { r M : r E R \ {0}} = {0}. Now we define a function f : M --+ HpE7~ Mp by: f ( a ) = (ap)pEp, where ap is as in (t). It is easy to see t h a t f is a h o m o m o r p h i s m . Moreover, f is one-one because if f ( a ) = (ap)pE p = f(b), t h e n for all p and n,
pn divides a - b = (a - ap) - (b - ap),
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so a - b belongs to M { r M ' r E R \ {0}} - {0}. To see that f is surjective, let (ap)pe~, E YIpe~, Mp, and consider the system of equations
{pnynp - x - ap" p E ~, n E w}. This system is finitely solvable because for any set { P l , . . . ,Pm} of m ap~, then for all n E w, pnlx-ap~ for i -- 1 , . . . m. primes, if x -- ~-~i=1 Therefore the system has a simultaneous solution, and if a is the "xvalue" of such a solution, then f(a) = (ap)pE p. So we have proved that M is isomorphic to HpET' Mp. All that remains is to show that each Mp is a pure-injective R(p)-module. For MB to be an R(p)-module, we need unique divisibility by elements q of R \ pR; but divisibility holds because of the definition of Mp and because of the isomorphism f; uniqueness holds because {x E Mp: qx = 0} C_ M{rM: r E R \ {0}} = {0}. Finally, MB is pure-injective as R-module since it is a direct s u m m a n d of the pureinjective module M; hence by 1.10 it is pure-injective as R(p)-module. E] 1.12 S u m m a r y . Let us sum up what we know about the structure of pure-injective R-modules for the case R = Z. In this case, an abelian group A is pure-injective if and only if A is isomorphic to a group of the form 1-Ip~p Ap 9 D where for each prime p, Ap is the completion of a group of the form
0 Z(p)
9
Z(#p) (,)
new
where Z(p n) is the cyclic group of order pn (= the cyclic Z(p)-module Z(p)/pnZ(p)). We can also replace Z(p) by its completion, Jp (cf. L3.#), and say that Ap is the completion of
0
z(p )
9
new
Moreover, D is of the form
0 z(p~162 9 pEP
J(#')
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133
(see L 2 for definitions). It follows from the proof of 1.7 and from the structure theory of divisible groups, that the cardinal numbers C~p, n, ~p, %, and 5 form a complete set of invariants for A.
1.13 T h e o r e m . If T is a torsion abelian group and C is any abelian group, then Hom(T, C) is a reduced pure-injective group. PROOF. Since T is the direct sum, T - ~ p E 7 ~ Tp of its p-primary parts, and Hom(T, C) ~- I-Ipep Hom(Tp, C), we can assume without loss of generality that T is a p-group for some prime p. Then Hom(T, C) is a Z(p)-module because for any f E Hom(T, C) and any m relatively prime to p, m fiT] = f[mT] = f[T] and the fact that f[T] is a p-group imply that there is a unique g E Hom(T, C) such that m g -- f . What we must show is that Hom(T, C) is a complete Z(p)-module. First of all, Hom(T, C) is Hausdorff in the p-adic topology: indeed, if f E pW Hom(T, C) and a E T, then pna = 0 for some n; now f = pug for some g E Hom(T, C), so f ( a ) = pUg(a) = g(pna) = O. Secondly, every Cauchy sequence in Hom(T, C) has a limit: indeed, let { f n : n E w} be a sequence in Hom(T, C) such that for all n, fn+l - fn = pngn for some gn; then define h: T -+ C by: h(a) - f l ( a ) + E ( f n + l
- fn)(a) - f l ( a ) + ~-~pngn(a);
n>l
n>l
for all a E T; this is a finite sum because a has finite order; then it is straightforward to check that h is the limit of the sequence. [-1 See Fuchs 1970 (section 46) for information about the cardinal invariants of Horn(T, C). 1.14 C o r o l l a r y . If D is a divisible abelian group and A is any abelian group, then Hom(A, D) is pure-injective. PROOF. Let T be the torsion part of A. The short exact sequence 0--+ T ~ A --+ A / T --+0
induces the Cartan-Eilenberg sequence
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0-+
H o m ( A / T , D) -~ Hom(A,D)--+ Hom(T, D) ~ E x t ( A / T , D) = O.
The last term is 0 because D is injective. We claim that H o m ( A / T , D) is divisible. If so, then the preceding exact sequence splits, and Hom(A, D) ~- H o m ( A / T , D ) @ Hom(T, D). Now Horn(T, D ) i s pure-injective by 1.13, so Hom(A, D) is pure-injective. Thus it remains to prove the claim. Let D ~ be a torsion-free divisible group containing A / T . We have the Cartan-Eilenberg sequence . . . - + Hom(D', D)--+ H o m ( A / T , D) --+ E x t ( D ' / ( A / T ) , D) - O. Moreover, Hom(D ~, D) is divisible since D ~ is torsion-free and divisible, so Horn(A/T, D) must be divisible. [:] Note that we have identified the divisible part, H o m ( A / T , D), of Hom(A, D) and the reduced part, Hom(T, D). 1.15 T h e o r e m . Let F be a filter on a set I such that there exist sets Xn E F such that ~new Xn - ~. Then for any abelian groups Ai(i E I), the reduced product 1-II A i / F is pure-injective. PROOF. (The definition of the reduced product I]I A i / F is given in II.3.1.) Let F and Xn be as in the statement of the theorem. Let A - 1-II A i / F . By 1.2 it suffices to show that A is Rl-algebraically compact. So let S - {Sn" n E w} be a system of equations over A which is finitely solvable in A. Thus each Sn is of the form m j=l
where m - re(n) may depend on n, rnj E Z, and an E ~ I Ai. Let Sn(i) denote the corresponding equation m -
j=l
in Ai. By hypothesis, for each k E w there is a solution in A of { S n ' n ~_ k}, so by definition of the reduced product and because
V.2 Cotorsion groups
135
F is closed under finite intersections, there exists a set Yk E F such that for all i E ]Irk, {Sn(i): n < k} has a solution in Ai. By replacing Yk with Y0 M . . . M Yk M Xk we can assume that ~kewYk -- ~ and Yn D_Yn+l. As well, we can assume Y0 = I. Now we are ready to define elements bj E I-II Ai (j E w) such t h a t yj - (bj)F is a solution to S in A. For each i, there is a unique k(i) so t h a t i E Yk(i) \ Yk(i)+l. Choose the bj(i) so t h a t they are a solution of the system {Sn(i)" n < k(i)}; this is possible since i E Yk(i). This will then give us the desired solution of S because for every n {i E I" the bj(i) solve Sn(i) in Ai} contains Yn+l and hence belongs to F. [:3 1.16 C o r o l l a r y . For any abelian group B, BW/B (W) is pure-injective. PROOF. In the theorem, let Ai = B, let F be the cofinite filter on I = w, and let Xn = w \ {n}. [:3
2
Cotorsion groups
T h r o u g h o u t this section we will be dealing with abelian groups, i.e., Z-modules, which we will refer to simply as groups. (Of course, our discussion extends routinely to a r b i t r a r y p.i.d.'s.) A group A is called cotorsion if E x t ( J , A) - 0 for all torsion-free groups J. T h a t is to say, A is cotorsion provided t h a t it is a direct s u m m a n d of every containing group B with the property t h a t B / A is torsion-free. Since B / A torsion-free implies t h a t A is pure in B, it follows t h a t every pure-injective group is cotorsion. Also, (see 2.2), a h o m o m o r p h i c image of a cotorsion group is cotorsion. Later we shall show t h a t the cotorsion groups are precisely the homomorphic images of pureinjective groups (see 2.6). This is of interest to us because of the following (which holds over a r b i t r a r y rings). 2.1 L e m m a . If M contains a non-zero homomorphic image of a pure-injective module, then M is not slender.
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V. P U R E - I N J E C T I V E M O D U L E S
PROOF. Suppose there is a homomorphism h: N is pure-injective and there exists a E N such that a homomorphism f" R (w) --+ N such that for all Since R (W) is pure in R W, f extends to g" R ~ ~ N. M shows that M is not slender. [21
-+ M such that N h(a) ~: O. There is n E w, f ( e n ) - a. Then h og. R W
2.2 L e m m a . For any short exact sequence 0 ~ A ~ B --+ C ~ O, if A and C are cotorsion then B is cotorsion. Conversely, if B is cotorsion then C is cotorsion. PROOF. Both assertions follow immediately from the exactness of the induced sequence E x t ( J , A ) --+ E x t ( J , B ) ~ Ext(J, C) --+ 0 for any (torsion-free) group J. [:3 Note that it is not true that every subgroup of a cotorsion group is cotorsion; in fact every group is a subgroup of a divisible, hence cotorsion, group. Here is an important way in which cotorsion groups arise: 2.3 P r o p o s i t i o n . For any groups A and B , Ext(A, B) is cotorsion. PROOF. Let D be a divisible group containing B. Then the short exact sequence 0--+ B ~ D ~ D / B ~ 0
induces the exact sequence Hom(A,D/B)-+
Ext(A, B) -~ Ext(A, D ) = 0.
The last term is 0 because D is injective. The first term is pureinjective (and hence cotorsion) by 1.14. Hence Ext(A, B) is the homomorphic image of a cotorsion group, and is thus cotorsion. V] E x a m p l e . Not every group which arises in this way is pure-injective. Let A be Q / Z and let B be a reduced p-group such that pWB ~ {0}. The short exact sequence 0-+ Z-+ Q-+ A-+ 0
V.2 Cotorsion groups
137
induces the exact sequence Hom(Q,B) Ext(Q, B)
--+ Hom(Z,B) -~ Ext(Z, B).
-+
Ext(A,B)
--+
The last term is 0 since Z is projective, and the first term is 0 since B is reduced. Thus B = Hom(Z, B) is isomorphic to a subgroup of Ext(A, B), and, in fact, B is the torsion part of Ext(A, B) because Ext(Q, B) is torsion-free. Therefore Ext(A, B) cannot be pureinjective since the p-torsion part, T, of a reduced pure-injective group satisfies p ' T = {0} (cf. 1.4 and 1.11). Here is a simple criterion for being cotorsion: 2.4 P r o p o s i t i o n . A is cotorsion if and only if Ext(Q, A) = 0. PROOF. Suppose that Ext(Q, A) = 0 and that J is a torsion-free group. Now J can be embedded in a divisible group, which is a direct sum of copies of Q. Thus by the Cartan-Eilenberg sequence we have an epimorphism Ext(Q (~) ,A) --+ E x t ( J , A) -+ 0 for some cardinal g. Since Ext(Q (~) , A) -~ Ext(Q,A) ~ -- O, conclude that Ext(J, A) = 0. V]
we
can
2.5 L e m m a . If A is reduced and cotorsion, then Ext(Q/Z, A) ~ A. PROOF. The short exact sequence
0-+ Z-+ Q + Q/z +0 induces the exact sequence Hom(Q, A) --+ Hom(Z, A) --+ Ext(Q/Z, A) ~ Ext(Q, A). The first group is 0 because A is reduced; the last is 0 because A is cotorsion. Therefore A ~ Hom(Z, A) -~ Ext(Q/Z, A). W1 2.6 T h e o r e m . If A is a cotorsion group, then A is the homomorphic image of a pure-injective group. Moreover, if A is torsion-free, then A is pure-injective.
V. P URE-INJECTIVE MODULES
138
PROOF. Let D be a divisible group containing A; if A is torsion-free choose D to be torsion-free. Then the short exact sequence
0--+ B ~ D --+ D / B --+0 induces the exact sequence
H o m ( A , D / B ) - ~ Ext(A, B) --+ Ext(A, D ) = 0. The last term is 0 since D is injective, and the next-to-last is isomorphic to A by 2.5. Hence A is a homomorphic image of Hom(Q/Z, D/A), which is pure-injective by 1.14. If A is torsion-free, then the first term is 0, so A is isomorphic to Horn(Q/Z, D/A). [:] 2.7 C o r o l l a r y . If A is a reduced torsion-free non-zero cotorsion group, then A contains a copy of Jp for some prime p. PROOF. This follows immediately from Theorem 2.6 and the structure theorem for pure-injectives (cf. 1.12). [:] We conclude with a brief discussion of a class of groups which arise, among other places, in the study of endomorphism rings of groups. (See section XIV.3.) 2.8 D e f i n i t i o n . A group A is called cotorsion-free if it does not contain any non-zero subgroups which are cotorsion. 2.9 T h e o r e m . For any group A the following are equivalent: (1) A is cotorsion-free; A
(2) Hom( , A)= O,
A
Z - 1-I,e, J,
(3)A does not contain a copy of Q, Z(p), or Jp for any prime p; (~)A does not contain any non-zero pure-injective subgroup; (5)A is reduced and torsion-free and does not contain a direct summand isomorphic to Jp for any prime p; (6)A is reduced and torsion-free and does not contain a subgroup isomorphic to Jp for any prime p.
V.2 Cotorsion groups
139 A
PROOF. (1) ~ (2) follows immediately from the facts that Z is pure-injective (cf. 1.9 and 1.11) and that a homomorphic image of a pure-injective group is cotorsion. (4) =v ( 3 ) i s easy, and (3) =, (4) follows from 2.7 because (3) clearly implies that A is reduced and torsion-free. (3) obviously implies (5), and (6) =~ (1) follows from 2.7. So it remains to prove (2) =v (3) and (5) =~ (6). Assume (2). If A contains a copy of Q, then there is a non-zero homomorphism from Z to A by the injectivity of Q. If A contains a copy of Z(p), we can obtain a non-zero hom omorphism of Z into A by composing the canonical surjection from Z onto Z / p Z with the projection of Z / p Z - which is a direct sum of copies of Z(p) onto one of its summands. Finally, A obviously cannot contain a copy of Jp. This proves (2) =v (3). Now assume (5), and suppose, to obtain a contradiction, that there is an embedding f : Jp --4 A. Since A is reduced, there is a maximal power, pk, of p which divides f(1). Define g" Jp -4 A by g(x) - p - k f (x) for all x e Jp. (Note that pk divides f (x) because there exists n e Z such that pkl(x -- n), so pk divides f ( x ) - n. f(1).) Clearly g is an embedding, so we will obtain a contradiction if we show that im(g) is pure in A, for then it will be a direct summand of A. But for any x E Jp there exists n E Z such that p divides ( x - n) in Jp. Since p divides ng(1) in A only if p divides n, we get that p divides g(x) in A only if p divides g(x) in im(g). V] h
As a consequence of the theorem and the results of Chapter III, we obtain immediately: 2.10 C o r o l l a r y . (i) If A has cardinality < 2 s~ then A is cotorsionfree if and only if A is reduced and torsion-free if and only if A is slender. (ii) If A is R i-free, then A is cotorsion-free. [3 By 2.1 and 2.6 we know that a necessary condition for A to be slender is that it be cotorsion-free. That this condition is not sufficient is seen by the example of Z ~, which is cotorsion-free. However, Nunke's theorem (IX.2.4) will tell us that Z W is the only additional obstacle to A being slender.
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V. P U R E - I N J E C T I V E M O D U L E S
EXERCISES In Exercises 1-3, D is an wl-incomplete ultrafilter on a cardinal ~, M is a group, and M* is the ultrapower M ~ / D of M with respect to D. (See II.3.1; note that here M* does not mean the dual of M.) By Theorem 1.15, M* is Rl-algebraically compact, hence pure-injective. Since M* is pure-injective, it is determined by cardinal invariants Olp, n, ~p, ")'p, 5, aS in 1.12. 1. Prove" ( a ) ~ p - l i m n ~ c ~ d i m p n M * / p n + l M * ; (b)'~p - l i m n ~ dimpnM*[p]; (c)(~ - l i m n ~ rank (n!M*). [Hint" the inequalities " _< " hold in any pure-injective group; the opposite inequalities need that M* is an ultrapower with respect to an wl-incomplete ultrafilter: e.g., in (a), use that p n M * / p n + l M * ~( p n M / p n + I M ) ~ / D and that ~p --dim(M*/T+pM*), where T is the torsion subgroup of M*.] 2. Prove that for any group A, the following are equivalent: (1) A ~- A W / D for every ultrafilter D on w; (2) A ~- M ~ / D for some wl-incomplete ultrafilter D on some ~; (3) A is pure-injective, and if C~p, n,/~p, 7p and ~ are the invariants of A, as in 1.12, then l(a), (b), and (c) hold as well as: (d) if a = C~p, n, ~p, or 7p, then either a is finite, or a W = a; (e) either (f = 0 or 5W = ~. [Hint: For (2) ::v (3), use the fact that IIW/DI = IIW/DI W if I is infinite (see Exercise II.4).] 3. If A ~- I-Inew B n / D for any groups Bn and some non-principal ultrafilter D on w, then A ~- A w / u for every ultrafilter U on w. [Hint" Use Exercise II.14 and Exercise 2 above.] 4. The cardinal invariants of Z w / z (W) a r e : ~,p - O ; ~ - 2 ~~ (cf. 1.12 and 1.16).
~p, n -
0; /~p -- 2R~
In Exercises 5-7, let F be a filter on I. Let us say that F is weakly t~-complete if the intersection of < t~ elements of F is non-empty. (The definition of ~-complete is given in II.2.4.) 5. Generalize Theorem 1.15 as follows" Let F be a ~-complete filter which is not weakly + -complete. Then for any modules Mi (i C I),
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141
the reduced product N = l-IieI Mi/F satisfies" any system of ~ linear equations over N is solvable in N whenever every subsystem of size < ~ is solvable in N.
6. (i) If F is a-complete and M is pure-injective, then N - M I / F satisfies: any system of a linear equations over N is solvable in N whenever every finite subsystem is solvable. (ii) Suppose F is weakly a+-complete and M is such t h a t there is a system of ~ linear equations over M which is not solvable in M but is such t h a t every subsystem of size < ~ is solvable. T h e n M I/F has the same property. 7. Let A be an abelian group. (i) If A is pure-injective, then for all filters F, A 1IF is pure-injective. (ii) If A is not pure-injective, then A 1IF is pure-injective if and only if F is not weakly wl-complete. [Hint: use 5 & 6.] 8. Suppose M is an R-module and a -- (IRI + b~0). Let I be the set of finite subsets of a and F the filter generated by {{X E I : a E X}" a < a}. Show t h a t the diagonal embedding 5M" M --+ M I/F is a pure embedding and every finitely solvable system of a equations whose p a r a m e t e r s are from 5M[M] is solvable in MI/F. (This can be used to give an alternate proof of 1.2 (2) ::~ (3).) 9. If A is cotorsion-free and ~: Z ~ --+ A, then for all x E A \ {0}, {n E w: ~(en) = x} is finite. [Hint: assume ~ and x form a counterexample. Let B - ~.x), and choose p so t h a t B is Hausdorff in the p-adic topology. Let B denote the p-adic closure of B in A. Choose E Zp SO t h a t ~x ~ B (and hence not in A). Show ~x E rge(p).]
NOTES The theory of pure-injective modules was developed by Kaplansky, Log, Maranda, and Fuchs, among others; see Fuchs 1970, Chapter VI, for more on this subject; see also Warfield 1969. The notion of a basic subgroup is due to Kulikov; see Fuchs 1970, Chapter VI. See Prest 1988 for an exposition of the important role of pure-injectivity in the model theory of modules. Our proofs of 1.2, 1.4 and 1.11 are inspired by model-theoretic approaches to the subject (particularly Eklof-Fisher 1972). Theorem 1.13 is due to Fuchs and
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V. P U R E - I N J E C T I V E M O D U L E S
Harrison. Theorem 1.15 is in Mycielski 1964 in more general form. (For modules over a general ring R, there are ultrafilters with respect to which arbitrary ultraproducts will be pure-injective; this is due to Keisler 1964b and Kunen 1972; see also Eklof 1977b (section 10).) Corollary 1.16 is due to Balcerzyk 1959. Cotorsion groups were discovered independently by Fuchs, Harrison and Nunke; for more on this subject see Fuchs 1970, Chapter IX. The notion of cotorsion-free comes from GSbel 1975, where it is called stout. The various parts of Theorem 2.9 are proved in GSbel 1975, GSbel-Wald 1979, and Dugas-GSbel 1982a. Exercises. 1-3:Eklof 1973; 4:Balcerzyk 1959; 5:Mycielski 1964; 7:Dugas-GSbel 1979a.