AbstractThe block variety VG,b(M) of a finitely generated indecomposable module M over the block algebra of a p-block b of a finite group G, introduced in (J. Algebra 215 (1999) 460), can be computed in terms of a vertex and a source of M. We use this to show that VG,b(M) is connected, and that every closed homogeneous subvariety of the affine variety VG,b defined by block cohomology H*(G,b) (cf. Algebras Rep. Theory 2 (1999) 107) is the variety of a module over the block algebra. This is analogous to the corresponding statements on Carlson's cohomology varieties in (Invent. Math. 77 (1984) 291)

Alperin

Benson

Broué

Carlson

Cartan

David J. Benson

Evens

Linckelmann

Markus Linckelmann

Puig

Thévenaz

English

Benson, David J.

Linckelmann, Markus

Elsevier - Publisher Connector 

Journal of Pure and Applied Algebra 197 (2005) 11–17
www.elsevier.com/locate/jpaa
Vertex and source determine the block variety of an
indecomposable module
David J. Bensona, Markus Linckelmannb,∗
aDepartment of Mathematics, University of Georgia, Athens, GA 30602, USA
bOhio State University, Math Tower 231, West 18th Avenue, Columbus, OH 43210, USA
Received 20 March 2002; received in revised form 21 June 2004
Available online 21 November 2004
Communicated by M. Broué
Abstract
Theblock varietyVG,b(M)of a ﬁnitely generated indecomposablemoduleM over the block algebra
of a p-block b of a ﬁnite group G, introduced in (J. Algebra 215 (1999) 460), can be computed in
terms of a vertex and a source ofM . We use this to show that VG,b(M) is connected, and that every
closed homogeneous subvariety of the afﬁne variety VG,b deﬁned by block cohomology H∗(G, b)
(cf. Algebras Rep. Theory 2 (1999) 107) is the variety of a module over the block algebra. This is
analogous to the corresponding statements on Carlson’s cohomology varieties in (Invent. Math. 77
(1984) 291).
© 2004 Elsevier B.V. All rights reserved.
MSC: 16E40; 20C20
1. Introduction
The theory of the cohomology variety VG(M) for a ﬁnitely generated moduleM over the
modular group algebra of a ﬁnite groupG was developed by Carlson [6–8] and others back
in the 1980s. It has played a major role in recent years in the modular representation theory
of ﬁnite groups. One of the problems with the theory has been that it has been much more
fruitful as a way of proving theorems about the principal block than about more general
blocks.
∗ Corresponding author.
E-mail address: linckelm@math.ohio-state.edu (M. Linckelmann).
0022-4049/$ - see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jpaa.2004.08.032
12 D.J. Benson, M. Linckelmann / Journal of Pure and Applied Algebra 197 (2005) 11–17
To address this problem, the second author [15] recently introduced the concept of the
block variety VG,b(M) of a ﬁnitely generated module M for a block b of a ﬁnite group
G. The deﬁnition of the ambient variety VG,b involves a variation of the Cartan–Eilenberg
stable element method [9] in ordinary cohomology for the Brauer pairs in the block. But the
deﬁnition of VG,b(M) involves going through the Hochschild cohomology of the block as
an intermediate step. In [17], the dependence onHochschild cohomology for the calculation
of block varieties is removed. It is shown in that paper that the block variety of M is the
image in VG,b of the variety at the level of the defect group. But ﬁrst, the module must be
cut down using the idempotent for the source algebra of the block, in the sense of Puig. An
example is given there, to show that this step is really necessary.
In this paper, we describe how to calculate the block variety of a module in terms of the
vertex and source of the module.We show that it is given as the image in VG,b of the variety
of the source ofM as a module for the vertex. Actually, there is a subtlety here: the vertex
and source are only well deﬁned up toG-conjugacy, and we must be careful how we choose
a P -conjugacy class.
Taken together with the stratiﬁcation theorem for VG,b [17] and Carlson’s theory of rank
varieties, our theorem implies that VG,b(M) can be calculated without knowing anything
about cohomology, just as was the case for VG(M).
Our theorem enables us to prove some basic properties forVG,b(M), including the analog
of Carlson’s theorem [8] that the variety of an indecomposable module is connected. We
also show that every closed homogeneous subvariety of VG,b can be realised as VG,b(N) for
a suitably chosen module N . Finally, we show that the block variety of a module survives
the Green correspondence with respect to a subgroup which controls fusion of Brauer pairs
in the block.
Throughout this paper, k is an algebraically closed ﬁeld of prime characteristic p. LetG
be a ﬁnite group. A block of G is a primitive idempotent b in Z(kG). Following Alperin
and Broué [1], a b-Brauer pair is a pair (Q, e) consisting of a p-subgroup Q of G and a
block e of CG(Q) such that BrQ(b)e = e, where BrQ : (kG)Q → CG(Q) is the Brauer
homomorphism [4]. It is shown in [1], that the set of b-Brauer pairs is a partially ordered
G-set such that G acts transitively on the set of maximal b-Brauer pairs. If (P, e) is a
maximal b-Brauer pair, then P is called a defect group of b (this notion is due to Brauer,
although he did not state it in these terms). If P is a defect group of b, then P is maximal
with respect to BrP (b) = 0; in particular, there is a primitive idempotent i ∈ (kGb)P such
that BrP (i) = 0. Such an idempotent is called a source idempotent of b (with respect to
P ). By Broué and Puig [5, Theorem 1.8], for any subgroupQ of P , there is a unique block
eQ of CG(Q) such that BrQ(i)eQ = 0. Then (Q, eQ) is a b-Brauer pair, and eQ is also the
unique block of CG(Q) such that (Q, eQ) ⊆ (P, eP ).
We denote by FG,b the category whose objects are the subgroups of P , and whose
morphisms are the sets of group homomorphisms  : Q → R for which there exists an
element x ∈ G satisfying x(Q, eQ) ⊆ (R, eR) and (u)= xux−1 for all u ∈ Q. The block
cohomology, introduced in [14], is deﬁned by
H ∗(G, b)= lim←
Q∈FG,b
H ∗(Q, k).
D.J. Benson, M. Linckelmann / Journal of Pure and Applied Algebra 197 (2005) 11–17 13
We identify H ∗(G, b) with the graded subalgebra of all “stable elements”  ∈ H ∗(P, k)
satisfying resPR()=res() for anymorphism : R→ P belonging to the categoryFG,b.
The algebra H ∗(G, b) is ﬁnitely generated graded commutative, and we denote by VG,b
the associated afﬁne variety, called the block variety of b (cf. [15]). Following [15], for
any ﬁnitely generated kGb-module M , there is a canonical graded algebra homomor-
phism H ∗(G, b) → Ext∗kGb(M,M), deﬁned using Hochschild cohomology of the block,
whose kernel determines the subvariety VG,b(M) of VG,b. If b is the principal block, then
H ∗(G, b) ∼= H ∗(G, k), and we write then VG and VG(M) instead of VG,b and VG,b(M),
respectively (that is, in the principal block case, the block varieties coincide with Carlson’s
cohomology varieties, introduced in [6,7]).
For any subgroup Q of P , the inclusion H ∗(G, b) ⊆ H ∗(P, k) followed by the re-
striction resPQ : H ∗(P, k) → H ∗(Q, k) is a graded algebra homomorphism, denoted by
rQ : H ∗(G, b)→ H ∗(Q, k). We denote by r∗Q : VQ → VG,b the induced map on varieties.
We refer to Benson [3] and Evens [10] for background material on cohomology varieties
of modules and Thévenaz [19] for block theory.
Theorem 1.1. LetG be a ﬁnite group, let b be a block of G, let P be a defect group of b, and
let i ∈ (kGb)P be a source idempotent. Let M be an indecomposable kGb-module. There
is a vertex Q of M contained in P and a source U of M such that U is a direct summand of
ResPQ(iM) and such that M is isomorphic to a direct summand of kGi⊗kQU. For any such
choice of a vertex and source of M we have
VG,b(M)= r∗Q(VQ(U)).
The proof of Theorem 1.1 will be given in Section 3.
Corollary 1.2. Foraﬁnitely generated indecomposable kGb-moduleM, the varietyVG,b(M)
is connected.
Proof. The variety VQ(U) is connected by Carlson’s theorem [8], and hence so is its image
under the map r∗Q, which is VG,b(M) by Theorem 1.1. 
Corollary 1.3. Every closed homogeneous subvariety V of VG,b is of the form VG,b(N) for
some ﬁnitely generated kGb-module N.
Proof. The inverse image in VP of V under the map r∗P : VP → VG,b is a closed homo-
geneous subvariety of VP ; thus, by Carlson [8], there is a ﬁnitely generated kP -module
U such that r∗P (VP (U)) = V . Using Green’s indecomposability theorem, we may write
U =⊕j∈J IndPQj (Uj ), where J is a ﬁnite indexing set, and where for any j ∈ J, Qj is a
subgroupofP andUj is an indecomposable kQj -module havingQj as vertex.Whenviewed
as kQj -kQj -bimodule, ikGi has a direct summand isomorphic to kQj , where j ∈ J (see
e.g. [16, 6.1(iii)]. Thus for any j ∈ J the left kGb-module kGi⊗kQj Uj has an indecompos-
able direct summandMj such that the restriction of iMj to kQj hasUj as direct summand.
Note that the kGb-module kGi⊗kQj Uj is a direct summand of kG⊗kQj Uj = IndGQj (Uj ).
14 D.J. Benson, M. Linckelmann / Journal of Pure and Applied Algebra 197 (2005) 11–17
Thus Mj has Qj as vertex and Uj as source, and VG,b(Mj ) = r∗Qj (VQj (Uj )) = r∗P
(VP (lndPQj (Uj ))) by Lemma 2.2 below. Thus N =
⊕
j∈J Mj satisﬁes VG,b(N)= V . 
The next corollary states that the block variety of a module survives the Green corre-
spondence with respect to a subgroup which controls fusion of Brauer pairs.
Corollary 1.4. Suppose the vertex Q of M is normal P. Set H = NG(Q) and denote by c
the unique block of H such that eQc= eQ. IfFG,b =FH,c then VG,b(M)= VH,c(f (M)),
where f (M) is the Green correspondent of M.
Proof. Since Q is normal in P , the pair (P, ep) is also a maximal c-Brauer pair. The
conditionFG,b =FH,c implies the equalityH ∗(G, b)=H ∗(H, c) as graded subalgebras
of H ∗(P, k). It is well known that the Green correspondent f (M) of M belongs to the
block c of H . Since every vertex and source of f (M) is also a vertex and source ofM , the
statement follows from Theorem 1.1. 
Note that if Q = P is abelian, we haveFG,b =FH,c, because in blocks with abelian
defect groups, the normaliser of a defect group controls fusion (cf. Alperin and Broué [1]).
Hence Corollary 1.4 applies in that situation. This particular case of 1.4 has independently
been proved by Kawai [12], who also observed [17, 2.1] (cf. 2.3 below).
2. Auxiliary results
Lemma 2.1. Let G be a ﬁnite group, let H be a subgroup of G and let M be a ﬁnitely
generated kG-module. We have (resGH )
∗(VH (resGH (M))) ⊆ VG(M).
Proof. Let  ∈ H ∗(G, k) such that ⊗ IdM is zero in Ext∗kG(M,M). Then resGH ()⊗ IdM
is zero in Ext∗kH (ResGH (M), Res
G
H (M)). Thus the annihilator of Ext
∗
kG(M,M) inH
∗(G, k)
is contained in the inverse image in H ∗(G, k) of the annihilator in H ∗(H, k) of
Ext∗kH (ResGH (M), Res
G
H (M)). Passing to varieties yields the statement. 
Lemma 2.2. Let G be a ﬁnite group, let H be a subgroup of G, and let N be a ﬁnitely
generated kH-module. We have
(resGH )
∗(VH (N))= VG(IndGH (N)).
Proof. This is a standard property of varieties for modules, and can be found for example
in Evens [10], Proposition 8.2.4 or Benson [2], Theorem 2.26.9. 
Theorem 2.3. Let G be a ﬁnite group, b a block of kG, P a defect group of b and i a source
idempotent of b in (kGb)P . The inclusion rP : H ∗(G, b) → H ∗(P, k) induces a ﬁnite
surjective morphism r∗P : VP → VG,b, and for any ﬁnitely generated kGb-module M we
have VG,b(M)= r∗P (VP (iM)), where iM is considered as kP-module.
D.J. Benson, M. Linckelmann / Journal of Pure and Applied Algebra 197 (2005) 11–17 15
Proof. See Linckelmann [17], Theorem 2.1 or Kawai [12]. 
It has been shown by Puig [18] that one can recover the categoryFG,b from the kP -kP -
bimodule structure of the source algebra ikGi of a block b (see also Linckelmann [16, 7.7]
for a proof). For any two subgroupsQ, R of P and any group homomorphism  : R→ Q
we denote by kQ the kR-kQ-bimodule kQ endowed with the regular action of kQ on the
right and the action of kR on the left obtained from restricting the regular action of kQ to
kR via .
Proposition 2.4. Let G be a ﬁnite group, let b be a block of G, let P be a defect group of b
and let i ∈ (kGb)P be a source idempotent. Let Q be a subgroup of P. Every indecomposable
direct summand of the source algebra ikGi as kP-kQ-bimodule is isomorphic to kP⊗kRkQ
for some subgroup R of P and some morphism  : R→ Q inFG,b.
Finally, we need the following statement from Linckelmann [13], which is about a certain
choice of vertices and sources of indecomposable modules (implying that vertices and
sources of indecomposable modules can be detected at the source algebra level).
Proposition 2.5. Let G be a ﬁnite group, let b be a block of G, let P be a defect group of b
and let i ∈ (kGb)P be a source idempotent of b. Any indecomposable kGb-module M has a
vertex Q contained in P and a kQ-source U such that U is a direct summand of ResPQ(iM)
and M is a direct summand of kGi⊗kQ U .
Proof. See 6.3 of Linckelmann [13]. 
3. Proof of Theorem 1.1
By Proposition 2.5 above there is a vertexQ ofM and a kQ-source U ofM such that U
is a direct summand of ResPQ(iM) and such thatM is a direct summand of kGi⊗kQ U . By
Proposition 2.4, we can write
ikGi ∼=
⊕
(R,)
kP ⊗
kR
kQ,
where (R,) runs over a family of pairs consisting of a subgroup R of P and a morphism
 : R→ Q belonging to the categoryFG,b.
Then iM is, as kP -module, a direct summand of
ikGi ⊗
kQ
U =
⊕
(R,)
IndPR(U),
where (R,) is as before. Thus,
VP (iM) ⊆ VP (ikGi ⊗
kQ
U)=
⋃
(R,)
VP (IndPR(U)).
16 D.J. Benson, M. Linckelmann / Journal of Pure and Applied Algebra 197 (2005) 11–17
By Lemma 2.2, we have VP (IndPR(U)) = (resPR)∗(VR(U)), for any (R,) as above.
Using Theorem 2.3 we get
VG,b(M)= r∗P (VP (iM)) ⊆
⊕
(R,)
r∗R(VR(U)).
For any (R,) occurring in this union, we have r∗R(VR(U)) = r∗(R)(ResQ(R)(U)) ⊆
r∗Q(VQ(U)), where the ﬁrst equality uses the stability of the elements in H ∗(G, b) with
respect to morphisms inFG,b, and where the second inclusion follows from Lemma 2.1.
Together, we get VG,b(M) ⊆ r∗Q(VQ(U)). The other inclusion follows trivially from the
fact that U is a direct summand of iM as kQ-module. 
4. A question
A theorem of Hida [11] gives a converse to Carlson’s connectedness theorem. It states
that given a connected closed homogeneous subvariety V of VG, there exists an indecom-
posable ﬁnitely generated kG-module N such that VG(N) = V . An obvious question is
whether the corresponding statement holds for a connected closed homogeneous subva-
riety V of the block variety VG,b. One might try to prove this statement by the method
used to prove Corollary 1.3. The problem is that it is not at all clear whether there is a
connected closed homogeneous subvariety of VP whose image in VG,b is equal to V . An
easy modiﬁcation of the proof of Corollary 1.3 shows that if V is the image of such a sub-
variety ofVP then there exists a ﬁnitely generated indecomposable kGb-moduleN such that
VG,b(N)= V .
References
[1] J.L. Alperin, M. Broué, Local methods in block theory, Ann. Math. 110 (1979) 143–157.
[2] D.J. Benson, Modular Representation Theory: New Trends and Methods, Lecture Notes in Mathematics,
Vol. 1081, Springer, Berlin, NewYork, 1984.
[3] D.J. Benson, Representations and Cohomology II, Cambridge University Press, Cambridge, UK, 1991.
[4] M. Broué, Brauer coefﬁcients of p-subgroups associated with a p-block of a ﬁnite group, J. Algebra 56
(1979) 365–383.
[5] M. Broué, L. Puig, Characters and local structure in G-algebras, J. Algebra 63 (1980) 306–317.
[6] J.F. Carlson, The complexity and varieties of modules, Lecture Notes in Math. 882 (1981) 415–422.
[7] J.F. Carlson, Varieties and the cohomology ring of a module, J. Algebra 85 (1983) 104–143.
[8] J.F. Carlson, The variety of an indecomposable module is connected, Invent. Math. 77 (1984) 291–299.
[9] H. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, NJ, 1956.
[10] L. Evens, The Cohomology of Groups, Oxford University Press, Oxford, 1991.
[11] A. Hida, Periodic modules and cohomology rings of ﬁnite groups, 2002, preprint.
[12] H. Kawai, Varieties for modules over a block of a ﬁnite group, 2001, preprint.
[13] M. Linckelmann, The source algebras of blocks with a Klein four defect group, J. Algebra 167 (1994)
821–854.
[14] M. Linckelmann, Transfer in Hochschild cohomology of blocks of ﬁnite groups, Algebras Rep.Theory 2
(1999) 107–135.
[15] M. Linckelmann, Varieties in block theory, J. Algebra 215 (1999) 460–480.
D.J. Benson, M. Linckelmann / Journal of Pure and Applied Algebra 197 (2005) 11–17 17
[16] M. Linckelmann, On splendid derived and stable equivalences between blocks of ﬁnite groups, J. Algebra
242 (2001) 819–843.
[17] M. Linckelmann, Quillen stratiﬁcation for block varieties, J. Pure Appl. Algebra 172 (2002) 257–270.
[18] L. Puig, Local fusions in block source algebras, J. Algebra (1986) 358–365.
[19] J. Thévenaz, G-algebras and Modular Representation Theory, Oxford Science Publications, Clarendon,
Oxford, 1995.


Vertex and source determine the block variety of an indecomposable module 

Crossref

Vertex and source determine the block variety of an indecomposable module

Benson, D. J.

Linckelmann, M.

City Research Online

Benson, D. J. & Linckelmann, M. (2005). Vertex and source determine the block variety of an indecomposable module. Journal of Pure and Applied Algebra, 197(1-3), 11 - 17. doi: 10.1016/j.jpaa.2004.08.032 <http://dx.doi.org/10.1016/j.jpaa.2004.08.032>City Research OnlineOriginal citation: Benson, D. J. & Linckelmann, M. (2005). Vertex and source determine the block variety of an indecomposable module. Journal of Pure and Applied Algebra, 197(1-3), 11 - 17. doi: 10.1016/j.jpaa.2004.08.032 <http://dx.doi.org/10.1016/j.jpaa.2004.08.032>Permanent City Research Online URL: http://openaccess.city.ac.uk/1906/ Copyright & reuseCity  University  London has developed City  Research Online  so that  its  users  may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders.  All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf you have any enquiries about any aspect of City Research Online, or if you wish to make contact  with the author(s) of this paper, please email the team at publications@city.ac.uk.VERTEX AND SOURCE DETERMINE THE BLOCKVARIETY OF AN INDECOMPOSABLE MODULEDavid J. Benson and Markus LinckelmannAbstract. The block variety VG,b(M) of a finitely generated indecomposable mod-ule M over the block algebra of a p-block b of a finite group G, introduced in [15],can be computed in terms of a vertex and a source of M . We use this to show thatVG,b(M) is connected, and that every closed homogeneous subvariety of the affinevariety VG,b defined by block cohomology H∗(G, b) (cf. [14]) is the variety of a mod-ule over the block algebra. This is analogous to the corresponding statements onCarlson’s cohomology varieties in [8].1. IntroductionThe theory of the cohomology variety VG(M) for a finitely generated module Mover the modular group algebra of a finite group G was developed by Carlson [6], [7],[8] and others back in the nineteen eighties. It has played a major role in recent yearsin the modular representation theory of finite groups. One of the problems with thetheory has been that it has been much more fruitful as a way of proving theoremsabout the principal block than about more general blocks.To address this problem, the second author [15] recently introduced the conceptof the block variety VG,b(M) of a finitely generated module M for a block b of afinite group G. The definition of the ambient variety VG,b involves a variation of theCartan–Eilenberg stable element method in ordinary cohomology for the Brauer pairsin the block. But the definition of VG,b(M) involves going through the Hochschild co-homology of the block as an intermediate step. In [17], the dependence on Hochschildcohomology for the calculation of block varieties is removed. It is shown in that paperthat the block variety ofM is the image in VG,b of the variety at the level of the defectgroup. But first, the module must be cut down using the idempotent for the sourcealgebra of the block, in the sense of Puig. An example is given there, to show thatthis step is really necessary.In this paper, we describe how to calculate the block variety of a module in termsof the vertex and source of the module. We show that it is given as the image inVG,b of the variety of the source of M as a module for the vertex. Actually, there isa subtlety here: the vertex and source are only well defined up to G-conjugacy, andwe must be careful how we choose a P -conjugacy class.Taken together with the stratification theorem for VG,b [17] and Carlson’s theory ofrank varieties, our theorem implies that VG,b(M) can be calculated without knowinganything about cohomology, just as was the case for VG(M).Our theorem enables us to prove some basic properties for VG,b(M), includingthe analog of Carlson’s theorem [8] that the variety of an indecomposable module isTypeset by AMS-TEX12 DAVID J. BENSON AND MARKUS LINCKELMANNconnected. We also show that every closed homogeneous subvariety of VG,b can berealised as VG,b(N) for a suitably chosen module N . Finally, we show that the blockvariety of a module survives the Green correspondence with respect to a subgroupwhich controls fusion of Brauer pairs in the block.Throughout this paper, k is an algebraically closed field of prime characteristic p.Let G be a finite group. A block of G is a primitive idempotent b in Z(kG). FollowingAlperin–Broue´ [1], a b-Brauer pair is a pair (Q, e) consisting of a p-subgroup Q ofG and a block e of CG(Q) such that BrQ(b)e = e, where BrQ : (kG)Q → CG(Q) isthe Brauer homomorphism [4]. It is shown in [1], that the set of b-Brauer pairs is apartially ordered G-set such that G acts transitively on the set of maximal b-Brauerpairs. If (P, e) is a maximal b-Brauer pair, then P is called a defect group of b (thisnotion is due to Brauer, although he did not state it in these terms). If P is a defectgroup of b, then P is maximal with respect to BrP (b) 6= 0; in particular, there is aprimitive idempotent i ∈ (kGb)P such that BrP (i) 6= 0. Such an idempotent is calleda source idempotent of b (with respect to P ). By Broue´–Puig [5, Theorem 1.8], forany subgroup Q of P , there is a unique block eQ of CG(Q) such that BrQ(i)eQ 6= 0.Then (Q, eQ) is a b-Brauer pair, and eQ is also the unique block of CG(Q) such that(Q, eQ) ⊆ (P, eP ).We denote by FG,b the category whose objects are the subgroups of P , and whosemorphisms are the sets of group homomorphisms ϕ : Q → R for which there existsan element x ∈ G satisfying x(Q, eQ) ⊆ (R, eR) and ϕ(u) = xux−1 for all u ∈ Q. Theblock cohomology, introduced in [14], is defined byH∗(G, b) = lim←−Q∈FG,bH∗(Q, k) .We identify H∗(G, b) with the graded subalgebra of all “stable elements” ζ ∈ H∗(P, k)satisfying resPR(ζ) = resϕ(ζ) for any morphism ϕ : R → P belonging to the categoryFG,b.The algebra H∗(G, b) is finitely generated graded commutative, and we denote byVG,b the associated affine variety, called the block variety of b (cf. [15]). Following[15], for any finitely generated kGb-module M , there is a canonical graded algebrahomomorphism H∗(G, b)→ Ext∗kGb(M,M), defined using Hochschild cohomology ofthe block, whose kernel detemines the subvariety VG,b(M) of VG,b. If b is the principalblock, then H∗(G, b) ∼= H∗(G, k), and we write then VG and VG(M) instead of VG,band VG,b(M), respectively (that is, in the principal block case, the block varietiescoincide with Carlson’s cohomology varieties, introduced in [6], [7]).For any subgroup Q of P , the inclusion H∗(G, b) ⊆ H∗(P, k) followed by therestriction resPQ : H∗(P, k) → H∗(Q, k) is a graded algebra homomorphism, denotedby rQ : H∗(G, b) → H∗(Q, k). We denote by r∗Q : VQ → VG,b the induced mapon varieties. We refer to Benson [3] and Evens [10] for background material oncohomology varieties of modules and The´venaz [20] for block theory.3Theorem 1.1. Let G be a finite group, let b be a block of G, let P be a defect groupof b, and let i ∈ (kGb)P be a source idempotent. Let M be an indecomposable kGb-module. There is a vertex Q of M contained in P and a source U of M such that U isa direct summand of ResPQ(iM) and such that M is isomorphic to a direct summandof kGi ⊗kQU . For any such choice of a vertex and source of M we haveVG,b(M) = r∗Q(VQ(U)).The proof of Theorem 1.1 will be given in Section 3.Corollary 1.2. For a finitely generated indecomposable kGb-module M , the varietyVG,b(M) is connected.Proof. The variety VQ(U) is connected by Carlson’s theorem [8], and hence so is itsimage under the map r∗Q, which is VG,b(M) by Theorem 1.1. Corollary 1.3. Every closed homogeneous subvariety V of VG,b is of the formVG,b(N) for some finitely generated kGb-module N .Proof. The inverse image in VP of V under the map r∗P : VP → VG,b is a closedhomogeneous subvariety of VP ; thus, by Carlson [8], there is a finitely generated kP -module U such that r∗P (VP (U)) = V . Using Green’s indecomposability theorem, wemay write U =⊕j∈J IndPQj(Uj), where J is a finite indexing set, and where for anyj ∈ J , Qj is a subgroup of P and Uj is an indecomposable kQj-module having Qj asvertex. When viewed as kQj-kQj-bimodule, ikGi has a direct summand isomorphicto kQj , where j ∈ J (see e.g. [16, 6.1(iii)]). Thus for any j ∈ J the left kGb-modulekGi ⊗kQjUj has an indecomposable direct summand Mj such that the restriction ofiMj to kQj has Uj as direct summand. Note that the kGb-module kGi ⊗kQjUj is adirect summand of kG ⊗kQjUj = IndGQj(Uj). Thus Mj has Qj as vertex and Uj assource, and VG,b(Mj) = r∗Qj(VQj (Uj)) = r∗P (VP (IndPQj(Uj))) by Lemma 2.2 below.Thus N =⊕j∈J Mj satisfies VG,b(N) = V . The next corollary states that the block variety of a module survives the Greencorrespondence with respect to a subgroup which controls fusion of Brauer pairs.Corollary 1.4. Suppose the vertex Q of M is normal P . Set H = NG(Q) and denoteby c the unique block of H such that eQc = eQ. If FG,b = FH,c then VG,b(M) =VH,c(f(M)), where f(M) is the Green correspondent of M .Proof. Since Q is normal in P , the pair (P, eP ) is also a maximal c-Brauer pair.The condition FG,b = FH,c implies the equality H∗(G, b) = H∗(H, c) as gradedsubalgebras of H∗(P, k). It is well known that the Green correspondent f(M) of Mbelongs to the block c of H. Since every vertex and source of f(M) is also a vertexand source of M , the statement follows from Theorem 1.1. 4 DAVID J. BENSON AND MARKUS LINCKELMANNNote that if Q = P is abelian, we have FG,b = FH,c, because in blocks withabelian defect groups, the normaliser of a defect group controls fusion (cf. Alperin–Broue´ [1]). Hence Corollary 1.4 applies in that situation. This particular case of 1.4has independently been proved by H. Kawai [12], who also observed [17, 2.1] (cf. 2.3below).2. Auxiliary resultsLemma 2.1. Let G be a finite group, let H be a subgroup of G and let M be a finitelygenerated kG-module. We have (resGH)∗(VH(resGH(M))) ⊆ VG(M).Proof. Let ζ ∈ H∗(G, k) such that ζ ⊗ IdM is zero in Ext∗kG(M,M). Then resGH(ζ)⊗IdM is zero in Ext∗kH(ResGH(M),ResGH(M)). Thus the annihilator of Ext∗kG(M,M) inH∗(G, k) is contained in the inverse image in H∗(G, k) of the annihilator in H∗(H, k)of Ext∗kH(ResGH(M),ResGH(M)). Passing to varieties yields the statement. Lemma 2.2. Let G be a finite group, let H be a subgroup of G, and let N be a finitelygenerated kH-module. We have(resGH)∗(VH(N)) = VG(IndGH(N)).Proof. This is a standard property of varieties for modules, and can be found forexample in Evens [10], Proposition 8.2.4 or Benson [2], Theorem 2.26.9. Theorem 2.3. Let G be a finite group, b a block of kG, P a defect group of b and i asource idempotent of b in (kGb)P . The inclusion rP : H∗(G, b)→ H∗(P, k) induces afinite surjective morphism r∗P : VP → VG,b, and for any finitely generated kGb-moduleM we have VG,b(M) = r∗P (VP (iM)) , where iM is considered as kP -module.Proof. See Linckelmann [17], Theorem 2.1 or Kawai [12]. It has been shown by Puig [19] that one can recover the category FG,b from the kP -kP -bimodule structure of the source algebra ikGi of a block b (see also Linckelmann[16, 7.7] for a proof). For any two subgroups Q, R of P and any group homomorphismϕ : R → Q we denote by ϕkQ the kR-kQ-bimodule kQ endowed with the regularaction of kQ on the right and the action of kR on the left obtained from restrictingthe regular action of kQ to kR via ϕ.Proposition 2.4. Let G be a finite group, let b be a block of G, let P be a defectgroup of b and let i ∈ (kGb)P be a source idempotent. Let Q be a subgroup of P .Every indecomposable direct summand of the source algebra ikGi as kP -kQ-bimoduleis isomorphic to kP ⊗kRϕkQ for some subgroup R of P and some morphism ϕ : R→ Qin FG,b.Finally, we need the following statement from Linckelmann [13], which is abouta certain choice of vertices and sources of indecomposable modules (implying that5vertices and sources of indecomposable modules can be detected at the source algebralevel).Proposition 2.5. Let G be a finite group, let b be a block of G, let P be a defectgroup of b and let i ∈ (kGb)P be a source idempotent of b. Any indecomposable kGb-module M has a vertex Q contained in P and a kQ-source U such that U is a directsummand of ResPQ(iM) and M is a direct summand of kGi ⊗kQU .Proof. See 6.3 of Linckelmann [13]. 3. Proof of Theorem 1.1By Proposition 2.5 above there is a vertex Q of M and a kQ-source U of M suchthat U is a direct summand of ResPQ(iM) and such that M is a direct summand ofkGi ⊗kQU . By Proposition 2.4, we can writeikGi ∼=⊕(R,ϕ)kP ⊗kRϕkQ ,where (R,ϕ) runs over a family of pairs consisting of a subgroup R of P and amorphism ϕ : R→ Q belonging to the category FG,b.Then iM is, as kP -module, a direct summand ofikGi ⊗kQU =⊕(R,ϕ)IndPR(ϕU) ,where (R,ϕ) is as before. ThusVP (iM) ⊆ VP (ikGi ⊗kQU) =⋃(R,ϕ)VP (IndPR(ϕU)) .By Lemma 2.2, we have VP (IndPR(ϕU)) = (resPR)∗(VR(ϕU)), for any (R,ϕ) as above.Using Theorem 2.3 we getVG,b(M) = r∗P (VP (iM)) ⊆⊕(R,ϕ)r∗R(VR(ϕU)) .For any (R,ϕ) occurring in this union, we have r∗R(VR(ϕU)) = r∗ϕ(R)(ResQϕ(R)(U)) ⊆r∗Q(VQ(U)), where the first equality uses the stability of the elements in H∗(G, b) withrespect to morphisms in FG,b, and where the second inclusion follows from Lemma2.1. Together, we get VG,b(M) ⊆ r∗Q(VQ(U)). The other inclusion follows triviallyfrom the fact that U is a direct summand of iM as kQ-module. 6 DAVID J. BENSON AND MARKUS LINCKELMANN4. A questionA theorem of Hida [11] gives a converse to Carlson’s connectedness theorem. Itstates that given a connected closed homogeneous subvariety V of VG, there existsan indecomposable finitely generated kG-module N such that VG(N) = V . An ob-vious question is whether the corresponding statement holds for a connected closedhomogeneous subvariety V of the block variety VG,b. One might try to prove thisstatement by the method used to prove Corollary 1.3. The problem is that it is notat all clear whether there is a connected closed homogeneous subvariety of VP whoseimage in VG,b is equal to V . An easy modification of the proof of Corollary 1.3 showsthat if V is the image of such a subvariety of VP then there exists a finitely generatedindecomposable kGb-module N such that VG,b(N) = V .References1. J. L. Alperin, M. Broue´, Local methods in block theory, Ann. Math. 110 (1979), 143–157.2. D. J. Benson, Modular Representation Theory: New Trends and Methods, Lecture Notes inMathematics 1081, Springer-Verlag, Berlin/New York, 1984.3. D. J. Benson, Representations and Cohomology II, Cambridge University Press, Cambridge, UK,1991.4. M. Broue´, Brauer coefficients of p-subgroups associated with a p-block of a finite group, J.Algebra 56 (1979), 365–383.5. M. Broue´, L. Puig, Characters and local structure in G-algebras, J. Algebra 63 (1980), 306–317.6. J. F. Carlson, The complexity and varieties of modules, in: Lecture Notes in Mathematics 882(1981), 415–422.7. J. F. Carlson, Varieties and the cohomology ring of a module, J. Algebra 85 (1983), 104–143.8. J. F. Carlson, The variety of an indecomposable module is connected, Invent. Math. 77 (1984),291–299.9. H. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press, 1956.10. L. Evens, The Cohomology of Groups, Oxford University Press, 1991.11. A. Hida, Periodic modules and cohomology rings of finite groups, preprint (2002).12. H. Kawai, Varieties for modules over a block of a finite Group, preprint (2001).13. M. Linckelmann, The source algebras of blocks with a Klein four defect group, J. Algebra 167(1994), 821–854.14. M. Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebras Rep.Theory 2 (1999), 107–135.15. M. Linckelmann, Varieties in block theory, J. Algebra 215 (1999), 460–480.16. M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, J.Algebra 242 (2001), 819–843.17. M. Linckelmann, Quillen stratification for block varieties, J. Pure Appl. Algebra 172 (2002),257–270.18. L. Puig, Pointed groups and construction of characters, Math. Z. 176 (1981), 265–292.19. L. Puig, Local fusions in block source algebras, J. Algebra (1986), 358–365.20. J. The´venaz, G-algebras and Modular Representation Theory, Oxford Science Publications,Clarendon, Oxford, 1995.7David J. BensonUniversity of GeorgiaDepartment of MathematicsAthens, Georgia 30602U.S.A.Markus LinckelmannOhio State UniversityMath Tower231, West 18th AvenueColumbus, Ohio 43210U.S.A.

file:///data/remote/core/dit/data/elsevier/pdf/ad8/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAyMjQwNDkwNDAwMTc4MQ==.pdf

Vertex and source determine the block variety of an indecomposable module

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Crossref

City Research Online

Elsevier - Publisher Connector