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Stable equivalence of self-injective algebras
JOURNALOFALGEBRA40, 63-74 (1976)
Stable Equivalence of Self-lnjective
Algebras
IDUN REITEN Department
of &lathematics,
University
of Trondheim, 7000 Trondheim, ATorway
Communicated by J. A. Green
Received July 3, 1974
Let A be an artin algebra, i.e., an artin ring that is a finitely generated module over its center R, which is an artin ring, and let mod A denote the category of finitely generated (left) A-modules. We denote by mod A/P the (projectively) stable category associated with mod A, whose objects are the same as those of mod A, denoted by M, for M in mod A. The morphisms are given by Horn& g) = Hom,(M, N)/P(M, N), where P(M, ;V) denotes the subgroup of Hom,(M, N) consisting of the maps f: JI1\’ that factor through a projective A-module. A and A’ are said to be stably equivalent if mod A/P and mod A’/P are equivalent. Stable equivalence of artin algebras, or more generally, of dualizing R-varieties, was studied in [l-6]. We know that two artin algebras A and A’ can be very different with respect to certain algebraic properties, as homological dimension, being commutative or not, and still be stably equivalent. In particular, if the Loewy length of A is at most 2, i.e., r2 = 0, where r denotes the radical of A, then A is stably equivalent to an hereditary algebra A’, also of Loewy length at most 2. In this paper, we study stable equivalence of self-injective algebras. Our Main Theorem is that if A is stably equivalent to a self-injective algebra, such that each indecomposable direct factor algebra has Loewy length greater than 2, then A is also self-injective. Results of Green on stable equivalence of group algebras show that two stably equivalent self-injective algebras (with no semisimple direct factor) are not necessarily Morita equivalent. We have divided the paper in two sections. In Section 1, we recall some results from [2], which we use to get some necessary conditions on the artin algebras stably equivalent to self-injective algebras. In Section 2, we recall some results on modules over generalized triangular matrix rings from [SJ, and use these to show that the class of algebras we describe in Section 1 consists exactly of products of self-injective algebras
63 Copyright -411 rights
IT’ 1976 by Academic Press, Inc. oi reproduction in any form reserved.
64
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and Takayama (generalized uniserial) algebras with Loewy length 2. This will prove our Main Theorem stated above. IVe also show that each Nakayama algebra with Loewy length 2 is stabl! equivalent to a self-injective algebra, so that we get the following structure theorem for the artin algebras stably equivalent to self-injective algebras. An artin algebra fl is stably equivalent to a self-injective algebra if and only if each indecomposable direct factor of 11 is self-injective, or Sakayama with Loewy length 2. All our rings will be artin algebras, which are basic, i.e., fl,‘r, where r denotes the radical of the artin algebra (1, is a finite product of division algebras. This is no loss of generality, since every artin algebra is XIorita equivalent to a basic artin algebra. All our modules will be finitely generated left modules unless otherwise stated, and mod fl will denote the category of finitely generated A-modules. D: mod fl --, mod /lot) will denote the usual duality for artin algebras.
Let /l be an artin algebra, and let mod/l denote the category of (finitely generated) left /l-modules. We start by recalling some results from [2]. Let mod A!P denote the category mod fl modulo projectives, and mod A/I’? the analogously defined category mod /l modulo injectives, where the objects are denoted by M (see [2]). W e d enote by mod(mod /l) the full subcategory of the additive contravariant functors from mod fl to abelian groups, whose objects are the functors which are cokernels of maps between representable functors ( , :W), with ill in mod /l. Let mod(mod fl) denote the full subcategory of mod(mod /l) whose objects are the functors that vanish on of mod A/P into projective objects. Th ere is a natural embedding mod(mod A), given by JJ’J--F ( , M), where the functor ( , IU) is defined by ( , M)(X) -= Horn@, NJ). This embedding induces an equivalence of categories between mod /l/P and the full subcategory of projective objects of mod(mod fl). There is further a natural embedding of mod /l/E, which is equivalent to mod n/P, into mod(mod A), given by Ii7-t Ext’( , M), which induces an equivalence between mod A/E and the full subcategor! of injective objects of mod(mod fl). From the above, it follows that (1 and il’ are stably equivalent if and only if mod(mod fl) and mod(mod A’) arc equivalent. Hence, the following result will be useful for deciding which algebras could be stably equivalent to self-injective algebras. PROPOSITION 1.I. (a) Every nonsimple projective indecomposable Amodule is injective if and onl>l if all projective objects in mod(mod A) are injecti,zve.
SELF-INJECTIVE
ALGEBRAS
(b) Every nonsimple indecomposable injective A-module if and only ;f all injective objects in mod(mod A) are projective.
65 is projective
Proof. (a) Assume that each nonsimple indecomposable projective fl-module is injective, and let ( , C) be an indecomposable projective object in mod(modfl), where C is an indecomposable nonprojective finitely generated A-module. Let P denote the projective cover of C, so that P,lrP g C/I-C, where r denotes the radical of (1. P can have no simple summand, for we would then have a map from C onto a simple projective fl-module. By our assumption, P is then an injective cl-module. Consider the exact sequence 0 - K--f P - C - 0. The sequence is clearly minimal, i.e., it has no split exact summands, since P is the projective cover of C. Then, we know from [2], that 0 ---f ( , C) + Extr( , K) - Extl( , P) is a minimal injective copresentation for ( , c) in mod(mod /l). Since Extl( , P) = 0, ( , C’) is injective. Assume conversely that each projective object in mod(mod fl) is injective. Let P be an indecomposable nonsimple projective A-module, and consider the exact sequence 0 + sot P ---f P----f Pjsoc P - 0, where sot P denotes the socle of P. Since sot PC rP, P is a projective cover for Pkoc P, so that the exact sequence has no split exact summands. Then 0 + ( , P/sot P) + Est’( , K)+ Extl( , P) is a minimal injective copresentation for ( , P/sot P) in __ mod -(mod /I). Since, by assumption on mod(mod A) ( , P/socP) is injective, we then can conclude that Extl( , P) is zero, which implies that P is injective. Part (b) can be proved in an analogous way, or by using that there is a duality between mod(mod /l) and mod(mod (1”“) [2]. We have the following consequence of this proposition, which is useful for our present purposes. COROLLARY 1.2. If an artin algebra A is stably equivalent to a self-injective algebra, then each nonsimple indecomposable projective A-module is injective, and each nonsimple indecomposable injective A-module is projective.
In the next section, we shall study more closely this new class of algebras arising from Corollary 1.2, by first studying the indecomposable algebras in the class.
I\Te start this section by recalling from [8] what the projective modules over a generalized triangular matrix ring
and injective
66
IDUN
REITEN
look like, where in our case R and S are artin algebras and M is a finitel! generated left S-module and right R-module. We consider the (left) flmodules as triples (iz, B,f), or written A
where il is an R-module, B an S-module and f: M @ 4 + B an S-homomorphism. A map between two A-modules A’
-4 and lf
1 f’
B’
B is a pair of maps (OL,,L3), where 0~:i-1 -A’ /3: B + B’ an S-homomorphism, such that
is an R-homomorphism
1 f
B
commutes.
The indecomposable
and
i’
---LB+
B’
projective
A-modules
are
0
(1)
j. ’ P with P an indecomposable
projective
S-module,
and
P
with P an indecomposable injective /l-modules are
projective E
0
R-module.
The
indecomposable
SELF-INJECTIVE
with E an indecomposable
injective
ALGEBRAS
R-module
Hom(M,
67
and
E)
with E an indecomposable injective S-module. n: M @ Hom(M, E) --f E is the natural map given by n(m @f) = f(m). We go on to investigate the class of rings /I with the following property, which we shall here call property (01): fl is indecomposable, every nonsimple indecomposable injective /l-module is projective, every nonsimple indecomposable projective /l-module is injective, and there is at least one simple injective il-module. By our results from Section 1, we know that if (1 is an indecomposable algebra that is a direct factor of an algebra stably equivalent to a self-injective algebra, but not itself self-injective, then /l must have property (a). We shall need the following. PROPOSITION
2.1.
Let
be an indecomposable algebra with k a division algebra. Then, A has property (a) if and only
is an indecomposable projective A-module that is not simple, and is consequently injective. Hence , ,M is injective and the natural map k - Horn, (L1Z,M) is an isomorphism of left k-modules. It is then easy to see that this
68
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is also an isomorphism of rings, so that M must be an indecomposable S-module. We now want to show that every nonsimple indecomposable injective S-module is projective. Let first E be a nonsimple indecomposable injective S-module that is not isomorphic to M. Then, HomS(dl, I +n I:’
E)
(2)
is a nonsimple indecomposable injective A-module, hence, also, projective by our assumption that /l has property (a). If Hom,(M, E) # (0), Eq. (2) must be isomorphic to
k
since this is the only indecomposable
projective
.I-module
B with .4 # (0). But this would imply that E and M are isomorphic. we conclude that Hom,(M, E) = (0), so that
Hence,
0
E is a projective /l-module. It now follows that E is a projective S-module. We further need to show that A4 is a simple S-module. Assume to the contrary that M is not simple. Let r, =- sot M and let T2 be a simple submodule of M/rM. Then, Hom,(,Z;I, E(T,)) + (0), where E(T,) denotes the injective envelope of T2 If T2 is itself injective, then
SELF-INJECTIVE
ALGEBRAS
69
would be an indecomposable nonsimple injective /l-module, and hence, also, projective. Then, as before, we would have T2 g M, and we would be done. If T, is not injective, i.e., E(T,) is not simple, we would also have E( T,) E 118 since Hom,(M, E(T,)) # (0), hence, Tl z T3 . But then, we would have a noninvertible homomorphism M+ M,‘rM+ Tl + M, in contradiction to the fact that Hom,(M, M) and k are isomorphic rings. We can now conclude that M is in fact a simple S-module. In particular, S has a simple injective module. To show that S is an indecomposable algebra, assume to the contrary that S is a product of two nonzero algebras S, and S, . Since M is a simple S-module, the action from one of the two components, say S, , would be zero. But then, we would get a nontrivial decomposition of /I as the product of the algebras
k (d’ls,
0 s,
1
and
S,,
a contradiction. This finishes the proof of the first implication. Assume conversely that S has property (a), M is a simple injective Smodule, and that the natural ring map k + Hom,(M, M) is an isomorphism. Ke first want to show that every indecomposable nonsimple projective /l-module is injective.
k 1 id M
is injective, since M is an injective S-module and k + Hom,(M, M) is an nonsimple isomorphism of k-modules. Let then (1) b e an indecomposable projective A-module. By our assumption on S, P is an injective S-module. As Jl!? is simple injective, we must have Hom,(M, P) = (0), so that (1) is an injective A-module. An indecomposable nonsimple injective /I-module must be of the form Eq. 2, where E is an indecomposable injective S-module. If E is not isomorphic to ~12, then Hom,(M, E) = (0). Hence, E can not be a simple S-module, and therefore, is projective. But then,
70
IDUN REITEN
is a projective
A-module.
is isomorphic
to
and hence, is projective.
Further
As A has the simple injective
A-module
k
and is obviously indecomposable, since S is and Al1 # (0), this finishes the proof that A has property (a). I;sing our proposition we can deduce the following. COROLLARY
2.2.
If the artin algebra fl has property
(cu), then gl. dim.
A < cc. Proof. \Ve assume, up to Morita equivalence, that A/r is a finite product of division algebras, and prove the result by induction on the number of nonisomorphic simple A-modules. If ?z = 1, the only simple A-module must be injective, so gl. dim. A =:I 0. Assume n > 1 and assume that A has property (CX)and that the result has been proved for algebras A with property (CC)and less than n nonisomorphic simple A-modules. Let Ti be a simple injective A-module. Then, Tz = D(T,) is a simple projective flopmodule. Denote by P the direct sum of one copy of each of the other indecomposable projective /lop-modules. Then, A is isomorphic to End(T) T, P)
i Hom,,,(
0 End(P) ) ’
in where HomAoB( T, P) is a right End(T)- module and a left End(P)-module the natural way. By Proposition 2.1, End(P) has property (a), and further, End(P) has II -~ 1 simple A-modules. Ry the induction assumption, gl. dim. End(P) < W. Since gl. dim. End(T) z= 0, we conclude that
SELF-INJECTIVE
ALGEBRAS
71
gl. dim. A < gl. dim. End(P) + gl. dim. End(T) + 1 < co [7], and the corollary is proved. We are now in the position to prove the following structure theorem. We recall that an artin algebra is a Nakayama algebra (generalized uniserial) if and only if the indecomposable injective and projective A-modules have a unique composition series. THEOREM 2.3. Let A be an indecomposable artin algebra. Then, every nonsimple indecomposable injective A-module is projective, and every nonsimple indecomposable projective A-module is injective if and only if A is either selfinjective or Nakayama with Loewy length at most 2.
Proof. Let A be an indecomposable artin algebra such that each nonsimple indecomposable injective A-module is projective and each nonsimple indecomposable projective A-module is injective, and assume that A is not self-injective. Th en, there must be a simple injective A-module, so that A has property (a). By Corollary 2.2, we have gl. dim. A < a. From [2], we then know that gl. dim. mod(mod A) < co. Since by Proposition 1.1, every injective object in mod(mod A) is projective, we have that gl. dim. mod(mod A) is 0 or co. Consequently, gl. dim. mod(mod A) = 0, and we then know that A is Nakayama with Loewy length at most 2. [2]. Assume conversely that A is Nakayama with Loewy length at most 2, and let E be an indecomposable nonsimple injective A-module. Since A is Nakayama, E has a unique composition series, so that E/rE is simple. Hence, the projective cover PE of E is indecomposable. Consider the exact sequence 0 + K--f Pn -+ E-j. 0. Since PE is the projective cover for E, we have KC rPE . Since PE has a unique composition series, rPE must be simple. If K # (0), we must have K = rPE , which gives the contradiction that E is simple. Hence, K = (0), so that E is projective. Since AOP is also Nakayama with Loewy length at most 2, each nonsimple indecomposable injective Aon-module is projective. The proof is then finished by using the usual duality D: mod A + mod Aon. Since we have shown in [4] that being of Loewy length at most 2 is a property preserved by stable equivalence, our main result of this paper is now an immediate consequence of Theorem 2.3 and our results in Section 1. THEOREM 2.4. If A is an artin algebra that is stably equivalent to a selfinjective artin algebra such that each indecomposable direct factor of A is of Loewiy length greater than 2, then A is itself self-in...ective.
\Fe have the following COROLLARY
2.5.
Theprojective
and injective objects coincide in mod(modA)
72
IDUN
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if a& only if the indecomposable direct factors of A aye self-injective, orNakaq,ama with Loeq length at most 2. We end the paper by giving a more detailed description of the artin algebras with property (a). By the above, what we are then really doing is describing the indecomposable Nakayama algebras of Loewy length 2 that are not self-injective, and the simple algebras. We shall then apply this to give a structure theorem for the artin algebras stably equivalent to selfinjective algebras. If A has property (a) and only one simple module up to isomorphism, we have already pointed out that A is isomorphic to a division algebra k. If A has at least two simple A-modules, we have seen in the proof of Corollary 2.2 that we can write
where k is a division algebra, and by Proposition 2.1, ,M is a simple injective module with Hom,(M, M) s k, and S has property (a!) and has one simple module. S is then a division algebra, and from Hom,(M, M) g k, we conclude that S s k, and consequently,
If A has property
(a) and 3 simple modules, we then get
Since Hom(;M, All) z k and M is simple
which is our notation for the factor ring of
injective,
we can conclude
that
SELF-INJECTIVE
ALGEBRAS
73
by the twosided ideal
Continuing form
this process, we get that all algebras with property
(a) are of the
with a similar notational convention as above. We can use this last more precise investigation to get a characterization of the artin algebras stably equivalent to selfinjective algebras. THEOREM 2.6. An artin algebra A is stably equivalent to a self-injective algebra if and only if each indecomposable direct factor of A is self-injective, or Nakayama with Loezcy length at most 2.
Proof. In view of what we have already shown, we need only establish that each algebra
where n > 2 denotes the number of nonzero idempotents for the algebra, is stably equivalent to a self-injective algebra. Since A, is Nakayama with Loewy length 2, we know from [2] that mod A,/P is semisimple. This is also easy to see directly since the only indecomposable nonprojective A,modules are the n - 1 simple nonprojective A,-modules, so that mod A,/P is isomorphic to a product of n - 1 copies of the field k. Denote by R cc JI the trivial extension of a ring R by a twosided R-module M, i.e., the elements are pairs (r, m), where Y E R, m EM, and multiplication is defined by (Y, m) (Y’, m’) =- (w’, rm’ -C mr’). Let r, = (,k, x ... x ,k,) ot &k, f Bki + 4k3 + . nk,_I + Iklz), where the iki are simple rings isomorphic to k, ikj denotes the two-sided Ik, x ‘.. x ,k,-module, where the structure as a left module is given by (x1 ,..., xi ,..., x,) x =L xix for x,. E .k, , x E ikj , and the structure as a right module is given in an analogous way. Let Si denote the simple left
74
IDUN
HEITES
r,,l-module that is isomorphic to ikj as a left module. If Pi denotes the projective cover of Si , it is not hard to see that rPi E S,+l for 1 < i 5: IZ - 1, and rP, z S, . In particular, sot r, g S, + ... f S,,, It is then easy to see that r, is self-injective: The dimension of r, as a left vector space over ii is the same as the dimension of r, as a right vector space over k. Let E be the direct sum of one copy of each of the indecomposable injective left rllmodules. 1Ve have a natural embedding of r, in E, and since by the ordinary duality between left and right P,-modules, r, and E have the same dimension over k, we must have r, g E, so that I’, is self-injective. It is easy to see that mod r,,/P is equivalent to the product of n copies of k. Hence, r,n m1is stably equivalent to A,! This finishes the proof of the theorem.
ACKNOWLEDGMENT I would like to thank Professor Maurice ment.
Auslander
for his inspiration
and encourage-
REFERENCES 1.
2. 3.
4.
5. 6.
7. 8.
%I. ALXSDER AND I. REITEN, Stable equivalence of artin algebras, ilt “Proc. Conf. on Orders, Group Rings, and Related Topics,” Springer Lecture Notes 353, Springer-Verlag, New York, 1973. M. ALX~ANDER AND I. REITEN, Stable equivalence of dualizing R-varieties, Admwccs ill Math. 12 (I 974), 306-366. >I. AUSLA~DER AND I. REITEN, Stable equivalence of dualizing R-nu?eties II: Hereditary dualizing R-varieties, Advances in Math. 17 (1975), 93-l 2 I. WI. A131,.4NDER AND I. REITEN, Stable equivalence of dualizing R-rarietles 111: Dualizing R-varieties stably equivalent to hereditary, dualizing R-varieties, Adr. in Math. 17 (1975), 122-142. R/I. AUSLANDER AND I. REITEN, Stable equivalence of dualizing R-varieties I\.: Higher global dimension, Adu. in Math. 17 (1975), 343-166. M. AVSLANDER AND 1. REITEN, Stable equivalence of dualizing R-varieties 1’: Artin algebras stably equivalent to hereditary algebras, Adu. in Math. 17 (1975), 167-195. K. L. FIELDS, On the global dimensional of residue rings, Pacific 1. Math. 32 (I 971), 345-349. R. Fossrr~r, P. GRIFFITH, XND I. REITEN, ‘I’he homological algebra of trivial estensions of abelian categories with application to ring theory, Springer Lecture Notes, 456 (1975).
Stable Equivalence of Self-lnjective
Algebras
IDUN REITEN Department
of &lathematics,
University
of Trondheim, 7000 Trondheim, ATorway
Communicated by J. A. Green
Received July 3, 1974
Let A be an artin algebra, i.e., an artin ring that is a finitely generated module over its center R, which is an artin ring, and let mod A denote the category of finitely generated (left) A-modules. We denote by mod A/P the (projectively) stable category associated with mod A, whose objects are the same as those of mod A, denoted by M, for M in mod A. The morphisms are given by Horn& g) = Hom,(M, N)/P(M, N), where P(M, ;V) denotes the subgroup of Hom,(M, N) consisting of the maps f: JI1\’ that factor through a projective A-module. A and A’ are said to be stably equivalent if mod A/P and mod A’/P are equivalent. Stable equivalence of artin algebras, or more generally, of dualizing R-varieties, was studied in [l-6]. We know that two artin algebras A and A’ can be very different with respect to certain algebraic properties, as homological dimension, being commutative or not, and still be stably equivalent. In particular, if the Loewy length of A is at most 2, i.e., r2 = 0, where r denotes the radical of A, then A is stably equivalent to an hereditary algebra A’, also of Loewy length at most 2. In this paper, we study stable equivalence of self-injective algebras. Our Main Theorem is that if A is stably equivalent to a self-injective algebra, such that each indecomposable direct factor algebra has Loewy length greater than 2, then A is also self-injective. Results of Green on stable equivalence of group algebras show that two stably equivalent self-injective algebras (with no semisimple direct factor) are not necessarily Morita equivalent. We have divided the paper in two sections. In Section 1, we recall some results from [2], which we use to get some necessary conditions on the artin algebras stably equivalent to self-injective algebras. In Section 2, we recall some results on modules over generalized triangular matrix rings from [SJ, and use these to show that the class of algebras we describe in Section 1 consists exactly of products of self-injective algebras
63 Copyright -411 rights
IT’ 1976 by Academic Press, Inc. oi reproduction in any form reserved.
64
IDLW
KEITEN
and Takayama (generalized uniserial) algebras with Loewy length 2. This will prove our Main Theorem stated above. IVe also show that each Nakayama algebra with Loewy length 2 is stabl! equivalent to a self-injective algebra, so that we get the following structure theorem for the artin algebras stably equivalent to self-injective algebras. An artin algebra fl is stably equivalent to a self-injective algebra if and only if each indecomposable direct factor of 11 is self-injective, or Sakayama with Loewy length 2. All our rings will be artin algebras, which are basic, i.e., fl,‘r, where r denotes the radical of the artin algebra (1, is a finite product of division algebras. This is no loss of generality, since every artin algebra is XIorita equivalent to a basic artin algebra. All our modules will be finitely generated left modules unless otherwise stated, and mod fl will denote the category of finitely generated A-modules. D: mod fl --, mod /lot) will denote the usual duality for artin algebras.
Let /l be an artin algebra, and let mod/l denote the category of (finitely generated) left /l-modules. We start by recalling some results from [2]. Let mod A!P denote the category mod fl modulo projectives, and mod A/I’? the analogously defined category mod /l modulo injectives, where the objects are denoted by M (see [2]). W e d enote by mod(mod /l) the full subcategory of the additive contravariant functors from mod fl to abelian groups, whose objects are the functors which are cokernels of maps between representable functors ( , :W), with ill in mod /l. Let mod(mod fl) denote the full subcategory of mod(mod /l) whose objects are the functors that vanish on of mod A/P into projective objects. Th ere is a natural embedding mod(mod A), given by JJ’J--F ( , M), where the functor ( , IU) is defined by ( , M)(X) -= Horn@, NJ). This embedding induces an equivalence of categories between mod /l/P and the full subcategory of projective objects of mod(mod fl). There is further a natural embedding of mod /l/E, which is equivalent to mod n/P, into mod(mod A), given by Ii7-t Ext’( , M), which induces an equivalence between mod A/E and the full subcategor! of injective objects of mod(mod fl). From the above, it follows that (1 and il’ are stably equivalent if and only if mod(mod fl) and mod(mod A’) arc equivalent. Hence, the following result will be useful for deciding which algebras could be stably equivalent to self-injective algebras. PROPOSITION 1.I. (a) Every nonsimple projective indecomposable Amodule is injective if and onl>l if all projective objects in mod(mod A) are injecti,zve.
SELF-INJECTIVE
ALGEBRAS
(b) Every nonsimple indecomposable injective A-module if and only ;f all injective objects in mod(mod A) are projective.
65 is projective
Proof. (a) Assume that each nonsimple indecomposable projective fl-module is injective, and let ( , C) be an indecomposable projective object in mod(modfl), where C is an indecomposable nonprojective finitely generated A-module. Let P denote the projective cover of C, so that P,lrP g C/I-C, where r denotes the radical of (1. P can have no simple summand, for we would then have a map from C onto a simple projective fl-module. By our assumption, P is then an injective cl-module. Consider the exact sequence 0 - K--f P - C - 0. The sequence is clearly minimal, i.e., it has no split exact summands, since P is the projective cover of C. Then, we know from [2], that 0 ---f ( , C) + Extr( , K) - Extl( , P) is a minimal injective copresentation for ( , c) in mod(mod /l). Since Extl( , P) = 0, ( , C’) is injective. Assume conversely that each projective object in mod(mod fl) is injective. Let P be an indecomposable nonsimple projective A-module, and consider the exact sequence 0 + sot P ---f P----f Pjsoc P - 0, where sot P denotes the socle of P. Since sot PC rP, P is a projective cover for Pkoc P, so that the exact sequence has no split exact summands. Then 0 + ( , P/sot P) + Est’( , K)+ Extl( , P) is a minimal injective copresentation for ( , P/sot P) in __ mod -(mod /I). Since, by assumption on mod(mod A) ( , P/socP) is injective, we then can conclude that Extl( , P) is zero, which implies that P is injective. Part (b) can be proved in an analogous way, or by using that there is a duality between mod(mod /l) and mod(mod (1”“) [2]. We have the following consequence of this proposition, which is useful for our present purposes. COROLLARY 1.2. If an artin algebra A is stably equivalent to a self-injective algebra, then each nonsimple indecomposable projective A-module is injective, and each nonsimple indecomposable injective A-module is projective.
In the next section, we shall study more closely this new class of algebras arising from Corollary 1.2, by first studying the indecomposable algebras in the class.
I\Te start this section by recalling from [8] what the projective modules over a generalized triangular matrix ring
and injective
66
IDUN
REITEN
look like, where in our case R and S are artin algebras and M is a finitel! generated left S-module and right R-module. We consider the (left) flmodules as triples (iz, B,f), or written A
where il is an R-module, B an S-module and f: M @ 4 + B an S-homomorphism. A map between two A-modules A’
-4 and lf
1 f’
B’
B is a pair of maps (OL,,L3), where 0~:i-1 -A’ /3: B + B’ an S-homomorphism, such that
is an R-homomorphism
1 f
B
commutes.
The indecomposable
and
i’
---LB+
B’
projective
A-modules
are
0
(1)
j. ’ P with P an indecomposable
projective
S-module,
and
P
with P an indecomposable injective /l-modules are
projective E
0
R-module.
The
indecomposable
SELF-INJECTIVE
with E an indecomposable
injective
ALGEBRAS
R-module
Hom(M,
67
and
E)
with E an indecomposable injective S-module. n: M @ Hom(M, E) --f E is the natural map given by n(m @f) = f(m). We go on to investigate the class of rings /I with the following property, which we shall here call property (01): fl is indecomposable, every nonsimple indecomposable injective /l-module is projective, every nonsimple indecomposable projective /l-module is injective, and there is at least one simple injective il-module. By our results from Section 1, we know that if (1 is an indecomposable algebra that is a direct factor of an algebra stably equivalent to a self-injective algebra, but not itself self-injective, then /l must have property (a). We shall need the following. PROPOSITION
2.1.
Let
be an indecomposable algebra with k a division algebra. Then, A has property (a) if and only
is an indecomposable projective A-module that is not simple, and is consequently injective. Hence , ,M is injective and the natural map k - Horn, (L1Z,M) is an isomorphism of left k-modules. It is then easy to see that this
68
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is also an isomorphism of rings, so that M must be an indecomposable S-module. We now want to show that every nonsimple indecomposable injective S-module is projective. Let first E be a nonsimple indecomposable injective S-module that is not isomorphic to M. Then, HomS(dl, I +n I:’
E)
(2)
is a nonsimple indecomposable injective A-module, hence, also, projective by our assumption that /l has property (a). If Hom,(M, E) # (0), Eq. (2) must be isomorphic to
k
since this is the only indecomposable
projective
.I-module
B with .4 # (0). But this would imply that E and M are isomorphic. we conclude that Hom,(M, E) = (0), so that
Hence,
0
E is a projective /l-module. It now follows that E is a projective S-module. We further need to show that A4 is a simple S-module. Assume to the contrary that M is not simple. Let r, =- sot M and let T2 be a simple submodule of M/rM. Then, Hom,(,Z;I, E(T,)) + (0), where E(T,) denotes the injective envelope of T2 If T2 is itself injective, then
SELF-INJECTIVE
ALGEBRAS
69
would be an indecomposable nonsimple injective /l-module, and hence, also, projective. Then, as before, we would have T2 g M, and we would be done. If T, is not injective, i.e., E(T,) is not simple, we would also have E( T,) E 118 since Hom,(M, E(T,)) # (0), hence, Tl z T3 . But then, we would have a noninvertible homomorphism M+ M,‘rM+ Tl + M, in contradiction to the fact that Hom,(M, M) and k are isomorphic rings. We can now conclude that M is in fact a simple S-module. In particular, S has a simple injective module. To show that S is an indecomposable algebra, assume to the contrary that S is a product of two nonzero algebras S, and S, . Since M is a simple S-module, the action from one of the two components, say S, , would be zero. But then, we would get a nontrivial decomposition of /I as the product of the algebras
k (d’ls,
0 s,
1
and
S,,
a contradiction. This finishes the proof of the first implication. Assume conversely that S has property (a), M is a simple injective Smodule, and that the natural ring map k + Hom,(M, M) is an isomorphism. Ke first want to show that every indecomposable nonsimple projective /l-module is injective.
k 1 id M
is injective, since M is an injective S-module and k + Hom,(M, M) is an nonsimple isomorphism of k-modules. Let then (1) b e an indecomposable projective A-module. By our assumption on S, P is an injective S-module. As Jl!? is simple injective, we must have Hom,(M, P) = (0), so that (1) is an injective A-module. An indecomposable nonsimple injective /I-module must be of the form Eq. 2, where E is an indecomposable injective S-module. If E is not isomorphic to ~12, then Hom,(M, E) = (0). Hence, E can not be a simple S-module, and therefore, is projective. But then,
70
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is a projective
A-module.
is isomorphic
to
and hence, is projective.
Further
As A has the simple injective
A-module
k
and is obviously indecomposable, since S is and Al1 # (0), this finishes the proof that A has property (a). I;sing our proposition we can deduce the following. COROLLARY
2.2.
If the artin algebra fl has property
(cu), then gl. dim.
A < cc. Proof. \Ve assume, up to Morita equivalence, that A/r is a finite product of division algebras, and prove the result by induction on the number of nonisomorphic simple A-modules. If ?z = 1, the only simple A-module must be injective, so gl. dim. A =:I 0. Assume n > 1 and assume that A has property (CX)and that the result has been proved for algebras A with property (CC)and less than n nonisomorphic simple A-modules. Let Ti be a simple injective A-module. Then, Tz = D(T,) is a simple projective flopmodule. Denote by P the direct sum of one copy of each of the other indecomposable projective /lop-modules. Then, A is isomorphic to End(T) T, P)
i Hom,,,(
0 End(P) ) ’
in where HomAoB( T, P) is a right End(T)- module and a left End(P)-module the natural way. By Proposition 2.1, End(P) has property (a), and further, End(P) has II -~ 1 simple A-modules. Ry the induction assumption, gl. dim. End(P) < W. Since gl. dim. End(T) z= 0, we conclude that
SELF-INJECTIVE
ALGEBRAS
71
gl. dim. A < gl. dim. End(P) + gl. dim. End(T) + 1 < co [7], and the corollary is proved. We are now in the position to prove the following structure theorem. We recall that an artin algebra is a Nakayama algebra (generalized uniserial) if and only if the indecomposable injective and projective A-modules have a unique composition series. THEOREM 2.3. Let A be an indecomposable artin algebra. Then, every nonsimple indecomposable injective A-module is projective, and every nonsimple indecomposable projective A-module is injective if and only if A is either selfinjective or Nakayama with Loewy length at most 2.
Proof. Let A be an indecomposable artin algebra such that each nonsimple indecomposable injective A-module is projective and each nonsimple indecomposable projective A-module is injective, and assume that A is not self-injective. Th en, there must be a simple injective A-module, so that A has property (a). By Corollary 2.2, we have gl. dim. A < a. From [2], we then know that gl. dim. mod(mod A) < co. Since by Proposition 1.1, every injective object in mod(mod A) is projective, we have that gl. dim. mod(mod A) is 0 or co. Consequently, gl. dim. mod(mod A) = 0, and we then know that A is Nakayama with Loewy length at most 2. [2]. Assume conversely that A is Nakayama with Loewy length at most 2, and let E be an indecomposable nonsimple injective A-module. Since A is Nakayama, E has a unique composition series, so that E/rE is simple. Hence, the projective cover PE of E is indecomposable. Consider the exact sequence 0 + K--f Pn -+ E-j. 0. Since PE is the projective cover for E, we have KC rPE . Since PE has a unique composition series, rPE must be simple. If K # (0), we must have K = rPE , which gives the contradiction that E is simple. Hence, K = (0), so that E is projective. Since AOP is also Nakayama with Loewy length at most 2, each nonsimple indecomposable injective Aon-module is projective. The proof is then finished by using the usual duality D: mod A + mod Aon. Since we have shown in [4] that being of Loewy length at most 2 is a property preserved by stable equivalence, our main result of this paper is now an immediate consequence of Theorem 2.3 and our results in Section 1. THEOREM 2.4. If A is an artin algebra that is stably equivalent to a selfinjective artin algebra such that each indecomposable direct factor of A is of Loewiy length greater than 2, then A is itself self-in...ective.
\Fe have the following COROLLARY
2.5.
Theprojective
and injective objects coincide in mod(modA)
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if a& only if the indecomposable direct factors of A aye self-injective, orNakaq,ama with Loeq length at most 2. We end the paper by giving a more detailed description of the artin algebras with property (a). By the above, what we are then really doing is describing the indecomposable Nakayama algebras of Loewy length 2 that are not self-injective, and the simple algebras. We shall then apply this to give a structure theorem for the artin algebras stably equivalent to selfinjective algebras. If A has property (a) and only one simple module up to isomorphism, we have already pointed out that A is isomorphic to a division algebra k. If A has at least two simple A-modules, we have seen in the proof of Corollary 2.2 that we can write
where k is a division algebra, and by Proposition 2.1, ,M is a simple injective module with Hom,(M, M) s k, and S has property (a!) and has one simple module. S is then a division algebra, and from Hom,(M, M) g k, we conclude that S s k, and consequently,
If A has property
(a) and 3 simple modules, we then get
Since Hom(;M, All) z k and M is simple
which is our notation for the factor ring of
injective,
we can conclude
that
SELF-INJECTIVE
ALGEBRAS
73
by the twosided ideal
Continuing form
this process, we get that all algebras with property
(a) are of the
with a similar notational convention as above. We can use this last more precise investigation to get a characterization of the artin algebras stably equivalent to selfinjective algebras. THEOREM 2.6. An artin algebra A is stably equivalent to a self-injective algebra if and only if each indecomposable direct factor of A is self-injective, or Nakayama with Loezcy length at most 2.
Proof. In view of what we have already shown, we need only establish that each algebra
where n > 2 denotes the number of nonzero idempotents for the algebra, is stably equivalent to a self-injective algebra. Since A, is Nakayama with Loewy length 2, we know from [2] that mod A,/P is semisimple. This is also easy to see directly since the only indecomposable nonprojective A,modules are the n - 1 simple nonprojective A,-modules, so that mod A,/P is isomorphic to a product of n - 1 copies of the field k. Denote by R cc JI the trivial extension of a ring R by a twosided R-module M, i.e., the elements are pairs (r, m), where Y E R, m EM, and multiplication is defined by (Y, m) (Y’, m’) =- (w’, rm’ -C mr’). Let r, = (,k, x ... x ,k,) ot &k, f Bki + 4k3 + . nk,_I + Iklz), where the iki are simple rings isomorphic to k, ikj denotes the two-sided Ik, x ‘.. x ,k,-module, where the structure as a left module is given by (x1 ,..., xi ,..., x,) x =L xix for x,. E .k, , x E ikj , and the structure as a right module is given in an analogous way. Let Si denote the simple left
74
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r,,l-module that is isomorphic to ikj as a left module. If Pi denotes the projective cover of Si , it is not hard to see that rPi E S,+l for 1 < i 5: IZ - 1, and rP, z S, . In particular, sot r, g S, + ... f S,,, It is then easy to see that r, is self-injective: The dimension of r, as a left vector space over ii is the same as the dimension of r, as a right vector space over k. Let E be the direct sum of one copy of each of the indecomposable injective left rllmodules. 1Ve have a natural embedding of r, in E, and since by the ordinary duality between left and right P,-modules, r, and E have the same dimension over k, we must have r, g E, so that I’, is self-injective. It is easy to see that mod r,,/P is equivalent to the product of n copies of k. Hence, r,n m1is stably equivalent to A,! This finishes the proof of the theorem.
ACKNOWLEDGMENT I would like to thank Professor Maurice ment.
Auslander
for his inspiration
and encourage-
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%I. ALXSDER AND I. REITEN, Stable equivalence of artin algebras, ilt “Proc. Conf. on Orders, Group Rings, and Related Topics,” Springer Lecture Notes 353, Springer-Verlag, New York, 1973. M. ALX~ANDER AND I. REITEN, Stable equivalence of dualizing R-varieties, Admwccs ill Math. 12 (I 974), 306-366. >I. AUSLA~DER AND I. REITEN, Stable equivalence of dualizing R-nu?eties II: Hereditary dualizing R-varieties, Advances in Math. 17 (1975), 93-l 2 I. WI. A131,.4NDER AND I. REITEN, Stable equivalence of dualizing R-rarietles 111: Dualizing R-varieties stably equivalent to hereditary, dualizing R-varieties, Adr. in Math. 17 (1975), 122-142. R/I. AUSLANDER AND I. REITEN, Stable equivalence of dualizing R-varieties I\.: Higher global dimension, Adu. in Math. 17 (1975), 343-166. M. AVSLANDER AND 1. REITEN, Stable equivalence of dualizing R-varieties 1’: Artin algebras stably equivalent to hereditary algebras, Adu. in Math. 17 (1975), 167-195. K. L. FIELDS, On the global dimensional of residue rings, Pacific 1. Math. 32 (I 971), 345-349. R. Fossrr~r, P. GRIFFITH, XND I. REITEN, ‘I’he homological algebra of trivial estensions of abelian categories with application to ring theory, Springer Lecture Notes, 456 (1975).