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Module extensions and blocks
JOURNAL OF ALGEBRA 5, 157-163 (1967)
Module Extensions and Blocks* IRVING Department
of Mathematics,
REINER
University
of Illinois,
Urbana, Illinois
Communicated by W. Feit Received
January
18, 1966
1. INTRODUCTION Let R be a Dedekind ring of characteristic zero, with quotient field K, and let G be a finite group. All RG-modules are assumed finitely generated and R-torsionfree. For each prime ideal P of R, let Rp denote the P-adic completion of R, with quotient field Kp . For convenience, we shall write Ext in place of Ext& or Extkro . If {Ext}, denotes the P-primary component of Ext, then for any pair of RG-modules M, N we have Ext(M, N) GZ c @WC
N)>P ,
P
where P ranges over the divisors of [G : 11. Furthermore, Mp = Rp OR M, then (Ext(M,
N,)P
cx
Ext(Mp
if we set
, NP).
For each KG-module U, denote by u(U) the collection of all RG-modules M such that K @R M s U. We shall say that two KG-modules U, V are linked at P if there exist modules M E u(U), NE u(V), such that {Ext(M, N)}p # 0. Call U and V linked if they are linked at some P, that is, if Ext(M, N) # 0 for some choice of M E u(U), NE u(V). Let U, be a KGi-module, i = 1,2, and let G = Gi x G, . We may form the KG-module U, # U, , defined as the K-space U, & U, , with the action of G given by (g1 >g2@l
0 u2) =gl%
og2u2
*
If [G1 : l] and [G, : l] are relatively prime, then as Vi ranges over all irreducible KG,-modules (i = 1,2), the product U, ## U, ranges over all irreducible KG-modules. * This research was supported
in part by the
157
National Science Foundation.
158
REINER
Let 2 denote the ring of rational integers, Q the rational field. Berman and Lichtman [I] recently proved THEOREM 1. Let G be a nilpotent group, and write G = GI x **. x G, , where GI ,..., G, are the Sylow subgroups of G belonging to the distinct prime divisors of [G : I]. For each i, 1 < i < r, let Vi and Vi be a pair of irreducible KG,-modules, and set
u= u, #- #UT,
v= VI #-- #V,.
Assume that the { Ui} differ from the {Vi} for at least one subscript i. Then U and V are linked if and only if the {Vi} differ from the (Vi} for exactly one subscri$ i. The object of the present note is to generalize this result, and at the same time to put it into a more natural setting involving block idempotents. As general reference for the techniques used herein, we may cite 121.
2. Let c&G)
DISJOINT MODULES
denote the center of RPG, and write 1 E c(RPG) as 1 = er + em*+ e, ,
a sum of orthogonal primitive idempotents in c(R,G). These {e#} are called the block idempotents of R,G, and are not necessarily primitive in c(KpG). For each i, 1 < i < t, let us write ei = C l 3 , 5
where the {Pi,} are orthogonal primitive idempotents in c(K,G). Then the decomposition of K,G into simple components is given by
and the irreducible KpG-modules are just the minimal left ideals in these simple components. Hence for each irreducible KPG-module X there is a unique block idempotent e such that ex = x, x E X. We say that X belongs to e; in this case, X is annihilated by all the other block idempotents. Let us call two KpG-modules X, Y disjoint if the set of block idempotents to which the composition factors of X belong has no elements in common with the corresponding set for Y.
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If in particular P+’ [G : 11, then each Eij lies in RPG, and SO the {Q} coincide with the {e,). Thus, in this case, if X and Y are KpG-modules having no common composition factor, then X and Y are disjoint. Returning to the general case, for any KG-module U, set Up = Kp & U. If M is an RG-module in cr(U), then Kp BRpMP s Up , that is, Mp E u( Up). Let us call two KG-modules U, V disjoint at P if Up and V, are disjoint. THEOREM
2. Let X and Y be disjoint KpG-modules. Then Ext(M, N) = 0 for all ME u(X),
NE u(Y).
(1)
Hence if U and V are KG-modules which are disjoint at P, then U and V are not linked at P. Proof. It suffices to prove the first assertion, which clearly implies the second. Let ME u(X), NE u(Y). Given any exact sequence 0+x,-+x-+x,-+0 of KpG-modules, we may find an exact sequence of RrG-modules O-+Ml-+M-+M,-+O in which MS E u(X,), i = 1, 2. Then Ext(Ms , N) + Ext(M, N) -+ Ext(M, , N) is exact, and thus it is enough to show that Ext(Mi , N) = 0, i = 1,2. By repeated use of this argument, we are reduced to proving that if X and Y are irreducible KpG-modules belonging to different block idempotents, then (1) holds. Suppose that X belongs to the block idempotent e; then em = m, m E M, whereas e annihilates N. Let us view Ext(M, N) as the group of binding functions from G to Hom,JM, N), modulo the subgroup of inner binding functions. Recall that a binding function is a map F : G -+ Hom(M, N), satisfying g,heG. F,, =gF, -I-F>, Call F inner if there exists I( E Hom(M, N) such that F,, =gu -ug,
g E G.
If F is any binding function, then eF and Fe are also binding functions. We claim that their difference is an inner binding function. Indeed, since Ext(KpM, KpN) = 0, there exists t E Hom(K,M, KJV) such that F L7= gt - tg,
gEG>
160 and for which (gt - tg)MCN,
gEG.
(2)
Setting u = et - te, we have eF, - F,e = gu - ug,
geG.
Furthermore, since e is an &-linear combination of group elements, it follows from (2) that u E Hom(M, N). Therefore eF -Fe is inner, as claimed. We have now shown that eF and Fe determine the same element of Ext(M, N). But Fe = F, since e acts as the identity on M; on the other hand, eF = 0 since e annihilates N. This shows that F is in the zero class, and completes the proof of the theorem. We shall need two easy observations.
LEMMA 1. Let U, V be KG-modules having no common composition factor, and supposethat P{ [G : 11. Then U and V are disjoint at P. Proof. Let U’ denote any composition factor of U, and V’ of V. Then U’ and V’ are distinct irreducible KG-modules, whence (U’), and (I”), have no common composition factor. Since P{ [G : I], this shows that (U’), and (V’), are disjoint, which implies that U and V are disjoint at P.
LEMMA2. Let G = GI x G, , and let U, , VI be KG,-modules which are disjoint at P. Let Us , V, be arbitrary KG,-modules. Then U, # Us and VI # V, are disjoint at P. Proof. The set of composition factors of (U, # U,), is the union of the sets of composition factors of X # (Us), , where X ranges over the composition factors of (U&I . If X belongs to the block idempotent e of RpGI , then e is a central idempotent of RpG which acts trivially on each composition factor of X # (Us), , and annihilates each composition factor of ( VI # V,), . This shows that (U, # U,), and (VI # V,), are disjoint, as claimed.
COROLLARY.Using the notation of Theorem 1, supposethat the {Vi> differ from the (Vi> at two or more places. Then U and V are not linked. Proof. Suppose that Vi and Vi are nonisomorphic, i = 1,2. We shall show that for each P, Vi and Vi are not linked at P. Since [Gr : I] and [G, : l] are relatively prime, either P 4’ [Gr : l] or P q [G, : 11. Without loss of generality, suppose P+’ [Gr : 11. By Lemma 1, U, and VI are disjoint at P, whence so are U and V by Lemma 2. The result now follows from Theorem 2.
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3. PRIMITIVE BLOCK IDEMPOTENTS In order to obtain a converse to Theorem 2, we shall have to restrict the types of block idempotents under consideration. If e is a block idempotent of R,G, then R,G * e is a two-sided ideal of RFG which cannot be decomposed into a direct sum of two-sided ideals. It may happen that in fact R,G * e cannot be decomposed into a direct sum of left ideals; in this case, e is a primitive idempotent not only in c(R~G), but in the ring RpG itself. This will occur if and only if the ring R,G * e is completely primary, that is, R,G * e/rad(R,G * e) is a skewfield. THEOREM 3. Let e be a block idempotent of RpG which is primitive in RFG. Let U and V be distinct irreducible KG-modules such that e acts as the identity on both U, and V, . Then U and V are linked at P.
Proof. Choose primitive idempotents e1, es in c(KG) such that U is a minimal left ideal in KG . l 1 , V in KG * l s . Then the RG-module RG * l 1 is a successive extension of modules in u(U), and likewise, RG * l p is a successive extension of modules in u(V). Hence if U and V are not linked at P, the left RpG-module RpG(cl + es) decomposes into a direct sum RpG(cl + Q) E R,G . q + R,G +c2 .
(3)
We shall show that this is impossible. Since e acts as identity on both U, and VP, we have eei = q , eEp= es . Setting E = e1 + es, we conclude that ee = E. The ring epimorphism R,G*e-+RpG*c given by x + XEinduces an epimorphism RpG * e/rad(RpG * e) -+ R,G * c/rad(R,G * e), and therefore RpG . E is also a completely primary ring. This shows that RPG * E is indecomposable as left RPG-module, which contradicts (3), and establishes the theorem. COROLLARY. Let G be a p-group, and let U, V be distinct irreducible KG-modules. Then for each P dividing p, U and V are linked at P.
Proof. The ring lipG is completely primary, and so we may take e = 1 in the above theorem. Remark. Theorem 3 may also be obtained from a result of Thompson ([3], Theorem 1). For if we set W = Kp( U + V) r\ RpG * e, and use his
162
REINER
approach, we see that W is an RP-pure RpG-submodule of R,G * e. Let bars denote reduction mod P; then W is an RG-submodule of RG * t? Since this latter module has a unique minimal submodule, Wmust be indecomposable, whence so is W. But on the other hand, if U and V are not linked at P, then W is decomposable. Next comes a lemma of independent interest. Suppose that E = G x H is a direct product, and let T be an RH-module. Given an exact sequence of RG-modules O--+N+L+M+O, we obtain an exact sequence of RE-modules: 0-N
# T-L
# T+M
# T-0.
Thus there exists a map vT : Ex&(M, LEMMA
3.
N) + Ext&(M
# T, N # T).
The map VT is nwnic.
Proof. If F : G --+ Hom,(M, N) is a binding function representing an element of Ext(M, N), then its image F’ under VT is given by gEG,hEH,mEM,tET.
F&,)(m 0 t) = F,m 0 t,
In order to show VT manic, it suffices to establish this fact when Rp is used in place of R. To avoid notational difficulties, we may instead restrict ourselves to the case where R is a principal ideal domain. We must show that if F’ is inner, then so is F. Now if F’ is inner, there exists a map w E HomR(M # T, N # T) such that J%.dm 0 t> = (g, h)w(m 0 4 - wu(gm0 W for all g, h, m, t. Setting h = 1, this becomes F,m @ t = (g, l)w(m @ t) - w(gm @ t).
(4)
Suppose that T = C@ Rt, , and set w(m 0 tt) = C4m)
0 4,
wsj E Hom(M, N).
Set t = ti in (4) and use (5), thereby obtaining
(5)
MODULE
BXTBNSIONS
AND
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163
for all g, m, i. But N # T = C@ N @ tj , and therefore F,m = gw&) - w&m) f or each g, m, i. Fixing i, we obtain F, = gwii - wig, proving that F is inner, and establishing the Lemma. COROLLARY 1. Let U and V be KG-modules, Y a KH-module, and set E = G x H. If U and V are linked at P, then U # Y and V # Y are also linked at P.
Proof. Since U and V are linked at P, there exist RG-modules M E u(U), NE u(V) such that {Ext(M, N)}P # 0. Choose any RH-module T E u(Y). Then by Lemma 3, {WM
# T, N # T>>P# 0.
Since M # T E u( U # Y) and N # T E u( V # Y), the Corollary is proved. COROLLARY 2. Using the notation of Theorem 1, supposethut the (US} and {Vi} differ at exactly one subscript i. Then U and V are linked.
Proof. Let us suppose that U, and V, are distinct, and that U, = V, for i = 2,..., Y. Choose P dividing [Gr : I] ; then U, and V, are linked at P, by theCorollarytoTheorem3.SetH=G2 x *a* x G,,Y= U, #*a- #U,, and use Corollary 1 above. Then U and V are also linked at P, and the proof is complete. RRFERENcE¶ I. BERMAN, S. D. AND LICHTMAN, A. I. Integral representations of finite nilpotent groups. Usp. Mat. Ah& 20 (1965), 186-188. 2. CURTIS, C. W. AND REINER, I. “Representation Theory of Finite Groups and Associative Algebras.” Wiley (Interscience), New York, 1962. 3. THOMPSON,J. G. “Vertices and Sources” (to be published).
Module Extensions and Blocks* IRVING Department
of Mathematics,
REINER
University
of Illinois,
Urbana, Illinois
Communicated by W. Feit Received
January
18, 1966
1. INTRODUCTION Let R be a Dedekind ring of characteristic zero, with quotient field K, and let G be a finite group. All RG-modules are assumed finitely generated and R-torsionfree. For each prime ideal P of R, let Rp denote the P-adic completion of R, with quotient field Kp . For convenience, we shall write Ext in place of Ext& or Extkro . If {Ext}, denotes the P-primary component of Ext, then for any pair of RG-modules M, N we have Ext(M, N) GZ c @WC
N)>P ,
P
where P ranges over the divisors of [G : 11. Furthermore, Mp = Rp OR M, then (Ext(M,
N,)P
cx
Ext(Mp
if we set
, NP).
For each KG-module U, denote by u(U) the collection of all RG-modules M such that K @R M s U. We shall say that two KG-modules U, V are linked at P if there exist modules M E u(U), NE u(V), such that {Ext(M, N)}p # 0. Call U and V linked if they are linked at some P, that is, if Ext(M, N) # 0 for some choice of M E u(U), NE u(V). Let U, be a KGi-module, i = 1,2, and let G = Gi x G, . We may form the KG-module U, # U, , defined as the K-space U, & U, , with the action of G given by (g1 >g2@l
0 u2) =gl%
og2u2
*
If [G1 : l] and [G, : l] are relatively prime, then as Vi ranges over all irreducible KG,-modules (i = 1,2), the product U, ## U, ranges over all irreducible KG-modules. * This research was supported
in part by the
157
National Science Foundation.
158
REINER
Let 2 denote the ring of rational integers, Q the rational field. Berman and Lichtman [I] recently proved THEOREM 1. Let G be a nilpotent group, and write G = GI x **. x G, , where GI ,..., G, are the Sylow subgroups of G belonging to the distinct prime divisors of [G : I]. For each i, 1 < i < r, let Vi and Vi be a pair of irreducible KG,-modules, and set
u= u, #- #UT,
v= VI #-- #V,.
Assume that the { Ui} differ from the {Vi} for at least one subscript i. Then U and V are linked if and only if the {Vi} differ from the (Vi} for exactly one subscri$ i. The object of the present note is to generalize this result, and at the same time to put it into a more natural setting involving block idempotents. As general reference for the techniques used herein, we may cite 121.
2. Let c&G)
DISJOINT MODULES
denote the center of RPG, and write 1 E c(RPG) as 1 = er + em*+ e, ,
a sum of orthogonal primitive idempotents in c(R,G). These {e#} are called the block idempotents of R,G, and are not necessarily primitive in c(KpG). For each i, 1 < i < t, let us write ei = C l 3 , 5
where the {Pi,} are orthogonal primitive idempotents in c(K,G). Then the decomposition of K,G into simple components is given by
and the irreducible KpG-modules are just the minimal left ideals in these simple components. Hence for each irreducible KPG-module X there is a unique block idempotent e such that ex = x, x E X. We say that X belongs to e; in this case, X is annihilated by all the other block idempotents. Let us call two KpG-modules X, Y disjoint if the set of block idempotents to which the composition factors of X belong has no elements in common with the corresponding set for Y.
MODULE
EXTENSIONS
AND
BLOCKS
159
If in particular P+’ [G : 11, then each Eij lies in RPG, and SO the {Q} coincide with the {e,). Thus, in this case, if X and Y are KpG-modules having no common composition factor, then X and Y are disjoint. Returning to the general case, for any KG-module U, set Up = Kp & U. If M is an RG-module in cr(U), then Kp BRpMP s Up , that is, Mp E u( Up). Let us call two KG-modules U, V disjoint at P if Up and V, are disjoint. THEOREM
2. Let X and Y be disjoint KpG-modules. Then Ext(M, N) = 0 for all ME u(X),
NE u(Y).
(1)
Hence if U and V are KG-modules which are disjoint at P, then U and V are not linked at P. Proof. It suffices to prove the first assertion, which clearly implies the second. Let ME u(X), NE u(Y). Given any exact sequence 0+x,-+x-+x,-+0 of KpG-modules, we may find an exact sequence of RrG-modules O-+Ml-+M-+M,-+O in which MS E u(X,), i = 1, 2. Then Ext(Ms , N) + Ext(M, N) -+ Ext(M, , N) is exact, and thus it is enough to show that Ext(Mi , N) = 0, i = 1,2. By repeated use of this argument, we are reduced to proving that if X and Y are irreducible KpG-modules belonging to different block idempotents, then (1) holds. Suppose that X belongs to the block idempotent e; then em = m, m E M, whereas e annihilates N. Let us view Ext(M, N) as the group of binding functions from G to Hom,JM, N), modulo the subgroup of inner binding functions. Recall that a binding function is a map F : G -+ Hom(M, N), satisfying g,heG. F,, =gF, -I-F>, Call F inner if there exists I( E Hom(M, N) such that F,, =gu -ug,
g E G.
If F is any binding function, then eF and Fe are also binding functions. We claim that their difference is an inner binding function. Indeed, since Ext(KpM, KpN) = 0, there exists t E Hom(K,M, KJV) such that F L7= gt - tg,
gEG>
160 and for which (gt - tg)MCN,
gEG.
(2)
Setting u = et - te, we have eF, - F,e = gu - ug,
geG.
Furthermore, since e is an &-linear combination of group elements, it follows from (2) that u E Hom(M, N). Therefore eF -Fe is inner, as claimed. We have now shown that eF and Fe determine the same element of Ext(M, N). But Fe = F, since e acts as the identity on M; on the other hand, eF = 0 since e annihilates N. This shows that F is in the zero class, and completes the proof of the theorem. We shall need two easy observations.
LEMMA 1. Let U, V be KG-modules having no common composition factor, and supposethat P{ [G : 11. Then U and V are disjoint at P. Proof. Let U’ denote any composition factor of U, and V’ of V. Then U’ and V’ are distinct irreducible KG-modules, whence (U’), and (I”), have no common composition factor. Since P{ [G : I], this shows that (U’), and (V’), are disjoint, which implies that U and V are disjoint at P.
LEMMA2. Let G = GI x G, , and let U, , VI be KG,-modules which are disjoint at P. Let Us , V, be arbitrary KG,-modules. Then U, # Us and VI # V, are disjoint at P. Proof. The set of composition factors of (U, # U,), is the union of the sets of composition factors of X # (Us), , where X ranges over the composition factors of (U&I . If X belongs to the block idempotent e of RpGI , then e is a central idempotent of RpG which acts trivially on each composition factor of X # (Us), , and annihilates each composition factor of ( VI # V,), . This shows that (U, # U,), and (VI # V,), are disjoint, as claimed.
COROLLARY.Using the notation of Theorem 1, supposethat the {Vi> differ from the (Vi> at two or more places. Then U and V are not linked. Proof. Suppose that Vi and Vi are nonisomorphic, i = 1,2. We shall show that for each P, Vi and Vi are not linked at P. Since [Gr : I] and [G, : l] are relatively prime, either P 4’ [Gr : l] or P q [G, : 11. Without loss of generality, suppose P+’ [Gr : 11. By Lemma 1, U, and VI are disjoint at P, whence so are U and V by Lemma 2. The result now follows from Theorem 2.
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3. PRIMITIVE BLOCK IDEMPOTENTS In order to obtain a converse to Theorem 2, we shall have to restrict the types of block idempotents under consideration. If e is a block idempotent of R,G, then R,G * e is a two-sided ideal of RFG which cannot be decomposed into a direct sum of two-sided ideals. It may happen that in fact R,G * e cannot be decomposed into a direct sum of left ideals; in this case, e is a primitive idempotent not only in c(R~G), but in the ring RpG itself. This will occur if and only if the ring R,G * e is completely primary, that is, R,G * e/rad(R,G * e) is a skewfield. THEOREM 3. Let e be a block idempotent of RpG which is primitive in RFG. Let U and V be distinct irreducible KG-modules such that e acts as the identity on both U, and V, . Then U and V are linked at P.
Proof. Choose primitive idempotents e1, es in c(KG) such that U is a minimal left ideal in KG . l 1 , V in KG * l s . Then the RG-module RG * l 1 is a successive extension of modules in u(U), and likewise, RG * l p is a successive extension of modules in u(V). Hence if U and V are not linked at P, the left RpG-module RpG(cl + es) decomposes into a direct sum RpG(cl + Q) E R,G . q + R,G +c2 .
(3)
We shall show that this is impossible. Since e acts as identity on both U, and VP, we have eei = q , eEp= es . Setting E = e1 + es, we conclude that ee = E. The ring epimorphism R,G*e-+RpG*c given by x + XEinduces an epimorphism RpG * e/rad(RpG * e) -+ R,G * c/rad(R,G * e), and therefore RpG . E is also a completely primary ring. This shows that RPG * E is indecomposable as left RPG-module, which contradicts (3), and establishes the theorem. COROLLARY. Let G be a p-group, and let U, V be distinct irreducible KG-modules. Then for each P dividing p, U and V are linked at P.
Proof. The ring lipG is completely primary, and so we may take e = 1 in the above theorem. Remark. Theorem 3 may also be obtained from a result of Thompson ([3], Theorem 1). For if we set W = Kp( U + V) r\ RpG * e, and use his
162
REINER
approach, we see that W is an RP-pure RpG-submodule of R,G * e. Let bars denote reduction mod P; then W is an RG-submodule of RG * t? Since this latter module has a unique minimal submodule, Wmust be indecomposable, whence so is W. But on the other hand, if U and V are not linked at P, then W is decomposable. Next comes a lemma of independent interest. Suppose that E = G x H is a direct product, and let T be an RH-module. Given an exact sequence of RG-modules O--+N+L+M+O, we obtain an exact sequence of RE-modules: 0-N
# T-L
# T+M
# T-0.
Thus there exists a map vT : Ex&(M, LEMMA
3.
N) + Ext&(M
# T, N # T).
The map VT is nwnic.
Proof. If F : G --+ Hom,(M, N) is a binding function representing an element of Ext(M, N), then its image F’ under VT is given by gEG,hEH,mEM,tET.
F&,)(m 0 t) = F,m 0 t,
In order to show VT manic, it suffices to establish this fact when Rp is used in place of R. To avoid notational difficulties, we may instead restrict ourselves to the case where R is a principal ideal domain. We must show that if F’ is inner, then so is F. Now if F’ is inner, there exists a map w E HomR(M # T, N # T) such that J%.dm 0 t> = (g, h)w(m 0 4 - wu(gm0 W for all g, h, m, t. Setting h = 1, this becomes F,m @ t = (g, l)w(m @ t) - w(gm @ t).
(4)
Suppose that T = C@ Rt, , and set w(m 0 tt) = C4m)
0 4,
wsj E Hom(M, N).
Set t = ti in (4) and use (5), thereby obtaining
(5)
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AND
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163
for all g, m, i. But N # T = C@ N @ tj , and therefore F,m = gw&) - w&m) f or each g, m, i. Fixing i, we obtain F, = gwii - wig, proving that F is inner, and establishing the Lemma. COROLLARY 1. Let U and V be KG-modules, Y a KH-module, and set E = G x H. If U and V are linked at P, then U # Y and V # Y are also linked at P.
Proof. Since U and V are linked at P, there exist RG-modules M E u(U), NE u(V) such that {Ext(M, N)}P # 0. Choose any RH-module T E u(Y). Then by Lemma 3, {WM
# T, N # T>>P# 0.
Since M # T E u( U # Y) and N # T E u( V # Y), the Corollary is proved. COROLLARY 2. Using the notation of Theorem 1, supposethut the (US} and {Vi} differ at exactly one subscript i. Then U and V are linked.
Proof. Let us suppose that U, and V, are distinct, and that U, = V, for i = 2,..., Y. Choose P dividing [Gr : I] ; then U, and V, are linked at P, by theCorollarytoTheorem3.SetH=G2 x *a* x G,,Y= U, #*a- #U,, and use Corollary 1 above. Then U and V are also linked at P, and the proof is complete. RRFERENcE¶ I. BERMAN, S. D. AND LICHTMAN, A. I. Integral representations of finite nilpotent groups. Usp. Mat. Ah& 20 (1965), 186-188. 2. CURTIS, C. W. AND REINER, I. “Representation Theory of Finite Groups and Associative Algebras.” Wiley (Interscience), New York, 1962. 3. THOMPSON,J. G. “Vertices and Sources” (to be published).