STABLE ENDOMORPHISM RINGS

OF IDEMPOTENT E-MODULES

Peteris Daugulis

Department of Mathematics, University of Georgia, Athens, Georgia 30603

February 20,1998

Abstract. Let G be a finite group, k an algebraically closed field of characteristic dividing |G|, C a thick subcategory of the stable category of finitely generated kG-modules corresponding to a collection of closed homogeneous subvarieties of the maximal ideal spectrum of H^{∗}(G, k) which is closed under finite unions and specialization, E_{C} the idempotent module corresponding to C. We show that the stable endomorphism ring of E_{C} is a local ring if C corresponds to a connected variety or an infinite collection of varieties which is connected in an appropriate sense. We show that noninvertible stable endomorphisms of E_{C} are locally nilpotent if C corresponds to a connected variety given by a sequence of homogeneous parameters. We relate the structure of the stable endomorphism ring of E_{C} to those of the finitely generated modules of the colimit and give a few examples for elementary abelian groups of low rank.

1. Introduction.

Let G be a finite group and k an algebraically closed field of characteristic di-

viding |G|. In this paper by M ⊗ N we always mean M ⊗_{k} N. Let V be a closed

homogeneous subvariety of the maximal ideal spectrum of H^{∗}(G, k). In [Rick]

Rickard gives a description of an exact sequence of kG-modules:

Key words and phrases. stable module categories, idempotent modules.

Typeset by AMS-TEX

0 ⟶ E_{V} ⟶ k ⊕ proj ⟶ F_{V} ⟶ 0

where E_{V} and F_{V} are (usually) infinitely generated nonprojective kG-modules having the properties expressed in the following theorem.

(1.1) Theorem.

(1) E_{V} and F_{V} are idempotent modules in the following sense:

E_{V} ⊗ E_{V} ≃ E_{V} ⊕ proj, F_{V} ⊗ F_{V} ≃ F_{V} ⊕ proj, E_{V} ⊗ F_{V} = proj.

(2) For any finitely generated kG module M satisfying V_{G}(M) ⊂ V, we have that M ⊗ E_{V} ≃ M ⊕ proj.

In [Rick] Rickard also gives a construction of E_{V} and F_{V} as colimits in the stable module category of finitely generated modules.

The idempotent modules have already played an important role in the solution of some problems of modular representation theory(see [BCR1], [BCR2], [BM], [W]).

In this paper we study the stable endomorphism rings of the E-modules. A motivation for this study is the property (2) of (1.1). It tells us that a stable endomorphism f of E_{V} induces a functor id ⊗ f in the thick subcategory C_{V} of modules having varieties contained in V. More precisely, given a finitely generated kG-module M in C_{V} and a stable map f ∈ \underline{End}_{kG}(E_{V}) we define the action of id ⊗ f on M as the stable map

M ⊗ E_{V} ⟶^{id_{M} ⊗f} M ⊗ E_{V} ,

STABLE ENDOMORPHISM RINGS OF IDEMPOTENT E-MODULES 3

on morphisms in C_{V} id ⊗ f is defined by sending a stable map M ⟶^{ϕ} N to the stable map represented by

M ⊗ E_{V} ⟶^{ϕ⊗f} N ⊗ E_{V}

We hope that these functors can provide new information about the thick subcategory C_{V} and serve as a source of invariants in representation theory.

The main general results of this paper are contained in the section ”Locality”. One of the main results is that \underline{End}_{kG}(E_{V}) is a local ring if V is a connected projective variety. The same property holds in the case when E_{C} is the idempotent E-module for a thick subcategory C which corresponds to a collection of subvarieties of V_{G}(k) which is in some sense connected. The idea behind the proofs of these results is to consider functors of form f ⊗ id_{M} where f ∈ \underline{End}_{kG}(E_{C}) and M ∈ C. For a suitable choice of an indecomposable M the functor f ⊗ id_{M} induces a noninvertible stable endomorphism of M and the proof is reduced to the fact that stable endomorphism rings of indecomposable finitely generated modules are local. The corresponding result for the infinite collection of varieties is proved using the same idea for a suitable infinite collection. The choice of M is based on a result of Akihiko Hida which we obtained independently. We also prove that under certain conditions noninvertible stable endomorphisms are locally nilpotent and the variety of the last element of the triangle corresponding to a noninvertible stable endomorphism is equal to the variety of E_{C}.

In the next section we prove a proposition relating the structure of the stable endomorphism ring of the E-module to those of the finitely generated modules occurring in the definition of the E-module as a colimit. More precisely, if compositions of any two noninvertible stable endomorphisms is equal to 0 for finitely

4 PETERIS DAUGULIS

generated limiting modules with sufficiently large indices then compositions of any two noninvertible stable endomorphisms is 0 for E_{C}.

In the section ”Examples” we give examples of the E-modules for abelian pgroups of p-rank 2 and 3 using our modification of diagrammatic technique. In these cases the stable endomorphism rings are either power series rings, tensor products of a power series ring and the two dimensional antisymmetric algebra or local rings with square-zero radical.

2. Background.

Let mod(kG) be the module category of finitely generated kG-modules and Mod(kG) be the category of all (not necessarily finitely generated) kG-modules.

This paper is written mostly in the language of the stable module categories stmod(kG) and StMod(kG) which have the same objects as mod(kG) and Mod(kG) but different morphisms: maps which factor through projectives are being ignored. It is customary to underline the stable morphism spaces, thus

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/P Hom_{kG}(M, N)

where P Hom_{kG}(M, N) is the set of kG-maps from M to N which factor through a projective(not necessarily finitely generated in the case of StMod(kG)) module. The stable categories have the structure of triangulated categories: instead of working with exact sequences we work with distinguished triangles (see [Ha] and [C1]).

Given two kG-modules M and N we study their n-fold extension groups Ext_{kG}^{∗}(M, N)

STABLE ENDOMORPHISM RINGS OF IDEMPOTENT E-MODULES 5

which alternatively are identified with \underline{Hom}_{kG}(Ω^{∗}(M), N). An important special case is Ext_{kG}^{∗}(k, k) which is identified with the cohomology ring H^{∗}(G, k).

Given a finitely generated kG-module M we consider the action of Ext_{kG}^{∗}(k, k) on Ext_{kG}^{∗}(M, M) by tensor mutiplication. The annihilator ideal of this action defines a closed homogeneous algebraic variety in the maximal ideal spectrum of Ext_{kG}^{∗}(k, k) or Ext_{kG}^{even}(k, k) called the variety of M and denoted by V_{G}(M). Note that V_{G}(k) can be identified with the whole maximal ideal spectrum of Ext_{kG}^{∗}(k, k) or Ext_{kG}^{even}(k, k). For an extended treatment see [B].

Alternatively V_{G}(M) can be computed using restrictions of M to certain subalgebras of kG isomorphic to the group algebra of the elementary abelian p group of rank one(cyclic shifted subgroups). Most easily this is done for elementary abelian groups: if G = E = (ℤ/p)^{r} = ⟨X_{1}, ..., X_{r}⟩ then we first define V_{E}^{♯} to be the linear subspace in Rad(kE) spanned by {X_{1} −1, ..., X_{r} −1⟩}, given a finite generated kEmodule M we define the rank variety V_{E}^{♯}(M) to be {0} ∪ {v ∈ V_{E}^{♯}|M ↓_{⟨1+v⟩} is not free}. It can be proved that V_{E}^{♯}(M) can be identified with V_{E}(M) and that V_{E}^{♯}(M) is isogenous to V_{E}(M) under that identification. For a group G which is not elementary abelian V_{G}(M) can be determined by looking at the restriction of the module to the elementary abelian subgroups of G by the Alperin-Evens Theorem(see [B]).

Given ζ ∈ Ext_{kG}^{n}(k, k) represented as a map ζ : Ω^{n}(k) ⟶ k we define L_{ζ} to be the kernel of ζ as shown:

0 ⟶ L_{ζ} ⟶ Ω^{n}(k) ⟶^{ζ} k ⟶ 0

A few years ago the notion of variety for modules was generalized by Benson, Carlson and Rickard to work for objects in Mod(kG) (see [BCR1]). It has been

6 PETERIS DAUGULIS

established that the right generalization of variety in the case of infinitely generated modules is a collection of closed homogeneous irreducible varieties in V_{G}(k). We denote it by V_{G}(M). We first give a list of properties of V:

(2.1) Theorem.

(0) If M is a finitely generated kG-module then

V_{G}(M) = {V ∈ P roj(V_{G}(k))|V ⊂ V_{G}(M)}

where by P roj(W) we mean the set of all closed homogeneous irreducible subvarieties of the maximal ideal spectrum of H^{even}(G, k) which are contained in W,

(1) V_{G}(M_{1} ⊕ M_{2}) = V_{G}(M_{1}) ∪ V_{G}(M_{2})

(2) If M_{1} ⟶ M_{2} ⟶ M_{3} ⟶ Ω^{−1}(M_{1}) is a triangle in StMod(kG) then

V_{G}(M_{1}) ⊂ V_{G}(M_{2}) ∪ V_{G}(M_{3})

(3) V_{G}(M_{1} ⊗ M_{2}) = V_{G}(M_{1}) ∩ V_{G}(M_{2})

(4) V_{G}(M) = V_{G}(Ω^{n}(M))) for any n ∈ ℤ

(5) A kG-module M is projective if and only if V_{G}(M) = ∅.

The properties (1), (2), (3), (4), (5) have their analogs for the classic variety V_{G}, in fact, V_{G} was defined so that, among other things, these properties hold. The classic variety should be thought as a special case of the general variety in an appropriate sense. In this paper we will refer to (2.1) even when talking about finitely generated modules.

For an elementary abelian group E = (ℤ/p)^{r} let V_{E}^{♯}(k) be the set of all nonzero closed homogeneous irreducible subvarieties of V_{E}(k). For an arbitrary kE-module

STABLE ENDOMORPHISM RINGS OF IDEMPOTENT E-MODULES 7

M we define V_{E}^{♯}(M)to be the set of all V ∈ V_{E}^{♯}(k) such that for any algebraically closed extension K of transcendence degree degree at least dim(V) and any generic point α ∈ V_{E}^{♯}(K) for V we have α ∈ V_{E}^{♯}(K ⊗ M). It is shown in [BCR1] that V_{E}^{♯}(M) does not depend on the field extension and is isogenous to V_{E}(M).

An important development of the last years in representation and cohomology theory of finite groups has been the discovery of idempotent modules. As we have mentioned in the introduction in [Rick] Rickard describes a construction of a triangle

E_{C} ⟶ k ⟶ F_{C} ⟶ Ω^{−1}(E_{C})

associated to a thick(closed under taking direct summands) triangulated subcategory C ⊂ stmod(kG). In particular we can construct such a triangle for any closed homogeneous subvariety V of V_{G}(k) since there exists a thick subcategory C_{V} = {M ∈ stmod(kG)|V_{G}(M) ∈ V }.

In his paper Rickard gives two constructions of the E_{C} and F_{C} modules. The first construction is rather abstract and works for arbitrary triangulated categories. Most of the properties of E_{C} and F_{C} modules are proved using the abstract construction. However, it would be hard, if not impossible to compute these modules in practice since the abstract construction involves consideration of all modules in the corresponding thick subcategory. Fortunately, for C corresponding to a closed homogeneous variety in V_{G}(k) there is an explicit construction using modules and maps. In particular, if ζ ∈ Ext_{kG}^{n}(k, k) is a nonnilpotent homogeneous element of degree n then F_{V_{ζ}} is defined as a colimit:

(2.2)  F_{V_{ζ}} = colim(k ⟶^{ζ} Ω^{−n}(k) ⟶^{ζ} Ω^{−2n}(k) ⟶^{ζ} Ω^{−3n}(k) ⟶ ...)

Note that F_{V_{ζ}} is defined together with a map η : k ⟶ F_{V_{ζ}}. Now we define E_{V_{ζ}} as the last element of the triangle starting with η:

E_{V_{ζ}} ⟶ k ⟶^{η} F_{V_{ζ}} ⟶ Ω^{−1}(E_{V_{ζ}})

If V = ⋂_{i=1}^{d} V_{ζ_{i}} then E_{V} = ⨂_{i=1}^{d} E_{V_{ζ}}.

The idempotent modules have a number of interesting and universal properties. It must be noted that except for special cases such as the case of p-rank equal to 1 or V = V_{G}(k) the idempotent modules are infinitely generated.

3. Locality.

In this section we first prove that the stable endomorphism ring of the E_{C} is local provided the variety or the collection of varieties of modules generating C is in a suitable sense connected. We also prove that endomorphisms representing noninvertible stable endomorphisms of E-modules under certain conditions are locally nilpotent.

We first state a result of Hida. For the sake of completeness we give a proof similar to that in [Hi].

(3.1) Lemma([Hi]). Let G be a finite group, k an algebraically closed field of characteristic dividing |G| and U a connected closed homogeneous subvariety of V_{G}(k). There exists an indecomposable finitely generated kG-module M such that V_{G}(M) = U.

Proof. Let U = ⋂_{i=1}^{d} V_{ζ_{i}} = ⋃_{j=1}^{e} W_{j} where V_{ζ_{i}}’s are the hypersurfaces and W_{j}’s are

STABLE ENDOMORPHISM RINGS OF IDEMPOTENT E-MODULES 9

the components of U. Let L = ⨂_{i=1}^{d} L_{ζ_{i}}. For any indecomposable direct summand N of L and for any i there is a power of ζ_{i} that annihilates the cohomology of N. We choose a sequence{n_{1}, ..., n_{d}} such that ζ_{i}^{n_{i}} annihilates the cohomology of each indecomposable direct summand of L. Let L′ = ⨂_{i=1}^{d} L_{ζ_{i}^{n_{i}}}, note that V_{G}(L′) = U. At this point we recall a result of Carlson:

(3.2) Lemma([C2]). Let M is a finitely generated kG-module, let ζ be an element in Ext_{kG}^{n}(k, k). If ζ annihilates the cohomology of M then

L_{ζ} ⊗ M ≃ Ω^{n}(M) ⊕ Ω(M) ⊕ proj.

In our case we have that for any indecomposable direct summand N of L

L_{ζ_{i}^{n_{i}}} ⊗ N ≃ Ω^{n_{i}deg(ζ_{i})}N ⊕ Ω(N) ⊕ proj.

Since L′ is a tensor product of L_{ζ_{i}^{n_{i}}} ’s we can obtain L′ ⊗ N by consecutively multiplying N by the tensor factors of L′. It follows that L′ ⊗ N is a direct sum of Heller translates of N and a projective module. For any indecomposable direct summand N′ of L′ we have that N′ ⊗ N is a direct summand of L′ ⊗ N and thus by the Krull-Schmidt theorem it is isomorphic to a(possibly empty) direct sum of Heller translates of N and a projective module. By the analogs of (3) and (4) of (2.1) for the varieties of finitely generated modules(see [B]) we have that either V_{G}(N) ⊂ V_{G}(N′) or V_{G}(N) ∩ V_{G}(N′) = ∅.

We remind the reader that A|B means ”A is isomorphic to a direct summand of B”. We remark that given a component W of U there exist indecomposable kGmodules N_{W} and N′_{W} such that N_{W} |L, N′_{W} |L′, W ⊂ V_{G}(N_{W}) and W ⊂ V_{G}(N′_{W}).

10 PETERIS DAUGULIS

This is because an irreducible variety cannot be expressed as a union of two varieties and V_{G}(L) = V_{G}(L′) = ⋃_{N} V_{G}(N) = ⋃_{N′} V_{G}(N′) where the indices run over the sets of indecomposable summands of L and L′.

If U has only one component then it is irreducible and every indecomposable nonprojective direct summand N of L has the property that V_{G}(N) = U. Suppose that U has at least two components, fix two components W and W′ and choose two indecomposable kG-modules N_{W} and N′_{W′} such that N_{W} |L, N′_{W′}|L′, W ⊂ V_{G}(N_{W}) and W′ ⊂ V_{G}(N′_{W′}). Since U is connected there exists a component W_{1} of U such that W_{1} ∩ V_{G}(N′_{W′}) ≠ ∅ and W_{1} ⊄ V_{G}(N′_{W′}). Choose an indecomposable direct summand N_{W_{1}} of L such that W_{1} ⊂ V_{G}(N_{W_{1}}), we have that neither V_{G}(N_{W_{1}}) ⊂ V_{G}(N′_{W′}) nor V_{G}(N_{W_{1}}) ∩ V_{G}(N′_{W′}) = ∅ which is a contradiction. Thus V_{G}(N_{W}) ⊂ V_{G}(N′_{W′}). Since W is arbitrary it follows that V_{G}(N′_{W′}) = U and the lemma is proved. QED.

(3.3) Remark. It was known for quite a long time that given any closed homogeneous subvariety V ⊂ V_{G}(k) there exists a finitely generated kG-module M with V_{G}(M) = V. (3.2) thus strenghthens this result.

Turning now to idempotent modules we first give an observation noticed some time ago by several people.

(3.4) Proposition. (1)\underline{End}_{kG}(E_{C}) is isomorphic as a ring to

a) the abelian group \underline{Hom}_{kG}(E_{C}, k) with the multiplicative structure given by the tensor multiplication of maps,

STABLE ENDOMORPHISM RINGS OF IDEMPOTENT E-MODULES 11

b) the abelian group \underline{End}_{kG}(E_{C}) with the multiplicative structure given by the tensor multiplication of maps. (2)\underline{End}_{kG}(F_{C}) is isomorphic as a ring to

a) the abelian group \underline{Hom}_{kG}(k, F_{C}) with the multiplicative structure given by the tensor multiplication of maps,

b) the abelian group \underline{End}_{kG}(F_{C}) with the multiplicative structure given by the tensor multiplication of maps.

Proof. It easily follows from [R] that \underline{End}_{kG}(E_{V}) and \underline{End}_{kG}(F_{V}) as abelian groups are isomorphic to \underline{Hom}_{kG}(E_{V} , k) and \underline{Hom}_{kG}(k, F_{V}). The maps \underline{End}_{kG}(E_{V}) ⟶ \underline{Hom}_{kG}(E_{V}), k) and \underline{End}_{kG}(F_{V}) ⟶ \underline{Hom}_{kG}(k, F_{V})) are given by compositions with ϵ and η, respectively. The inverses are given by the universal properties of the idempotent modules.

For any f ∈ \underline{End}_{kG}(E_{V}) and g ∈ \underline{End}_{kG}(F_{V}) we have commutative squares in StMod(kG):

[figure: E_{V} ⊗ E_{V}; id_{E_{V}} ⊗ϵ; f⊗id_{E_{V}}; E_{V} ⊗ E_{V}; id_{E_{V}} ⊗ϵ; E_{V} ⊗ k; f⊗id_{k}; E_{V} ⊗ k]

and

[figure: F_{V} ⊗ F_{V}; g⊗id_{F_{V}}; F_{V} ⊗ F_{V}; F_{V} ⊗ k; id_{F_{V}} ⊗η; g⊗id_{k}; F_{V} ⊗ k; id_{F_{V}} ⊗η]

with all vertical maps being stable isomorphisms. It follows that f and g are stably

12 PETERIS DAUGULIS

isomorphic to f ⊗id_{E_{V}} and g⊗id_{F_{V}}, respectively. Thus for any two elements f_{1} and f_{2} in \underline{End}_{kG}(E_{V}) and any two g_{1} and g_{2} in \underline{End}_{kG}(F_{V}) we have that f_{2}∘f_{1} ≃ f_{1}⊗f_{2} and g_{2} ∘ g_{1} ≃ g_{1} ⊗ g_{2} in StMod(kG). Finally we note that

E_{V} ⊗ E_{V} ⟶^{ϵ⊗ϵ} k ⊗ k

and

k ⊗ k ⟶^{η⊗η} F_{V} ⊗ F_{V}

are stably isomorphic to ϵ and η. QED.

It immediately follows from (3.1) and commutativity of tensor products that stable endomorphism ring of both idempotent modules are commutative(see [BG]).

Using (3.1) we are able to prove the following result.

(3.5) Theorem. If U is a connected closed homogeneous subvariety of V_{G}(k) then \underline{End}_{kG}(E_{U}) is a local ring.

Proof. To prove that \underline{End}_{kG}(E_{U}) is local it is sufficient to show that the sum of any two noninvertible elements of \underline{End}_{kG}(E_{U}) is noninvertible.

Let f_{1} and f_{2} be noninvertible elements in \underline{End}_{kG}(E_{U}). The last elements of the two triangles

(3.6)  E_{U} ⟶^{f_{i}} E_{U}⟶Z_{i}⟶Ω^{−1}(E_{U})

are not projective, note that V_{G}(Z_{i}) ⊂ V_{G}(E_{U}) by (2) of (2.1). By (3.2) there exists an indecomposable finitely generated kG-module M such that V_{G}(M) = U and V_{G}(M) = {W ⊂ V_{G}(k)|W ⊂ U} . Tensoring (3.6) by M we get triangles

(3.7)  M ⊗ E_{U} ⟶^{id_{M} ⊗f_{i}} M ⊗ E_{U}⟶M ⊗ Z_{i}⟶M ⊗ Ω^{−1}(E_{U})

By (3) of (2.1) we have that V_{G}(M ⊗ Z_{i}) = V_{G}(M) ∩ V_{G}(Z_{i}) ≠ ∅ and thus id_{M} ⊗ f_{i} is not stably invertible. Since M ⊗E_{U} ≃ M ⊕proj it follows that id_{M} ⊗f_{i} represents a noninvertible element of \underline{End}_{kG}(M). If f_{1} + f_{2} is invertible then id_{M} ⊗ (f_{1} + f_{2}) is invertible. This is a contradiction since the stable endomorphism ring of M is a local ring. QED.

The second proposition deals with infinite collections of closed homogeneous subvarieties which are in some sense connected and for which there exist the idempotent modules.

(3.8) Definition. A collection of subvarieties will be called connected provided the union of its elements is connected.

(3.9) Example. Let V_{G}(k) have projective dimension D. Let W_{d} be the collection of closed homogeneous subvarieties of V_{G}(k) of projective dimension d < D. It is easy to see that W is connected in the sense of (3.8) since any two points in V_{G}(k) can be joined by a linear variety of positive dimension.

(3.10) Theorem. Let C be a collection of closed homogeneous subvarieties of V_{G}(k) which is closed under specialization and finite unions and satisfies the property that given any two elements of C there is a finite connected subcollection of C containing these elements. Let E_{C} be the E-module corresponding to the thick subcategory generated by C. Then \underline{End}_{kG}(E_{C}) is a local ring.

14 PETERIS DAUGULIS

Proof. Since C is closed under specialization and finite unions the subcategory of all finitely generated kG-modules with varieties contained in elements of C is a thick subcategory and thus there is a distinguished triangle(see [BCR2]):

(3.11)  E_{C} ⟶ k ⟶ F_{C} ⟶ Ω^{−1}(E_{C})

As in the proof of (3.4) we consider two noninvertible maps f_{1} and f_{2} and the corresponding triangles

(3.12)  E_{C} ⟶^{f_{i}} E_{C}⟶Z_{i}⟶Ω^{−1}(E_{C})

with Z_{i} nonprojective.

Since C satisfies the specified property there is a finite connected subcollection D such that ⋃_{W ∈D} W ∩ V_{G}(Z_{i}) ≠ ∅ for each i. Denote by V_{D} the set of all closed homogeneous subvarieties contained in the union of elements of D. By (3.2) there is an indecomposable finitely generated kG-module such that V_{G}(M) = ⋃_{W ∈D} W and thus V_{G}(M) = V_{D}. Now we procceed as in the proof of (3.5). Since V_{G}(M) ∩ V_{G}(Z_{i}) ≠ ∅ by construction for both values of i we have that id_{M} ⊗ f_{i} represent noninvertible elements of \underline{End}_{kG}(M). If f_{1} + f_{2} is invertible then id_{M} ⊗ (f_{1} + f_{2}) has to invertible which is a contradiction. QED.

(3.13) Remark. We note at this point that the proofs of (3.5) and (3.9) do not work for the idempotent F_{V} -modules since there does not exist a finitely generated kG module M such that V_{G}(M) = V_{G}(F_{V}). Moreover, \underline{End}_{kG}(F_{V}) fails to be local in

STABLE ENDOMORPHISM RINGS OF IDEMPOTENT E-MODULES 15

the case when G = (ℤ/2)^{2} and V is a single point, in that case it isomorphic to the polynomial ring in one variable over k.

We remind the reader that an endomorphism f of a kG-module M is called locally nilpotent provided f^{n_{v}}(v) = 0 for any v ∈ M and some n_{v} ∈ ℕ. We next prove that under certain conditions on V endomorphisms of E_{V} -modules representing noninvertible stable endomorphisms are locally nilpotent. Before proving the proposition we need to state a lemma of Hida(see [Hi]).

(3.14) Lemma([Hi]). Let ζ_{1}, ..., ζ_{s} be a homogeneous set of parameters for Ext_{kG}^{∗}(k, k), i.e. Ext_{kG}^{∗}(k, k) is finitely generated as a module over the polynomial subalgebra k[ζ_{1}, ..., ζ_{s}]. If V = ⋂_{i=1}^{d} V_{ζ_{i}} (with d < s) is connected then ⨂_{i=1}^{d} L_{ζ_{i}} is stably indecomposable.

(3.15) Theorem. If V is a connected closed homogeneous subvariety of V_{G}(k) satisfying conditions of (3.10) then any noninvertible endomorphism f of E_{V} is locally nilpotent.

Proof. Let V = ⋂_{i=1}^{d} V_{ζ_{i}}. By (3.14) we have that the nonprojective direct summand of

(3.16)  ⨂_{i=1}^{d} Ω^{−n_{i}deg(ζ_{i})}(L_{ζ_{i}^{n_{i}}})

is indecomposable for any choice of the sequence {n_{1}, ..., n_{d}} thus

V_{G}(⨂_{i=1}^{d} Ω^{−n_{i}deg(ζ_{i})}(L_{ζ_{i}^{n_{i}}})) = V.

16 PETERIS DAUGULIS

We can express E_{V} as a colimit X_{1} ⟶ X_{2} ⟶ ... where each X_{i} is a finitely generated module of form (3.16). It follows that for any noninvertible map f ∈ End_{kG}(E_{V}) the map

E_{V} ⊗ X_{i} ⟶^{f⊗id_{X_{i}}} E_{V} ⊗ X_{i}

is stably noninvertible and thus for some integer n_{f,i} we have that

(f^{n_{f,i}} ⊗ id_{X_{i}})(E_{V} ⊗ X_{i}) = 0

in stmod(kG). Since E_{V} is a colimit of the X_{i}’s it follows that the restriction of f to any submodule X′_{i} of E_{V} equal to the image under the defining map of some X_{i} factors through a projective after a finite number of steps. Since the image of a projective map is contained in the radical of the target it follows that after adding an additional finite power of f the image of X′_{i} is zero in Mod(kG). QED.

(3.17) Corollary. If V is a connected closed homogeneous subvariety of V_{G}(k) satisfying conditions of (3.14) then for any b ∈ E_{V} there is n_{b} ∈ ℕ such that Rad^{n_{b}}(\underline{End}_{kG}(E_{V})) annihilates b.

(3.18) Remark. The proposition (3.15) may be useful trying to establish whether the E-modules are pure injective(see [Cr-B]). More precisely, a kG-module M is pure injective if it has a locally nilpotent endomorphism with a finite-dimensional kernel. Under the conditions of (3.14) all endomorphisms representing noninvertible stable endomorphisms are locally nilpotent but, unfortunately, in the interesting cases, i.e. when the p-rank is more than 2, they have infinite-dimensional kernels (see (5.5)).

STABLE ENDOMORPHISM RINGS OF IDEMPOTENT E-MODULES 17

Given a stable endomorphism f ∈ \underline{End}_{kG}(E_{V}) with V satisfying the conditions of (3.11) we are now able to determine the variety of the last element X_{f} of the triangle

E_{V} ⟶^{f} E_{V} ⟶ X_{f} ⟶ Ω^{−1}(E_{V}).

(3.20) Theorem. Let {ζ_{1}, ..., ζ_{n}} be a regular sequence for a connected closed homogeneous variety V. Let

E_{V} ⟶^{f} E_{V} ⟶ X_{f} ⟶ Ω^{−1}(E_{V})

be a distinguished triangle with f a noninvertible endomorphism. We have that V_{G}(X_{f}) = P roj(V).

Proof. Since f is noninvertible it follows from (3.15) that restrictions of f to all proper subalgebras of kG as well as to cyclic shifted subalgebras over transcendentally extended fields are noninvertible. Thus res_{p}(f) is stably zero for any p ∈ V_{G}(E_{V}) and res_{p}(X_{f}) is not projective which implies that V_{G}(X_{f}) ⊆ V_{G}(E_{V}). The opposite inclusion follows from fact that the X_{f} is an element of a distinguished triangle with the other two elements being E_{V}. QED.

4. Square-zero radical.

Next we give a proposition relating the structure of \underline{End}_{kG}(E) to those of the finitely generated limiting modules.

18 PETERIS DAUGULIS

(4.1) Definition. Given a nonprojective kG-module M and a nonzero map f ∈ \underline{Hom}_{kG}(M, k) we will call a stable map g ∈ \underline{End}_{kG}(M) f-essential provided f∘g ≠ 0 in \underline{Hom}_{kG}(M, k).

Note that the definition makes sense since for any map p ∈ P Hom_{kG}(M, M) and any nonzero map f ∈ Hom_{kG}(M, k) we have that f ∘ p = 0 in the stable category.

Given a closed homogeneous subvariety V = ⋂_{i=1}^{d} V_{ζ_{i}} of V_{G}(k) we have that E_{V} ≃ ⨂ E_{V_{ζ_{i}}} in StMod(kG)(see [Rick]). Thus E_{V} = colim(X_{1} ⟶ X_{2} ⟶ ...) where each X_{i} is a tensor product of Heller translates of L_{ζ_{b}^{a}} for some values of a and b, for example, we can choose X_{i} = ⨂_{m=1}^{d} Ω^{−i∗deg(ζ_{m})}(L_{ζ_{m}^{i}}). Note that for each X_{i} there is a distinguished map ϵ_{i} ∈ \underline{Hom}_{kG}(X_{i}, k) which is the tensor product of the distiguished maps occuring in the triangle defining each L_{ζ_{b}^{a}}. Also note that a kG-map f : E_{V} ⟶ E_{V} is stably nonzero if and only if f is ϵ-essential.

(4.2) Theorem. Let V be an arbitrary closed homogeneous variety in V_{G}(k). Let X_{1} ⟶ X_{2} ⟶ ... be a colimit defining E_{V}. If there is an index i such that for all j > i there are no nonzero ϵ_{j}-essential compositions of noninvertible maps in \underline{End}_{kG}(X_{j}) then there are no nonzero compositions of noninvertible maps in \underline{End}_{kG}(E_{V}).

Proof. If a kG-map f : E_{V} ⟶ E_{V} is stably nonzero then the composition

E_{V} ⊗ E_{V} ⟶^{f⊗id} E_{V} ⊗ E_{V} ⟶^{ϵ⊗ϵ} k ⊗ k

is nonzero. Since E_{V} = colim_{i}(X_{i}) it follows that starting from some i we have that

E_{V} ⊗ X_{i} ⟶^{f⊗id_{X_{i}}} E_{V} ⊗ X_{i} ⟶^{ϵ⊗ϵ_{i}} k ⊗ k

is nonzero.

Let E_{V} ⟶^{f_{1}} E_{V} ⟶^{f_{2}} E_{V} be a nonzero composition of two noninvertible and and stably nonzero maps, note that for each i id_{X_{i}} ⊗ f_{1,2} is noninvertible. In such a case there is an index i such that for all j > i id_{X_{j}} ⊗ (f_{2} ∘ f_{1}) is stably nonzero and ϵ_{j}-essential which is a contradiction. QED.

(4.3) Remark. In the case when \underline{End}_{kG}(E) is local and the hypothesis of (4.1) holds (4.1) says that Rad^{2}(\underline{End}_{kG}(E)) = 0.

(4.4) Remark. Hypothesis of (4.4) holds if G = (ℤ/2)^{3}, k is an algebraically closed field of characteristic 2 and V is a hypersurface in V_{G}(k)(see example (5.5)). Since in this case \underline{End}_{kG}(E_{V}) is a local ring (4.2) implies that \underline{End}_{kG}(E_{V}) has a squarezero radical. Examples seem to indicate that the hypothesis may hold for G an arbitrary elementary abelian p-group and V a hypersurface in V_{G}(k).

(4.5) Lemma. If ζ ∈ Ext_{kG}^{n}(k, k) then ϵ-nonessential elements in \underline{End}_{kG}(Ω^{−n}(L_{ζ})) and thus in \underline{End}_{kG}(Ω^{−m}(L_{ζ})) for any m ≥ 0 form a two sided ideal.

Proof. Nonessential elements clearly form an abelian subgroup and a right ideal. To prove that they form a left ideal we choose f ∈ \underline{End}_{kG}(Ω^{−n}(L_{ζ})) such that ϵ ∘ f = 0 and arbitrary g ∈ \underline{End}_{kG}(Ω^{−n}(L_{ζ})). We look at the following diagram in stmod(kG):

[figure: Ω^{−n+1}(k); ρ; Ω^{−n}(L_{ζ}); ϵ; f; k; Ω^{−n}(ζ); Ω^{−n}(k); Ω^{−n+1}(k); ρ; Ω^{−n}(L_{ζ}); ϵ; g; k; Ω^{−n}(ζ); Ω^{−n}(k); Ω^{−n+1}(k); ρ; Ω^{−n}(L_{ζ}); ϵ; k; Ω^{−n}(ζ); Ω^{−n}(k)]

where the horizontal rows are distinguished triangles. Since ϵ ∘ f = 0 it follows that f factors through ρ:

[figure: Ω^{−n+1}(k); ρ; ϕ∘ρ; Ω^{−n}(L_{ζ}); ϕ; ϵ; f; k; Ω^{−n}(k); Ω^{−n+1}(k); ρ; Ω^{−n}(L_{ζ}); ϵ; g; k; Ω^{−n}(k); Ω^{−n+1}(k); ρ; Ω^{−n}(L_{ζ}); ϵ; k; Ω^{−n}(k)]

We can complete f to a map of triangles with the other two maps being zero since Ω^{−m}(k) is endotrivial for any m. It follows that f ∘ ρ = 0. This implies that g ∘ f factors through ϵ:

[figure: Ω^{−n+1}(k); ρ; ϕ∘ρ; Ω^{−n}(L_{ζ}); ϕ; ϵ; f; k; ψ; Ω^{−n}(k); Ω^{−n+1}(k); ρ; Ω^{−n}(L_{ζ}); ϵ; g; k; Ω^{−n}(k); Ω^{−n+1}(k); ρ; Ω^{−n}(L_{ζ}); ϵ; k; Ω^{−n}(k)]

It follows that g ∘ f can be completed to a map of triangles with the other two maps equal to zero and the claim is proved. QED.

(4.6) Corollary. Let X_{1} ⟶ X_{2} ⟶ ... ⟶ X_{n} ⟶ be the colimit defining E_{V} for V a connected hypersurface in V_{G}(k). (4.2) can be formulated in terms of quotient rings of the stable endomorphism rings of X_{i}’s: if the stable endomorphism rings modulo ideal of nonessential maps has square-zero radical for sufficiently large i’s then the stable endomorphism ring of E_{V} has square-zero radical.

Proof. The claim immediately follows from (4.5) since it makes the quotient rings of the stable endomorphism rings for the limiting modules well defined. QED.

5. Examples.

In this section we give a few examples of idempotent modules for low rank pgroups in varying degrees of detalization using the diagrammatic technique. We first give our definition of module diagrams(for the previous definitions see [BC],

[C1] and [Rin]).

We suggest a natural and simple definition of module diagram which contains all the previous definitions, applies unconditionally to all associative algebras and arbitrary modules over these algebras and makes many results in representation theory transparent. Diagrams in the sense of Alperin-Benson-Carlson(see [BC]) can be obtained from our diagrams as suitable quotients.

(5.1) Definition. Let A be an associative algebra over a field k. Let A be given by a set of generators {x_{1}, .., x_{n}}. Let a (not necessarily finitely generated) A-module

22 PETERIS DAUGULIS

M be given by n matrices X_{1}, ..., X_{n}. We use the convention that the matrices act from the left. We define the diagram D(M) to be an oriented graph with edges labeled by a pair of indices (i, λ), λ ≠ 0 so that for each pair of vertices there is at most one edge having the first label of the pair equal to i for each i. The vertices of D(M) are indexed by the chosen basis elements with respect to which the matrices are given. There is an edge from a vertex v to a vertex v′ having the label (i, λ) provided X_{i} has a nonzero entry equal to λ at the intersection of the column corresponding to v and the row corresponding to v′. By default we omit the second label if the matrix entry is equal to 1. In a similar way we give digrammatic interpretation to A-homomorphisms.

The correspondence M ⟶ D(M) should be thought as the generalized inverse of the map G ⟶ M(G) which sends a graph G to its adjacency matrix M(G).

(5.2) Example. Let G = (ℤ/3)^{2} = ⟨X, Y ⟩, k an algebraically closed field of characteristic 3, let x = X − 1, y = Y − 1.

Since kG as a vector space over k is generated by the basis

{1, x, y, x^{2}, xy, y^{2}, x^{2}y, xy^{2}, x^{2}y^{2}}

with obvious action of the nilpotent generators x and y we represent kG as the diagram on F ig.1.

[figure: •; x; y; •; x; y; •; x; y; •; y; •; x; y; •; x; •; y; •; x; •]

Fig.1.

The even cohomology ring of G is the polynomial ring in two variables over k generated by elements in degree 2 so we need to compute Ω^{2}(k) which looks like F ig.2.

[figure: a_{2}; x; y; a_{1}; y; •; x; y; •; x; y; a_{3}; x; •; y; •; x; y; •; x; •; •]

Fig.2.

24 PETERIS DAUGULIS

With respect to this diagram the nonnilpotent generators of Ext_{kG}^{∗}(k, k) correspond to maps supported on a_{1} and a_{3} while the nilpotent generator of the antisymmetric algebra corresponds to the map supported on a_{2}. This may be established, for example, by considering tensor products of the maps.

Let V be a point in V_{G}(k) ≃ ℙ^{1} corresponding to the map ζ ∈ Hom_{kG}(Ω^{2}(k), k) supported on a_{3}, i.e defined by

ζ(a_{1}) = ζ(a_{2}) = 0, ζ(a_{3}) = 1 :

and shown on F ig.3.

[figure: a_{2}; x; y; a_{1}; y; •; x; y; •; x; y; a_{3}; x; ζ; •; •; y; •; x; y; •; x; •; •]

Fig.3.

To construct F_{V} we need to compute Ω^{−2n}(ζ) for any positive n. The diagram for Ω^{−2}(ζ) is shown on F ig.4.

[figure: •; x; y; •; x; y; •; x; y; •; x; y; •; x; y; •; Ω^{−2}(ζ); •; •; y; •; x; •; •]

Fig.4.

The diagram for Ω^{−4}(ζ) which we give on F ig.5 essentially allows us to guess the general form of Ω^{−2n}(ζ)(dotted arrows represent the map).

\xymatrix{ &  & • \ar@{.>}[r] \ar[dl] \ar[dr] & • \ar[dl] \ar[dr] & • \ar@{.>}[r] \ar[dl] \ar[dr] & • \ar[dl] \ar[dr] &  &  &  & • \ar[dl] \ar[dr] &  & • \ar[dl] \ar[dr] \\  & • \ar@{.>}[r] \ar[dl] \ar[dr] & • \ar[dl] \ar[dr] & • \ar@{.>}[r] \ar[dl] \ar[dr] & • \ar[dl] \ar[dr] & • \ar@{.>}[r] \ar[dl] \ar[dr] & • \ar[dl] \ar[dr] &  & • \ar[dl] \ar[dr] &  & • \ar[dl] \ar[dr] &  & • \ar[dl] \ar[dr] \\ • \ar@{.>}[r] & • & • \ar@{.>}[r] \ar[dr] & • \ar[dr] & • \ar@{.>}[r] \ar[dl] & • \ar[dl] & • \ar@{.>}[r] & • &  & • \ar[dr] &  & • \ar[dl] &  & • \\  &  &  & • \ar@{.>}[r] & • &  &  &  &  &  & •}

Fig.5.

Putting together all maps of form Ω^{−2n}(ζ) we obtain F_{V} with the distinguished map η shown on F ig.6.

[figure: •; x; y; •; x; y; •; x; y; •; x; y; •; x; y; •; x; y; •; x; y; •; x; y; •; x; y; •; x; y; •; η; •; •; y; •; x; •; •; y; •; x; •; •; •]

Fig.6.

Computing Ω(Coker(η)) we obtain the diagram for the E_{V} module together with the distinguished map ϵ shown on F ig.7.

[figure: a; y; x; •; y; x; •; v_{0}; ϵ; y; •; y; x; •; y; x; b; y; x; •; y; x; •; y; x; •; y; c; y; x; •; x; •; y; •; y; x; •; x; •; •; •; •]

Fig.7.

To determine the structure of the stable endomorphism ring of E_{V} we have to recall that there is a bijection of abelian groups:

\underline{End}_{kG}(E_{V}) ≃ \underline{Hom}_{kG}(E_{V} , k).

This means that we have to try to factor kG-maps from E_{V} to k through the distinguished map ϵ. We can easily see that the map supported on b factors through the shift t = \overrightarrow{bv_{0}} of the diagram to the left which sends b to v_{0}. t generates a power series subring in \underline{End}_{k}G(E_{V}) which corresponds to elements in \underline{Hom}_{kG}(E_{V} , k) supported on basis elements obtained from b by translations to the right. We also have to deal with the elements of \underline{Hom}_{kG}(E_{V} , k) supported on a and its translates to the right. It is easy to see that the map supported on a factors through the endomorphism n = \overrightarrow{av_{0}} which is defined by

n(a) = v_{0}, n(b) = c.

Clearly n^{2} = 0 in StMod(kG). Thus we have that \underline{End}_{kG}(E_{V}) ≃ k[[t]]⊗(k[n]/⟨n^{2}⟩), i.e any stable endomorphism of E_{V} is in form n ∘ f + g with f and g elements of k[[t]].

(5.3) Example. The construction of (5.2) is easy to generalize for G = (ℤ/p)^{2} = ⟨X, Y ⟩ for any prime p and V a point in V_{G}(k). The diagram for kG is a rectangle subdivided in (p − 1)^{2} squares, etc. On F ig.8 we give a short piece of the diagram for G = (ℤ/5)^{2}.

[figure: a; y; x; •; y; x; •; y; x; •; y; x; •; y; x; •; y; x; •; v_{0}; ϵ; y; •; y; x; •; y; x; •; y; x; •; y; x; b; x; •; y; •; y; x; •; y; x; •; y; x; •; x; •; y; •; y; x; •; y; x; •; x; •; y; •; y; x; •; x; •; •]

Fig.8.

Clearly the difference between E_{V} -modules for different p amounts to the difference of the size of submodules generated by the basis element a and its translates to the right and does not affect the structure of the stable endomorphism rings of E_{V} which is isomorphic k[[t]] ⊗ (k[n]/⟨n^{2}⟩) for any p.

(5.4) Example. Let G = (ℤ/p^{n})×(ℤ/p^{m}). In this case again we have V_{G}(k) ≃ ℙ^{1}. In this case kG is a subdivided rectangle having sides of possibly different lengths. In the case of V = {point} we have that E_{V} -modules have almost the same structure as above except that the submodule generated by the basis element a has the diagram equal to a subdivided rectangle with sides of different lengths and one vertex removed. Again we have that \underline{End}_{kG}(E_{V}) ≃ k[[t]] ⊗ (k[n]/⟨n^{2}⟩).

(5.5) Example. Let G = (ℤ/2)^{3} = ⟨X, Y, Z⟩, let k be an algebraically closed field of characteristic 2. Let x = X − 1, y = Y − 1, z = Z − 1 be nilpotent generators.

We have that Ext_{kG}^{∗}(k, k) is isomorphic to the polynomial ring in 3 variables over k so we can take V to be a linear hypersurface. We consecutively compute a diagram for Ω^{−1}(k), choose a nonzero element ζ ∈ Ext_{kG}^{1}(k, k) and compute Ω^{−n}(ζ) for a few small values of n as in the previous examples. Skipping the explicit computations we compute a diagram for E_{V} which is shown on Fig.9. and claim that \underline{End}_{kG}(E_{V}) is isomorphic to an infinitely-generated local k-algebra with a square-zero radical. We note that the submodules M_{i} generated by {v_{0}, a_{1,1}, ..., a_{i,1}, ..., a_{i,i}} for any i are isomorphic to Ω^{−i−1}(L_{ζ^{i+1}}), inspection of structures of these modules verifies the claim made in (4.3).

Taking V to be a point we compute E_{V} as a tensor product of two E-modules corresponding to linear hypersurfaces of the above kind. The diagram is shown on Fig.10. In this case \underline{End}_{kG}(E_{V}) ≃ k[[t_{1}, t_{2}]]. On Fig.10 the vertices are labeled so that the kG-maps supported on the vertices correspond to elements of \underline{End}_{kG}(E_{V}) in the obvious way.

Acknowledgements

We thank Jon F.Carlson for guidance and help. We also thank David J. Benson and Wayne W.Wheeler for useful conversations.

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