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Annihilators of cohomology modules
JOURNAL
OF ALGEBRA
69, 150-154 (1981)
Annihilators
of Cohomology
Modules
GEORGES. AVRUNIN Department
of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003, and Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903 Communicated by W. Feit Received May 5, 1980
Let G be a finite group, k a field of characteristic p > 0, and U a kGmodule. Recently, Alperin and Evens [l] proved that the complexity of U, a numerical invariant defined in terms of the minimal projective resolution of U, is determined by the restrictions of U to elementary abelian p-subgroups. This result has also been obtained by Carlson [2], as a consequenceof a theorem concerning the dimension of the sum of the nonprojective components of such a restriction. In this paper we focus on annihilators of modules for the cohomology ring H*(G, k) and show how to obtain the Alperin-Evens theorem and some of its corollaries from a decomposition theorem for certain ideals.
1. COMPLEXITY Let G be a finite group and let k be a field of characteristic p. Let U and V be kG-modules, and regard k as kG-module with trivial G-action. The Hopf algebra structure of kG leads to a homomorphism Ext,m(k, k) 0 Ext;,(U, V) -, Ext;,tm(U, P’) (see [5, Sect. 8.41, for example). When U = I’= k, this is the cup product of the cohomology ring H*(G, k) = Ext&(k, k), and in general it makes Ext&(U, I’) a module for the cohomology ring. Now let H be a subgroup of G and let V be a kH-module. Shapiro’s lemma gives isomorphisms sl: Ext&( V I’, M) + Ext$(V, M IH) and s2: Ext&(M, V 1”) -+ Ext&(M IH, I’). Each Ext$(U, I’) becomes an H*(G, k)module via the restriction map, and it is easy to check that this makes s, and s2 H*(G, k)-isomorphisms. 150 0021.8693/81/030150-059602.00/0 Copyright 0 1981 by Academic Press, Inc. Ail rights of reproduction in any form reserved.
ANNIHILATORS OF COHQMOLOGY ~~~UL~~
151
For a finite-dimensional kG-module U, define a module ~ohomology ring by setting i@(G, U) = IJ Ext&(L, U), w taken over the isomorphism classesof irreducible kG-rnod~~csL, by H’*(G, k) the even-dimensional cohomology ring, an module M, we let Ann,(M) be the annihilator of M in define the complexity of U, written c,(U), to be the H’*(G, k)/(Ann&(G,
U) f?
This definition is equivalent to that given b of the minimal projective resolution. We sketch a proof here, altkough this fact will not be used except in the proof of the Alpe~~~-~~e~stheorem itself. It f~ilows from [ 1, Lemma 5.41 that the complexity of U7 in the AIperin-Evens sense,is the growth of Ii?(G, U), where the growth y(V) of a graded k-space Y is the smallest nonnegative integer g such that there is a positive number c with dim, V, < cnge-l for all azsu~c~e~t~y large. Each Ext&(k, U) = H*(G, Wom,(L, U)) is a finitely generated (G, k)-modde by [4], and then, since H*(G, k) is a finitely generat module over H2*(G, k), ,@(G, U) is a finitely generated., faithful H**(G, k)l Ann,fi(G, U) n H’*(G, k))-module. Thi implies that y(lR(G, 2*(G, k)/(Ann&f(G, v) n I-12*(G,k))). ut this quotient ring has Imension c,(U), and so is a finitely gen ed module over a polyn subalgebra in c&U) variables over k. Therefore its growth is cc(U), and the two definitions are indeed the same.
2. THE MAIN THEOREM
In order to prove a decomposition theorem for the radical of Ann,fi(G, U), we need to introduce another module for N*(G, k). (6 u) = IJ Extf,#, u>, where the sum ranges over the ~sorn~r~b~s of finite-dimensional indecomposable kG-modules, U) 2 Ann,M(G, U). @(G U>< M(G, u?, so Ann&G, each indecomposable X has a finite cornpos~t~o~series, an argument from the long exact Ext sequence shows that Ann rad(Ann,M(G, U)), where rad(I) denotes the radical of an i rad(Ann,n(G, !I)) = rad(Ann,M(G, U)). Conversations with Jon Carlson indicate that rad( rad(Ann, Ext&( U, U)), as well. LEMMA.
Let f be a p-subgroup of G for which U is ~e~~t~ve~~ projectiw.
Then rad(Ann,M(G, U) = rad(Ann,i@(G, U)) = rad(A~~~ rad(A~~~~(~~ U IP)).
152
GEORGE S.AVRUNIN
ProoJ: From the preceding discussion, the two outer equalities are clear. Since P is a P-group, a(P, VIP) = Ext&(k, U Ip), which is isomorphic to Ext&(k IG,u) by Shapiro’s lemma. Thus, rad(Ann,#(P, U Ip)) 1 rad(Ann,M(G, U)). Shapiro’s lemma also implies that rad(Ann,M(P, U Ip) c rad(Ann,M(G, UI, I”)). But U is relatively P-projective, so U is a summand of UI, IG* Then we have rad(Ann,M(P, U Ip)) 5 rad(Ann,M(G, U)). This completes the proof. THEOREM. With above, rad(Ann,@G, U)) = the notation n rad(Ann,i@(E, U IE)), where the intersection is taken over representatives
E of the conjugacy classes of elementary abelian p-subgroups of G. Proof. Since every kG-module is relatively projective for a Sylow psubgroup of G, the lemma implies that it is sufficient to consider the case in which G is a p-group. The theorem is trivial if G is elementary abelian, so assume not. We proceed by induction on the order of G, assuming the theorem for groups of smaller order. Since G is a p:group, @G, U) = ExttG(k, U). Suppose x E Ann,@(G, U). Then a power of x annihilates M(G, U) and therefore annihilates Ext&(k, U IE)g ExtzG(k IG,U) for each elementary abelian subgroup E. Thus rad(Ann,M(G, U)) c fi rad(Ann, M(E, U IE)). Suppose x E n rad(Ann,@(E, VIE)). By induction, a power of x annihilates @H, U lH) = ExttH(k, U lH) = H*(H, U IH) for any proper subgroup H of G, and we may replace x by this power. Let G, be a maximal subgroup of G. A nontrivial homomorphism from the cyclic group G/G, to the prime field IF, determines a nonzero element y of H’(G, k); let py E H*(G, k) be the Bockstein of y. Recall that H*(G, U) has a filtration {F’H*(G, U)} associated with the spectral sequence of the group extension l+G,+G+G/G,+l. pH*(G, U) = H*(G, U) and F’H*(G, U) is the kernel of the restriction map H*(G, u)+H*(G,, uIG1). Since x annihilates H*(G,, UIGI), we have x . H*(G, U) sF’H*(G, U) and xi . H*(G, U) sF%T*(G, U) for all i > 1. By [8] or [l, Lemma 4.11, /3y . FH*(G, U) = P+*H*(G, U), and by a theorem of Serre [9] there is a sequence G, ,..., Gj of maximal subgroups of G and associated elements yr,..., yj E H’(G, k) such that each /Iyi is nonzero but By, . By2 . . . . . pyj = 0. Thus x*j . H*(G, U) -c,f?yI . ,!?y2. . . . . byjH*(G, U) = 0, and x E rad(Ann,M(G, U)). This completes the proof. We remark that, as is apparent from the proof, the intersection may be taken over representatives of the G-conjugacy classes of the elementary abelian subgroups of a vertex of U.
ANNXHILATORS
OF COHOMOLGGY 3.
~Q~~L~~
153
APPLICATIONS
e now prove the Alperin-Evens theore I Alperin and Evens also obtained a number of known results as corollaries; we give alternate proofs of two of these from our point of view. et E be an elementary abelian subgroup The theorem implies that ruil dimension of H’*(G, k)/(Ann, urn of the Krull dimensions of H’*(G, k)) for elementary abelian p-subgroup is a finitely generated module over H*(G, generated over H’*(G, k). It follows t 2*(G, k)/(A~n~~(E, U IE)n H’*(G, k)) 2t(~, k)/(Ann,i@(E,
CTIE) n H”*(E, k)).
THEOREM (Alperin-Evens). If U is a kG-mod~~e~ then cG(U) = max cE(U 1,) as E ranges over the elementary ~be~~a~~-~~bg~~~~§ of 6. Thus gives a relation between the dimension of t.he variety associated to 2*(G3k)/(Ann~~(G, U) n N’*(G, k)) and the imensions of the varieties of the 2*(E, k)l(Ann,@C U lE)n N2*(E, k)). It would be interesti have an analog of the stratification of Theorem 3.7 (of [O] in th In the case U = k, we know that N’*(E, k)/(A~~~~(~~ k) n a polynomial algebra in rank E variables over k, and thus has ual to the rank of E. Our theorem then ~rn~~~es the f~~~~~~~~ THEOREM (Quillen). The Krull dimension sf maximum rank of an elementary abelian p-subggrotip If U is a projective kG-module. it is clear that cc cc(U) = 0, H’*(G, k)/(A nn,fi(G, U) n H’*(G, k)) has finite k-dirn~~s~o~~ Since li;4i(GgU) is a finitely generated module for this ~~ot~e~t, it tos has finite k-dimension, and then Ewtz,(L, U) = 0 for some n and all i~re~~~~b~~ kG-modules L. This implies that U is injective, and therefore projective, since kC is a Frobenius algebra. We then have THEQREM (Chouinard [3]). A kG-module hi is projective If and on/y if U lE is projective for each eiementary abelian ~-~~bg~o~~ E of
Added ia
prooJ
The author
also proved the main theorem.
has been informed by .I. L. Alperin rhat be and Evms have
154
GEORGE S.AVRUNIN ACKNOWLEDGMENT
The author is grateful to Leonard L. Scott for a number of helpful conversations.
REFERENCES 1. J. L. ALPERIN AND L. EVENS, Representations, resolutions and Quillen’s dimension
theorem, to appear. 2. J. F. CARLSON, The dimensions of modules and their restrictions over modular group
algebras, to appear. 3. L. G. CHOUINARD,Projectivity and relative projectivity over group rings, .I. Pure Appl. Algebra 7 (1976), 287-302. 4. L. EVENS,The cohomology ring of a finite group, Trans. Amer. Math. Sot. 101 (1961), 224-239. 5. S. MACLANE, “Homology,” Springer, New York, 1963. 6. D. QUILLEN, A cohomological criterion for p-nilpotence, J. Pure Appl. Algebra 1 (1971), 361-372. 7. D. QUILLEN,The spectrum of an equivariant cohomology ring, I, Ann. of Math. 94 (1971), 549-572. 8. D. QLJILLENAND B. B. VENKOV, Cohomology of finite groups and elementary abelian subgroups, Topology 11 (1972), 3 17-3 18. 9. J.-P. SERRE,Sur la dimension cohomologique des groupes profinis, Topology 3 (1965) 413-420.
OF ALGEBRA
69, 150-154 (1981)
Annihilators
of Cohomology
Modules
GEORGES. AVRUNIN Department
of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003, and Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903 Communicated by W. Feit Received May 5, 1980
Let G be a finite group, k a field of characteristic p > 0, and U a kGmodule. Recently, Alperin and Evens [l] proved that the complexity of U, a numerical invariant defined in terms of the minimal projective resolution of U, is determined by the restrictions of U to elementary abelian p-subgroups. This result has also been obtained by Carlson [2], as a consequenceof a theorem concerning the dimension of the sum of the nonprojective components of such a restriction. In this paper we focus on annihilators of modules for the cohomology ring H*(G, k) and show how to obtain the Alperin-Evens theorem and some of its corollaries from a decomposition theorem for certain ideals.
1. COMPLEXITY Let G be a finite group and let k be a field of characteristic p. Let U and V be kG-modules, and regard k as kG-module with trivial G-action. The Hopf algebra structure of kG leads to a homomorphism Ext,m(k, k) 0 Ext;,(U, V) -, Ext;,tm(U, P’) (see [5, Sect. 8.41, for example). When U = I’= k, this is the cup product of the cohomology ring H*(G, k) = Ext&(k, k), and in general it makes Ext&(U, I’) a module for the cohomology ring. Now let H be a subgroup of G and let V be a kH-module. Shapiro’s lemma gives isomorphisms sl: Ext&( V I’, M) + Ext$(V, M IH) and s2: Ext&(M, V 1”) -+ Ext&(M IH, I’). Each Ext$(U, I’) becomes an H*(G, k)module via the restriction map, and it is easy to check that this makes s, and s2 H*(G, k)-isomorphisms. 150 0021.8693/81/030150-059602.00/0 Copyright 0 1981 by Academic Press, Inc. Ail rights of reproduction in any form reserved.
ANNIHILATORS OF COHQMOLOGY ~~~UL~~
151
For a finite-dimensional kG-module U, define a module ~ohomology ring by setting i@(G, U) = IJ Ext&(L, U), w taken over the isomorphism classesof irreducible kG-rnod~~csL, by H’*(G, k) the even-dimensional cohomology ring, an module M, we let Ann,(M) be the annihilator of M in define the complexity of U, written c,(U), to be the H’*(G, k)/(Ann&(G,
U) f?
This definition is equivalent to that given b of the minimal projective resolution. We sketch a proof here, altkough this fact will not be used except in the proof of the Alpe~~~-~~e~stheorem itself. It f~ilows from [ 1, Lemma 5.41 that the complexity of U7 in the AIperin-Evens sense,is the growth of Ii?(G, U), where the growth y(V) of a graded k-space Y is the smallest nonnegative integer g such that there is a positive number c with dim, V, < cnge-l for all azsu~c~e~t~y large. Each Ext&(k, U) = H*(G, Wom,(L, U)) is a finitely generated (G, k)-modde by [4], and then, since H*(G, k) is a finitely generat module over H2*(G, k), ,@(G, U) is a finitely generated., faithful H**(G, k)l Ann,fi(G, U) n H’*(G, k))-module. Thi implies that y(lR(G, 2*(G, k)/(Ann&f(G, v) n I-12*(G,k))). ut this quotient ring has Imension c,(U), and so is a finitely gen ed module over a polyn subalgebra in c&U) variables over k. Therefore its growth is cc(U), and the two definitions are indeed the same.
2. THE MAIN THEOREM
In order to prove a decomposition theorem for the radical of Ann,fi(G, U), we need to introduce another module for N*(G, k). (6 u) = IJ Extf,#, u>, where the sum ranges over the ~sorn~r~b~s of finite-dimensional indecomposable kG-modules, U) 2 Ann,M(G, U). @(G U>< M(G, u?, so Ann&G, each indecomposable X has a finite cornpos~t~o~series, an argument from the long exact Ext sequence shows that Ann rad(Ann,M(G, U)), where rad(I) denotes the radical of an i rad(Ann,n(G, !I)) = rad(Ann,M(G, U)). Conversations with Jon Carlson indicate that rad( rad(Ann, Ext&( U, U)), as well. LEMMA.
Let f be a p-subgroup of G for which U is ~e~~t~ve~~ projectiw.
Then rad(Ann,M(G, U) = rad(Ann,i@(G, U)) = rad(A~~~ rad(A~~~~(~~ U IP)).
152
GEORGE S.AVRUNIN
ProoJ: From the preceding discussion, the two outer equalities are clear. Since P is a P-group, a(P, VIP) = Ext&(k, U Ip), which is isomorphic to Ext&(k IG,u) by Shapiro’s lemma. Thus, rad(Ann,#(P, U Ip)) 1 rad(Ann,M(G, U)). Shapiro’s lemma also implies that rad(Ann,M(P, U Ip) c rad(Ann,M(G, UI, I”)). But U is relatively P-projective, so U is a summand of UI, IG* Then we have rad(Ann,M(P, U Ip)) 5 rad(Ann,M(G, U)). This completes the proof. THEOREM. With above, rad(Ann,@G, U)) = the notation n rad(Ann,i@(E, U IE)), where the intersection is taken over representatives
E of the conjugacy classes of elementary abelian p-subgroups of G. Proof. Since every kG-module is relatively projective for a Sylow psubgroup of G, the lemma implies that it is sufficient to consider the case in which G is a p-group. The theorem is trivial if G is elementary abelian, so assume not. We proceed by induction on the order of G, assuming the theorem for groups of smaller order. Since G is a p:group, @G, U) = ExttG(k, U). Suppose x E Ann,@(G, U). Then a power of x annihilates M(G, U) and therefore annihilates Ext&(k, U IE)g ExtzG(k IG,U) for each elementary abelian subgroup E. Thus rad(Ann,M(G, U)) c fi rad(Ann, M(E, U IE)). Suppose x E n rad(Ann,@(E, VIE)). By induction, a power of x annihilates @H, U lH) = ExttH(k, U lH) = H*(H, U IH) for any proper subgroup H of G, and we may replace x by this power. Let G, be a maximal subgroup of G. A nontrivial homomorphism from the cyclic group G/G, to the prime field IF, determines a nonzero element y of H’(G, k); let py E H*(G, k) be the Bockstein of y. Recall that H*(G, U) has a filtration {F’H*(G, U)} associated with the spectral sequence of the group extension l+G,+G+G/G,+l. pH*(G, U) = H*(G, U) and F’H*(G, U) is the kernel of the restriction map H*(G, u)+H*(G,, uIG1). Since x annihilates H*(G,, UIGI), we have x . H*(G, U) sF’H*(G, U) and xi . H*(G, U) sF%T*(G, U) for all i > 1. By [8] or [l, Lemma 4.11, /3y . FH*(G, U) = P+*H*(G, U), and by a theorem of Serre [9] there is a sequence G, ,..., Gj of maximal subgroups of G and associated elements yr,..., yj E H’(G, k) such that each /Iyi is nonzero but By, . By2 . . . . . pyj = 0. Thus x*j . H*(G, U) -c,f?yI . ,!?y2. . . . . byjH*(G, U) = 0, and x E rad(Ann,M(G, U)). This completes the proof. We remark that, as is apparent from the proof, the intersection may be taken over representatives of the G-conjugacy classes of the elementary abelian subgroups of a vertex of U.
ANNXHILATORS
OF COHOMOLGGY 3.
~Q~~L~~
153
APPLICATIONS
e now prove the Alperin-Evens theore I Alperin and Evens also obtained a number of known results as corollaries; we give alternate proofs of two of these from our point of view. et E be an elementary abelian subgroup The theorem implies that ruil dimension of H’*(G, k)/(Ann, urn of the Krull dimensions of H’*(G, k)) for elementary abelian p-subgroup is a finitely generated module over H*(G, generated over H’*(G, k). It follows t 2*(G, k)/(A~n~~(E, U IE)n H’*(G, k)) 2t(~, k)/(Ann,i@(E,
CTIE) n H”*(E, k)).
THEOREM (Alperin-Evens). If U is a kG-mod~~e~ then cG(U) = max cE(U 1,) as E ranges over the elementary ~be~~a~~-~~bg~~~~§ of 6. Thus gives a relation between the dimension of t.he variety associated to 2*(G3k)/(Ann~~(G, U) n N’*(G, k)) and the imensions of the varieties of the 2*(E, k)l(Ann,@C U lE)n N2*(E, k)). It would be interesti have an analog of the stratification of Theorem 3.7 (of [O] in th In the case U = k, we know that N’*(E, k)/(A~~~~(~~ k) n a polynomial algebra in rank E variables over k, and thus has ual to the rank of E. Our theorem then ~rn~~~es the f~~~~~~~~ THEOREM (Quillen). The Krull dimension sf maximum rank of an elementary abelian p-subggrotip If U is a projective kG-module. it is clear that cc cc(U) = 0, H’*(G, k)/(A nn,fi(G, U) n H’*(G, k)) has finite k-dirn~~s~o~~ Since li;4i(GgU) is a finitely generated module for this ~~ot~e~t, it tos has finite k-dimension, and then Ewtz,(L, U) = 0 for some n and all i~re~~~~b~~ kG-modules L. This implies that U is injective, and therefore projective, since kC is a Frobenius algebra. We then have THEQREM (Chouinard [3]). A kG-module hi is projective If and on/y if U lE is projective for each eiementary abelian ~-~~bg~o~~ E of
Added ia
prooJ
The author
also proved the main theorem.
has been informed by .I. L. Alperin rhat be and Evms have
154
GEORGE S.AVRUNIN ACKNOWLEDGMENT
The author is grateful to Leonard L. Scott for a number of helpful conversations.
REFERENCES 1. J. L. ALPERIN AND L. EVENS, Representations, resolutions and Quillen’s dimension
theorem, to appear. 2. J. F. CARLSON, The dimensions of modules and their restrictions over modular group
algebras, to appear. 3. L. G. CHOUINARD,Projectivity and relative projectivity over group rings, .I. Pure Appl. Algebra 7 (1976), 287-302. 4. L. EVENS,The cohomology ring of a finite group, Trans. Amer. Math. Sot. 101 (1961), 224-239. 5. S. MACLANE, “Homology,” Springer, New York, 1963. 6. D. QUILLEN, A cohomological criterion for p-nilpotence, J. Pure Appl. Algebra 1 (1971), 361-372. 7. D. QUILLEN,The spectrum of an equivariant cohomology ring, I, Ann. of Math. 94 (1971), 549-572. 8. D. QLJILLENAND B. B. VENKOV, Cohomology of finite groups and elementary abelian subgroups, Topology 11 (1972), 3 17-3 18. 9. J.-P. SERRE,Sur la dimension cohomologique des groupes profinis, Topology 3 (1965) 413-420.