COMPLEXITY AND QUOTIENT CATEGORIES

FOR GROUP ALGEBRAS

Jon F. Carlson, Department of Mathematics, University of Georgia, Athens, GA 30602 USA E-mail: jfc@sloth.math.uga.edu

Peter W. Donovan, School of Mathematics, The University of New South Wales, Kensington, NSW 2033 Australia E-mail: peterd@hydra.maths.unsw.oz.au

Wayne W. Wheeler, Mathematics, IAS, Australian National University, GPO Box 4, Canberra, ACT 2601 Australia E-mail: wheeler@pell.anu.edu.au

Introduction

One of the main points of this paper is that some aspects of the structure of modules and of module theory in general can be detected by various categorical constructions on the module category. Indeed, for any particular property possessed by modules it seems likely that a category theory can be constructed that characterizes or at least reveals some facet of the property. This idea certainly is not new; it is an underlying theme in the work of Auslander and others, where the main emphasis is on the role of functors in the module theory. Our offering here, however, has a very different flavor. We shall concentrate on quotients of subcategories of the module category of a group algebra. The quotient construction is given by a localization process that turns out to be closely tied to the structure of cohomology rings.

Our main results can be summarized as follows. Let G be a finite group, and let k be an algebraically closed field of characteristic p > 0. We shall work entirely within the stable category \overline{mod}-kG of finitely generated kG-modules modulo projectives. The complexity of a kG-module M is a nonnegative integer giving the polynomial rate of growth of the minimal projective resolution of M; it can also be described as the dimension of the variety of M. The subcategory M_{c} of \overline{mod}-kG consisting of all kG-modules of complexity at most c is a thick subcategory of the full stable category, and hence it makes sense to take the quotient Q_{c} = M_{c}/M_{c−1}. Then Q_{c} is triangulated, and its objects are expressible as finite direct sums of indecomposable objects. In general, however, the category has no Krull-Schmidt Theorem. If M and N are kG-modules in M_{c}, then Hom_{Q_{c}}(M, N) is obtained by localizing Êxt_{kG}^{∗}(M, N) at a certain multiplicative subset of H^{∗}(G, k) and then taking the elements of degree zero. The multiplicative subset is determined by a general position criterion on the varieties of the modules and the cohomology elements. This result is explicitly stated in Theorem 3.5. One consequence is that if r is the p-rank of G, then the endomorphism ring of the trivial module in Q_{r} is the direct sum of t local rings, where t is the number of components of the maximal ideal spectrum of H^{∗}(G, k) of maximum dimension.

In the last section we present some examples. Given the previous results, it is relatively easy to compute the endomorphism ring of the trivial module for some well known groups, such as the elementary abelian p-groups. The case of the dihedral group is particularly interesting. Here it can be shown that the direct sum of two copies of the trivial module is isomorphic in the quotient category Q_{2} to the direct sum of modules induced from the maximal elementary abelian subgroups. A look at the endomorphism rings in the quotient category shows why it is necessary for the cohomology rings of the induced modules to be noncommutative in an essential way. This fact was considered rather surprising when it was first discovered a few years ago, and the quotient construction may shed new light on such phenomena. A similar sort of thing should happen for all of the extraspecial p-groups, but the analysis of these cases remains incomplete. None of the results proved here provide new information about the module category, but the indications are that such theorems may not be far away.

Finally, we should note that there are many possible variations on the construction given here. Indeed, the one we give may not be the best or most natural. Another approach might be to take the quotient of the stable category \overline{mod}-kG by the subcategory of all modules M whose cohomological support variety V_{G}(M) contains no component of V_{G}(k). If s is the number of conjugacy classes of maximal elementary abelian p-subgroups of G, then Quillen’s Dimension Theorem implies that the endomorphism ring of the trivial module would be a direct sum of s local rings. Alternatively, we could concentrate on a single component of the maximal ideal spectrum V_{G}(k) and take the quotient by the subcategory of all modules whose varieties fail to contain that component. In this case the endomorphism ring of k would be local. These and other results can be derived using exactly the same techniques as in this paper.

§1. Preliminaries

Let k be an algebraically closed field of characteristic p > 0, and let G be a finite group such that p divides |G|. The cohomology ring H^{∗}(G, k) is a finitely generated, graded k-algebra, and if ζ and η are homogeneous elements of H^{∗}(G, k), then ζη = (−1)^{(deg ζ)(deg η)}ηζ. In particular, H^{∗}(G, k) is commutative if p = 2. If p > 2, then all elements of odd degree are nilpotent, so they lie in the radical Rad H^{∗}(G, k). Thus it is always true that H^{∗}(G, k)/ Rad H^{∗}(G, k) is a commutative ring, and the maximal ideal spectrum V_{G}(k) of H^{∗}(G, k) is a homogeneous affine variety. If ζ is any element of H^{∗}(G, k), then V_{G}(ζ) denotes the subvariety consisting of all maximal ideals containing ζ. If M is a kG-module, then Ext_{kG}^{∗}(M, M) ≅ H^{∗}(G, Hom_{k}(M, M)) is a finitely generated module over H^{∗}(G, k). Let J(M) denote the annihilator of Ext_{kG}^{∗}(M, M) in H^{∗}(G, k), and let V_{G}(M) ⊆ V_{G}(k) denote the subvariety consisting of all maximal ideals containing J(M). The complexity cx_{G}(M) of the module M is the dimension of V_{G}(M).

For each kG-module M fix an injective hull ι_{M} : M → I(M), and let Ω^{−1}(M) = Coker ι_{M}. If n > 1, then Ω^{−n}(M) is defined inductively by setting Ω^{−n}(M) = Ω^{−1}(Ω^{−n+1}(M)). The modules Ω^{n}(M) are defined by a dual procedure, beginning with a projective cover of M.

If ζ is a nonzero element of H^{n}(G, k), then we will write ζ̂ : Ω^{n}(k) → k for the map representing ζ. Set L_{ζ} = Ker ζ̂ so that there is a short exact sequence

0  wL_{ζ}  wΩ^{n}(k)  w ζ̂ k  w0.

The first proposition lists some of the basic properties of varieties that will be needed in later sections. The reader is referred to [1] or [3] for proofs of the results.

Proposition 1.1.

(1) V_{G}(M) = {0} if and only if M is projective.

(2) If 0 → M_{1} → M_{2} → M_{3} → 0 is an exact sequence of kG-modules, then V_{G}(M_{i}) ⊆ V_{G}(M_{j}) ∪ V_{G}(M_{m}) for {i, j, m} = {1, 2, 3}. In particular, the two largest complexities are equal.

(3) V_{G}(M ⊕ N) = V_{G}(M) ∪ V_{G}(N).

(4) V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N).

(5) If W ⊆ V_{G}(k) is a closed homogeneous subvariety, then there is a kG-module M with V_{G}(M) = W.

(6) If ζ is a nonzero element of H^{n}(G, k), then V_{G}(L_{ζ}) = V_{G}(ζ).

(7) V_{G}(M) = V_{G}(M^{∗}) = V_{G}(Ω^{n}(M)) for all n.

Let mod-kG denote the category of all finitely generated right kG-modules. If M and N are kG-modules, let PHom_{kG}(M, N) ⊆ Hom_{kG}(M, N) denote the subspace of all kG-homomorphisms that factor through a projective module. The stable category \overline{mod}-kG has the same objects as mod-kG, but morphisms are defined by setting \overline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/ PHom_{kG}(M, N). If f : M → N is a kGhomomorphism, then \overline{f} will denote the corresponding morphism in \overline{mod}-kG. Although the stable category is not abelian, it is a triangulated category by Theorem I.2.6 of [5], and the translation functor is given by Ω^{−1}.

Let M and N be kG-modules, and consider a doubly infinite projective resolution

P_{∗} : ⋯  wP_{2}  wP_{1}  wP_{0}  w ∂_{0} P_{−1}  w⋯

of M. In other words, P_{∗} is an exact sequence of projective modules with Im ∂_{0} ≅ M. Then Êxt_{kG}^{n}(M, N) is defined for any n ∈ ℤ by setting

Êxt_{kG}^{n}(M, N) = H^{n}(Hom_{kG}(P_{∗}, N)).

In particular, we have Êxt_{kG}^{n}(M, N) = Ext_{kG}^{n}(M, N) for n > 0, and Êxt_{kG}^{n}(M, N) ≅ \overline{Hom}_{kG}(Ω^{n}(M), N) for all n. For our purposes the most important difference between Êxt and Ext is that Êxt_{kG}^{0}(M, N) = \overline{Hom}_{kG}(M, N), whereas Ext_{kG}^{0}(M, N) = Hom_{kG}(M, N).

The following well-known result describes the connection between short exact sequences in mod-kG and triangles in \overline{mod}-kG.

Proposition 1.2. Suppose that there is a short exact sequence

0  wM′  w f M  w g M′′  w0

in mod-kG. Then there is a triangle in \overline{mod}-kG of the form

M′ w \overline{f} M w \overline{g} M′′ w Ω^{−1}M′.

Moreover, every triangle in \overline{mod}-kG is isomorphic to one that arises from a short exact sequence in this way.

Proof: The first assertion is Lemma I.2.7 of [5], and the second follows easily from Lemma I.2.3 of [5]. □

One consequence of Proposition 1.2 is that many results can be stated either in terms of short exact sequences in mod-kG or in terms of triangles in \overline{mod}-kG. For example, since the variety of a projective module is {0}, Proposition 1.1(2) could be rephrased to say that if M_{1} → M_{2} → M_{3} → Ω^{−1}M_{1} is a triangle in \overline{mod}-kG, then V_{G}(M_{i}) ⊆ V_{G}(M_{j})∪V_{G}(M_{m}) for {i, j, m} = {1, 2, 3}. Whenever two such statements are possible, we have chosen to present the result in terms of short exact sequences. It will sometimes be useful, however, to use the corresponding results for triangles in \overline{mod}-kG, and we will do so without further comment.

Now suppose that C is a triangulated category and E is a full triangulated subcategory of C. Assume that for any triangle

\begin{matrix}M  w α \\ N  w \\ U  w \\ T M \end{matrix}

in C such that U is an object of E and α factors through an object in E, the objects M and N also lie in E. Then E is said to be a thick (or épaisse) subcategory of C. By Proposition 1.3 of [7], a full triangulated subcategory of C is thick if and only if it is closed under direct summands in C.

Let r be the p-rank of G. For each integer c with 1 ≤ c ≤ r let M_{c} denote the full subcategory of the stable category consisting of the objects of complexity at most c.

Proposition 1.3. M_{c−1} is a thick subcategory of M_{c} for 1 ≤ c ≤ r.

Proof: It is clear that M_{c−1} is an additive category. If \overline{f} : M → N is a morphism in M_{c−1}, then there is a triangle

M  w \overline{f} N  wU  wΩ^{−1}M

in \overline{mod}-kG. Because cx_{G}(U) ≤ max\{cx_{G}(M), cx_{G}(N)\}, it follows that U also lies in M_{c−1}. Thus M_{c−1} is a triangulated subcategory of \overline{mod}-kG and also of M_{c}. Finally, if M is a module with cx_{G}(M) ≤ c − 1, then any summand of M has complexity at most c − 1, so M_{c−1} is thick. □

Definition 1.4. Let M and N be objects of M_{c}. Suppose that f : M → N is a kG-homomorphism such that there is a triangle

M  w \overline{f} N  wU  wΩ^{−1}M

with U in M_{c−1}. Then we will say that \overline{f} is a complexity isomorphism in M_{c}. When the particular complexity c is understood, we will simply say that \overline{f} (or f) is a complexity isomorphism.

The advantage of a thick subcategory is that the corresponding quotient category inherits a triangulated structure. Thus we set Q_{c} = M_{c}/M_{c−1}, and we proceed to study the categories Q_{c}. To be more explicit, the category Q_{c} is obtained by localizing the category M_{c} with respect to the complexity isomorphisms. Thus a morphism from M to N in Q_{c} is represented by a pair of maps \overline{s} : U → M and \overline{f} : U → N in M_{c} such that \overline{s} is a complexity isomorphism. If \overline{s}′ : U′ → M and \overline{f}′ : U′ → N are two other such maps in M_{c}, then the pairs (\overline{s}, \overline{f}) and (\overline{s}′, \overline{f}′) represent the same morphism in Q_{c} if and only if there is an object X in M_{c} with maps such the the diagram

U [ [ [ [ ^\overline{s} _{’ ’ ’ ’)} \overline{f}

M  X u \overline{t}  w u u  N

U′^{4 4 4 47}^{\overline{s}′} ^{h h hh j}^{\overline{f}′}

commutes and \overline{t} is a complexity isomorphism.

Roughly speaking, the idea behind this construction is to invert all complexity isomorphisms in M_{c}. Thus the maps \overline{s}, \overline{s}′, and \overline{t} above represent isomorphisms in Q_{c}. It is often useful to think of the morphism (\overline{s}, \overline{f}) : M → N in Q_{c} as a composite \overline{f}\overline{s}^{−1} : M → N, even though \overline{s} may not actually be invertible in M_{c}. For further information on localization and quotient categories the reader is referred to [4] or [6].

Proposition 1.5. Let M, N, and U be objects of M_{c}, and suppose that \overline{f} : M → N, \overline{g} : N → U, and \overline{h} = \overline{g}\overline{f} : M → U are morphisms. If any two of \overline{f}, \overline{g}, and \overline{h} are complexity isomorphisms, then so is the third.

Proof: Consider the triangles

M  w \overline{f} N  wC_{1}  wΩ^{−1}M,
N  w \overline{g} U  wC_{2}  wΩ^{−1}N,

and

M  w \overline{h} U  wC_{3}  wΩ^{−1}M

in M_{c}. By the Octahedral Axiom there is a triangle

C_{1}  wC_{3}  wC_{2}  wΩ^{−1}C_{1}.

If \overline{f} and \overline{g} are complexity isomorphisms, then cx_{G}(C_{1}) < c and cx_{G}(C_{2}) < c, so cx_{G}(C_{3}) < c. Thus \overline{h} is a complexity isomorphism. A similar argument applies in the other cases. □

Proposition 1.6. Let M and N be objects of M_{c}, and suppose that M ≅ N in Q_{c}. Then V_{G}(M) and V_{G}(N) have the same c-dimensional components.

Proof: It suffices to consider the case in which there is a triangle in M_{c} of the form U → M → N → Ω^{−1}U with cx_{G}(U) < c. Since V_{G}(N) ⊆ V_{G}(U) ∪ V_{G}(M) and dim V_{G}(U) < c, any c-dimensional component of V_{G}(N) must also be a component of V_{G}(M). Similarly, every c-dimensional component of V_{G}(M) is a component of V_{G}(N). □

§2. Direct Sum Decompositions

The purpose of this section is to consider the way in which objects in the quotient categories Q_{c} decompose as direct sums. The main result is that each object can be expressed as a finite direct sum of indecomposables. As an example in Section 5 will show, however, the Krull–Schmidt Theorem can fail.

Begin by fixing an integer c with 1 ≤ c ≤ r. The main arguments that follow for the category Q_{c} are variations of the procedure used to construct a subadditive function on the Auslander–Reiten quiver of a group algebra in [9]. Thus if M is an object of Q_{c} and

\begin{aligned}⋯ & w \\ & P_{1}  w \\ & P_{0}  w \\ & M  w \\ & 0 \end{aligned}

is a minimal projective resolution of M, consider the Poincaré series

η_{M}(t) = ∑_{n=0}^{∞} t^{n} dim P_{n}.

For any simple kG-module S, the number of times the projective cover P_{S} of S occurs as a summand of P_{n} is given by dim Ext_{kG}^{n}(M, S). Hence

dim P_{n} = ∑_{S}(dim P_{S})(dim Ext_{kG}^{n}(M, S)).

Because Ext_{kG}^{∗}(M, S) ≅ H^{∗}(G, M^{∗} ⊗ S) is a finitely generated module for H^{∗}(G, k), it follows by Proposition 1.8.2 of [1] that there exist a polynomial f and nonnegative integers k_{1}, …, k_{n} such that

η_{M}(t) = \frac{f(t)}{∏_{i=1}^{n}(1 − t^{k_{i}})}.

In fact, k_{1}, …, k_{n} are the degrees of a set of homogeneous generators for H^{∗}(G, k). By Proposition 1.8.3 of [1] the pole of η_{M} at t = 1 has order cx_{G}(M), and the value of the analytic function (∏_{i} k_{i})η_{M}(t)(1 − t)^{cx_{G}(M)} at t = 1 is a positive integer. Let η′(M) denote this value, and set

η(M) = \{ \begin{matrix}η′(M) & if cx_{G}(M) = c \\ 0 & if cx_{G}(M) < c. \end{matrix}

Thus η(M) is the value of the analytic function (∏_{i} k_{i})η_{M}(t)(1 − t)^{c} at t = 1.

Lemma 2.1. Let 0 → M′ → M → M′′ → 0 be a short exact sequence of modules of complexity at most c. Then η(M) ≤ η(M′) + η(M′′).

Proof: We may assume without loss of generality that M′ and M′′ have no projective summands since these split off from the sequence without affecting the quantity η(M′)+η(M′′)−η(M). If P′_{∗} → M′ and P′′_{∗} → M′′ are minimal projective resolutions, then M has a projective resolution

(1)  ⋯  wP′_{1} ⊕ P′′_{1}  wP′_{0} ⊕ P′′_{0}  wM  w0.

Let f, f′, and f′′ be polynomials such that η_{M}(t) = f(t)/ ∏(1 − t^{k_{i}}), η_{M′}(t) = f′(t)/ ∏(1 − t^{k_{i}}), and η_{M′′}(t) = f′′(t)/ ∏(1 − t^{k_{i}}). Set

F (t) = η_{M′}(t) + η_{M′′}(t) − η_{M}(t) = \frac{f′(t) + f′′(t) − f(t)}{∏(1 − t^{k_{i}})}

so that F is a rational function. The projective resolution of M given in (1) is the direct sum of a minimal resolution of M and a projective resolution Q_{∗} → 0 of 0. Thus

F (t) = \frac{f′(t) + f′′(t) − f(t)}{∏(1 − t^{k_{i}})} = ∑_{n=0}^{∞} t^{n} dim Q_{n}.

If c′ is the order of the pole of F at t = 1, then certainly c′ ≤ c. Because the value of (∏ k_{i})F (t)(1 − t)^{c′} at t = 1 is a positive integer, it follows that the value of (∏ k_{i})F (t)(1 − t)^{c} at t = 1 is nonnegative. Hence η(M′) + η(M′′) − η(M) ≥ 0, as desired. □

Proposition 2.2. The function η is constant on isomorphism classes of Q_{c}.

Proof: It suffices to consider the case in which M and N are kG-modules of complexity c and there is a triangle

(1)  M  wN  wU  wΩ^{−1}M

in M_{c} with cx_{G}(U) < c. Then there is also a triangle

(2)  ΩU  wM  wN  wΩ^{−1}(ΩU).

Since η vanishes on projective modules, the previous result also applies to triangles in M_{c}. Applying the lemma to (1) and (2) gives η(N) ≤ η(M) + η(U) = η(M) and η(M) ≤ η(ΩU) + η(N) = η(N). Hence η(M) = η(N), as desired. □

Corollary 2.3. Any object in Q_{c} is a finite direct sum of indecomposable objects.

Proof: Suppose that the corollary is false. Among all objects in Q_{c} that cannot be expressed as a finite direct sum of indecomposables, choose a module M with η(M) as small as possible. Since M is not isomorphic to 0, it follows that η(M) > 0. Moreover, M is not itself indecomposable, so M ≅ M′ ⊕ M′′ for two nonzero objects M′ and M′′ of Q_{c}. Then η(M) = η(M′) + η(M′′), so it follows that 0 < η(M′) < η(M) and 0 < η(M′′) < η(M). Thus the modules M′ and M′′ are both finite direct sums of indecomposables in Q_{c} and hence so is M. This completes the proof. □

§3. Morphisms in the Quotient Categories

The process of localization in a category is usually considered to be analogous to localization in ring theory. In the case of the quotient categories Q_{c}, however, the connection is more than a mere analogy; the main result of this section shows that there is an isomorphism between Hom_{Q_{c}}(Ω^{n}(M), N) and the elements of degree n in a certain localization of Êxt_{kG}^{∗}(M, N). We will also see that the categories Q_{c} are periodic in the sense that there is an integer n > 0 such that Ω^{n}(M) ≅ M in Q_{c} for every object M. Although this property is reminiscent of the stable category of a block with a cyclic defect group, there are important differences. Indeed, if G has p-rank r and 1 ≤ c < r, then the category Q_{c} has infinitely many isomorphism classes of indecomposable objects.

Lemma 3.1. Suppose that

E : 0  wU  wM′  w s M  w0

is an exact sequence and η ∈ H^{m}(G, k) is an element such that V_{G}(U) ⊆ V_{G}(η). Then for t sufficiently large there is a homomorphism g : Ω^{mt}(k) ⊗ M → M′ such that the composition sg : Ω^{mt}(k) ⊗ M → M is a map representing η^{t} · Id_{M} ∈ Ext_{kG}^{mt}(M, M).

Proof: Since V_{G}(U) ⊆ V_{G}(η), by definition there is an integer N such that if t ≥ N, then η^{t} annihilates Ext_{kG}^{∗}(U, U). Because Ext_{kG}^{∗}(M, U) is a finitely generated module over Ext_{kG}^{∗}(U, U), it follows that η^{t} also annihilates Ext_{kG}^{∗}(M, U). Let h : Ω^{mt}(k) → k be the map representing η^{t}, and consider the exact sequence

0  wHom_{kG}(Ω^{mt}(k) ⊗ M, U)  wHom_{kG}(Ω^{mt}(k) ⊗ M, M′)  w s_{∗}

Hom_{kG}(Ω^{mt}(k) ⊗ M, M) w δ Ext_{kG}^{1}(Ω^{mt}(k) ⊗ M, U) w · · · .

The short exact sequence E determines an element cls(E) ∈ Ext_{kG}^{1}(M, U), and the connecting homomorphism δ is given by cup product with cls(E). Hence

δ(h ⊗ Id_{M}) = cls(E) · η^{t} · Id_{M} = 0,

and it follows that there is a homomorphism g : Ω^{mt}(k)⊗M → M′ such that h⊗Id_{M} = s_{∗}(g) = sg. This completes the proof. □

Lemma 3.2. Suppose that 0 → U → M′ → M → 0 is an exact sequence such that cx_{G}(U) < cx_{G}(M). Then there exists η ∈ H^{m}(G, k) for some m ≥ 0 such that V_{G}(U) ⊆ V_{G}(η) and dim(V_{G}(M) ∩ V_{G}(η)) < dim V_{G}(M).

Proof: Write V_{G}(M) ∪ V_{G}(M′) = W ∪ V_{1} ∪ ⋯ ∪ V_{t}, where V_{1}, …, V_{t} are all of the distinct components of maximum dimension and W is the union of all the remaining components. Since V_{G}(U) is a subvariety of V_{G}(M) ∪ V_{G}(M′) with dim V_{G}(U) < dim V_{G}(M) = dim V_{G}(M′), it follows that dim(V_{G}(U) ∩ V_{i}) < dim V_{i} for 1 ≤ i ≤ t. Set V′_{i} = V_{G}(U)∪(∪_{j≠i}V_{j}), and let x_{i} ∈ V_{i}−V′_{i}. Choose a homogeneous polynomial f_{i} such that f_{i}(x_{i}) ≠ 0 and f_{i} vanishes on V′_{i}. By taking appropriate powers if necessary, we may assume that f_{1}, …, f_{n} all have the same degree. Then f = f_{1} + ⋯ + f_{n} is a homogeneous polynomial, V_{G}(U) ⊆ V_{G}(f), and f(x_{i}) = f_{i}(x_{i}) ≠ 0. Thus dim(V_{G}(f) ∩ V_{i}) < dim V_{i}, and it follows that

dim(V_{G}(f) ∩ V_{G}(M)) ≤ dim(V_{G}(f) ∩ (V_{G}(M) ∪ V_{G}(M′))) < dim V_{G}(M).

Taking η to be the cohomology class corresponding to f, we get the desired result. □

Proposition 3.3. Suppose that s : M′ → M is a complexity isomorphism. Then there is an integer n ≥ 0 and a complexity isomorphism g : Ω^{n}(k) ⊗ M → M′ such that the composite sg represents ζ ∪ Id_{M} ∈ Ext_{kG}^{n}(M, M) for some cohomology class ζ ∈ H^{n}(G, k) with dim(V_{G}(ζ) ∩ V_{G}(M)) < dim V_{G}(M). In particular, sg : Ω^{n}(k) ⊗ M → M is a complexity isomorphism.

Proof: Because s is a complexity isomorphism, there is a triangle in M_{c} of the form

U  wM′  w \overline{s} M  wΩ^{−1}U

with cx_{G}(U) < cx_{G}(M) = cx_{G}(M′). By Lemma 3.2 there exist an integer m ≥ 0 and a cohomology class η ∈ H^{m}(G, k) such that V_{G}(U) ⊆ V_{G}(η) and dim(V_{G}(M) ∩ V_{G}(η)) < dim V_{G}(M). Then Lemma 3.1 implies that for t sufficiently large there is a homomorphism g : Ω^{mt}(k) ⊗ M → M′ such that sg is a map representing η^{t} · Id_{M} ∈ Ext_{kG}^{mt}(M, M). Now consider the short exact sequence

0  wL_{η}  wΩ^{m}(k)  w η̂ k  w0.

Tensoring with M gives a sequence

0  wL_{η} ⊗ M  wΩ^{m}(k) ⊗ M  w η̂ ⊗ Id_{M} k ⊗ M  w0.

Because dim(V_{G}(L_{η} ⊗ M)) = dim(V_{G}(L_{η}) ∩ V_{G}(M)) = dim(V_{G}(η) ∩ V_{G}(M)) < dim V_{G}(M), we see that η̂ ⊗ Id_{M} : Ω^{m}(k) ⊗ M → k ⊗ M is a complexity isomorphism and hence so is η̂^{t} ⊗ Id_{M} : Ω^{mt}(k) ⊗ M → k ⊗ M. Set n = mt and ζ = η^{t}. Because sg and η̂^{t} ⊗ Id_{M} both represent ζ · Id_{M}, it follows that sg is a complexity isomorphism and hence so is g. This completes the proof. □

The importance of Proposition 3.3 is that it allows us to simplify our description of morphisms in Q_{c}. Indeed, if \overline{s} : M′ → M is a complexity isomorphism and f : M′ → N is any homomorphism, then we claim that the morphism (\overline{s}, \overline{f}) : M → N in Q_{c} is equivalent to one with M′ replaced by Ω^{n}(M) for some n ≥ 0. This assertion is a consequence of Proposition 3.3, which gives the following commutative diagram:

M′ N N N N N Q \overline{s} _{ȷ ȷ ȷ ȷȷ fi} \overline{f}

M  Ω^{n}(M) u ^{\overline{s}\overline{g}}  w^{\overline{f}\overline{g}} u \overline{g}  N

Ω^{n}(M).^{A A A A D}^{\overline{s}\overline{g}} ^{````}^{ˆ\overline{f}\overline{g}}

Indeed, Proposition 3.3 even shows that \overline{s}\overline{g} represents ζ · Id_{M} ∈ Ext_{kG}^{n}(M, M) for an appropriate cohomology class ζ ∈ H^{n}(G, k).

If ζ ∈ Êxt_{kG}^{n}(M, N), then we write ζ̂ ∈ \overline{Hom}_{kG}(Ω^{n}(M), N) for the stable map corresponding to ζ under the isomorphism Êxt_{kG}^{n}(M, N) ≅ \overline{Hom}_{kG}(Ω^{n}(M), N). Note that there are no projective homomorphisms Ω^{n}(k) → k, so this notation does not conflict with that introduced previously in the case where M = N = k and ζ ∈ H^{n}(G, k) = Êxt_{kG}^{n}(k, k).

Now let M and N be objects of Q_{c}, and set

S_{M} = {ζ ∪ Id_{M} | ζ ∈ H^{∗}(G, k) is homogeneous and dim(V_{G}(ζ) ∩ V_{G}(M)) < c}.

Then S_{M} is a multiplicative subset of the center of Êxt_{kG}^{∗}(M, M), and the localization Êxt_{kG}^{∗}(M, N)S_{M}^{−1} is a graded module over the graded ring Êxt_{kG}^{∗}(M, M)S_{M}^{−1}. We write homogeneous elements of Êxt_{kG}^{∗}(M, N)S_{M}^{−1} as fractions α/β, where α ∈ Êxt_{kG}^{∗}(M, N) and β ∈ S_{M} are homogeneous. Because of the noncommutativity of Êxt_{kG}^{∗}(M, M), it is important to bear in mind that α/β = α · (1/β), but α/β ≠ (1/β) · α. In fact, the expression (1/β) · α is not even defined since Êxt_{kG}^{∗}(M, N)S_{M}^{−1} is not a left module for Êxt_{kG}^{∗}(M, M)S_{M}^{−1}.

Although it is clear that Êxt_{kG}^{∗}(M, N)S_{M}^{−1} is a right Êxt_{kG}^{∗}(M, M)S_{M}^{−1}-module, it is perhaps less obvious that it is a graded Êxt_{kG}^{∗}(N, N)S_{N}^{−1}-Êxt_{kG}^{∗}(M, M)S_{M}^{−1}- bimodule. The following lemma will be used to describe the left module structure.

Lemma 3.4. Let M and N be objects of Q_{c}. If α ∈ Êxt_{kG}^{∗}(M, N) and η ∈ S_{N} are homogeneous, then there are homogeneous elements α′ ∈ Êxt_{kG}^{∗}(M, N) and η′ ∈ S_{M} such that αη′ = ηα′.

Proof: Let m = deg α and n = deg η, and consider the triangle

Ω^{n}(N)  w η̂ N  w \overline{f} U  wΩ^{−1}Ω^{n}(N).

Since η ∈ S_{N}, it follows that η̂ is a complexity isomorphism so that dim V_{G}(U) < c. Hence \overline{f}α̂ : Ω^{m}(M) → U is a map with (1, \overline{f}α̂) = 0 in Q_{c}, and Proposition 1.20 of [4] implies that there exist an object M′ and a complexity isomorphism \overline{g} : M′ → Ω^{m}(M) such that \overline{f}α̂\overline{g} = 0. By Proposition 3.3 we may assume that M′ = Ω^{ℓ}(M) for some ℓ ≥ m and that \overline{g} = Ω^{m}η̂′ for some η′ ∈ S_{M}. Because \overline{f}α̂(Ω^{m}η̂′) = 0, it follows that there is a map \overline{h} : Ω^{ℓ}(M) → Ω^{n}(N) such that α̂(Ω^{m}η̂′) = η̂\overline{h}. Let α′ be the element of Êxt_{kG}^{ℓ−n}(M, N) with Ω^{n}α̂′ = \overline{h}. Then the equation α̂(Ω^{m}η̂′) = η̂(Ω^{n}α̂′) implies that αη′ = ηα′. This completes the proof. □

Using Lemma 3.4, we can define the structure of a graded left Êxt_{kG}^{∗}(N, N)S_{N}^{−1}- module on Êxt_{kG}^{∗}(M, N)S_{M}^{−1}. Indeed, suppose that α/β ∈ Êxt_{kG}^{∗}(M, N)S_{M}^{−1} and ζ/η ∈ Êxt_{kG}^{∗}(N, N)S_{N}^{−1} are homogeneous. Then by Lemma 3.4 there are homogeneous elements α′ ∈ Êxt_{kG}^{∗}(M, N) and η′ ∈ S_{M} such that αη′ = ηα′. Define

\frac{ζ}{η} · \frac{α}{β} = \frac{ζα′}{η′β} = \frac{ζα′}{βη′} .

It is a straightforward exercise to show that this action is well-defined and that Êxt_{kG}^{∗}(M, N)S_{M}^{−1} is a graded Êxt_{kG}^{∗}(N, N)S_{N}^{−1}-Êxt_{kG}^{∗}(M, M)S_{M}^{−1}-bimodule.

More generally, if M, N, and U are objects of Q_{c}, then the usual composition product ∘ : Êxt_{kG}^{∗}(U, N) ⊗ Êxt_{kG}^{∗}(M, U) → Êxt_{kG}^{∗}(M, N) gives a well-defined product ∘ : Êxt_{kG}^{∗}(U, N)S_{U}^{−1} ⊗ Êxt_{kG}^{∗}(M, U)S_{M}^{−1} → Êxt_{kG}^{∗}(M, N)S_{M}^{−1}. This product is defined on homogeneous elements by exactly the same procedure as that used above: if α/β ∈ Êxt_{kG}^{∗}(M, U)S_{M}^{−1} and ζ/η ∈ Êxt_{kG}^{∗}(U, N)S_{U}^{−1} are homogeneous, let α′ ∈ Êxt_{kG}^{∗}(M, U) and η′ ∈ S_{M} be homogeneous elements such that αη′ = ηα′, and set

\frac{ζ}{η} ∘ \frac{α}{β} = \frac{ζ ∘ α′}{η′β} = \frac{ζ ∘ α′}{βη′} .

The following theorem shows that if m is any integer, then Hom_{Q_{c}}(Ω^{m}(M), N) can be identified with the elements of degree m in Êxt_{kG}^{∗}(M, N)S_{M}^{−1}.

Theorem 3.5. Let M and N be objects of Q_{c}. Then for any integer m there is a vector space isomorphism between Hom_{Q_{c}}(Ω^{m}(M), N) and the subspace of elements of degree m in Êxt_{kG}^{∗}(M, N)S_{M}^{−1}, and this isomorphism commutes with compositions.

Proof: Let E^{m}(M, N) ⊆ Êxt_{kG}^{∗}(M, N)S_{M}^{−1} denote the subspace of elements of degree m, and supose that α/β ∈ E^{m}(M, N). Set n = deg β so that deg α = m + n. Since β ∈ S_{M}, it follows that β̂ : Ω^{n}(M) → M is a complexity isomorphism, and (Ω^{m}β̂, α̂) ∈ Hom_{Q_{c}}(Ω^{m}(M), N). It is straightforward to check that the map Φ : E^{m}(M, N) → Hom_{Q_{c}}(Ω^{m}(M), N) given by Φ(α/β) = (Ω^{m}β̂, α̂) is a well-defined homomorphism of vector spaces. Moreover, Φ is an epimorphism by Proposition 3.3.

Now suppose that (Ω^{m}β̂, α̂) = Φ(α/β) = 0. Since the map (Ω^{m}β̂, 1) : Ω^{m}(M) → Ω^{m+n}(M) is invertible in Q_{c} and (Ω^{m}β̂, α̂) = (1, α̂) ∘ (Ω^{m}β̂, 1), it follows that (1, α̂) : Ω^{m+n}(M) → N is zero. By Proposition 1.20 of [4] we see that there exist a module M′ and a complexity isomorphism \overline{g} : M′ → Ω^{m+n}(M) such that α̂\overline{g} = 0. Proposition 3.3 allows us to assume that M′ = Ω^{i}(M) for some i ≥ m + n and that \overline{g} = Ω^{m+n}γ̂ for some γ ∈ S_{M}. Since α̂ ∘ (Ω^{m+n}γ̂) = 0 in \overline{Hom}_{kG}(Ω^{i}(M), N), it follows that αγ = 0 in Êxt_{kG}^{i}(M, N), and α/β = αγ/βγ = 0. Hence Φ is an isomorphism.

Finally, we must show that Φ commutes with compositions. Let U be another object of Q_{c}. Suppose that α/β ∈ E^{m}(M, U) and ζ/η ∈ E^{n}(U, N), and let α′ ∈ Êxt_{kG}^{∗}(M, U) and η′ ∈ S_{M} be homogeneous elements such that αη′ = ηα′. Let i = deg β and j = deg η, and set ℓ = deg η′ + i + m + n. Since αη′ = ηα′, it follows that α̂(Ω^{i+m}η̂′) = η̂(Ω^{j}α̂′), and the diagram

Ω^{ℓ}(M) ` ` ` `˜ Ω^{i+m+n}η̂′ _{A A AA C} Ω^{j+n}α̂′

Ω^{i+m+n}(M) ` ` ` `˜ Ω^{m+n}β̂  _{A A AA C} Ω^{n}α̂  Ω^{j+n}(U) ` ` ` `Ω̃^{n}η̂  _{A A AA C} ζ̂

Ω^{m+n}(M)  Ω^{n}(U)  N.

shows that

Φ (\frac{ζ}{η} ) ∘ Φ (\frac{α}{β}) = (Ω^{n}η̂, ζ̂) ∘ (Ω^{m}β̂, α̂)

= (Ω^{m+n}β̂ ∘ Ω^{i+m+n}η̂′, ζ̂ ∘ Ω^{j+n}α̂′)

= Φ (\frac{ζ ∘ α′}{βη′} )

= Φ (\frac{ζ}{η} ∘ \frac{α}{β}) .

This completes the proof. □

Let ζ_{1}, …, ζ_{r} be homogeneous elements of H^{∗}(G, k) such that k[ζ_{1}, …, ζ_{r}] is a polynomial subring of H^{∗}(G, k). If H^{∗}(G, k) is finitely generated as a module over k[ζ_{1}, …, ζ_{r}], then ζ_{1}, …, ζ_{r} is said to be a homogeneous set of parameters for H^{∗}(G, k). An equivalent condition is that the homogeneous hypersurfaces V_{G}(ζ_{i}) of V_{G}(k) intersect in {0} and that the number of hypersurfaces is exactly dim V_{G}(k). By the Noether Normalization Lemma a homogeneous set of parameters always exists.

Proposition 3.6. There is an integer n > 0 such that Ω^{n}(M) ≅ M in Q_{c} for every object M.

Proof: Let ζ_{1}, …, ζ_{r} be a homogeneous set of parameters for H^{∗}(G, k). Set n = lcm{deg ζ_{i}|1 ≤ i ≤ r} and η_{i} = ζ_{i}^{n/ deg ζ_{i}} so that η_{1}, …, η_{r} all have degree n. Let W be the vector subspace of H^{n}(G, k) spanned by η_{1}, …, η_{r}.

If M is a module of complexity c, let V_{1}, …, V_{t} denote all of the distinct components of V_{G}(M) of dimension c, and let I_{j} ⊆ H^{∗}(G, k) denote the ideal of the variety V_{j} for 1 ≤ j ≤ t. If there is a j such that W ∩ I_{j} = W, then η_{i} ∈ I_{j} for all i and hence V_{j} ⊆ V_{G}(η_{i}). But then V_{j} ⊆ ⋂_{i} V_{G}(η_{i}) = {0}, which is impossible. Thus W ∩ I_{j} is a proper subspace of W for all j. Since the field k is infinite, it follows that

W ≠ ⋃_{j=1}^{t}(W ∩ I_{j}).

Choose η ∈ W − ⋃_{j}(W ∩ I_{j}). Since η ∉ I_{j} for 1 ≤ j ≤ t, it follows that V_{j} ⊈ V_{G}(η) and hence dim(V_{G}(η) ∩ V_{G}(M)) < dim V_{G}(M). If f : Ω^{n}(M) → M represents η · Id_{M} ∈ Ext_{kG}^{n}(M, M), then f is a complexity isomorphism. Thus Ω^{n}(M) ≅ M in Q_{c}, as desired. □

To provide an interpretation of Proposition 3.6, let B be a block of kG, and suppose that r(B) is the p-rank of a defect group of B. If 1 ≤ c ≤ r(B), then we can define the quotient category Q_{c}(B) to be the full subcategory of Q_{c} consisting of modules lying in the block B. Let e_{c} be the smallest positive integer such that Ω^{2e_{c}}(M) ≅ M for every object M of Q_{c}(B). This defines a sequence of integers (e_{1}, …, e_{r(B)}).

Now suppose that B′ is a block of another group algebra over k and that B and B′ are stably equivalent. Since cx_{G}(M) is the growth rate of Ext_{kG}^{∗}(M, M), it is easy to see that the equivalence preserves complexities, so it induces equivalences between Q_{c}(B) and Q_{c}(B′) for all c. Hence the sequence of integers (e_{1}, …, e_{r(B)}) is an invariant of stable equivalence.

Note that if B is a block with a cyclic defect group, then r(B) = 1 so that the above procedure defines a single integer e = e_{1}. It is well-known that in this case the integer e is the number of edges in the Brauer tree corresponding to B; equivalently, it is the number of simple kG-modules lying in the block.

Lemma 3.7. Suppose that M and N are kG-modules of complexity at most c. If dim(V_{G}(M) ∩ V_{G}(N)) < c, then Hom_{Q_{c}}(M, N) = 0.

Proof: As we have seen, any morphism from M to N in Q_{c} can be expressed as α/β, where α ∈ Êxt_{kG}^{n}(M, N) and β ∈ S_{M} are homogeneous of the same degree. Let α′ : Ω^{n}(M) → N be a kG-homomorphism representing α. We will show that α′ factors through an object of complexity less than c so that α/β = 0.

Consider the maps f : Ω^{n}(M) → Ω^{n}(M) ⊗ Hom_{k}(Ω^{n}(M), N) and g : Ω^{n}(M) ⊗ Hom_{k}(Ω^{n}(M), N) → N given by f(x) = x⊗α′ and g(x⊗λ) = λ(x). Because α′ is a Ginvariant element of Hom_{k}(Ω^{n}(M), N), it follows that f and g are kG-homomorphisms with gf = α′. Thus α′ factors through the module Ω^{n}(M) ⊗ Hom_{k}(Ω^{n}(M), N) ≅ Ω^{n}(M) ⊗ Ω^{n}(M)^{∗} ⊗ N, and Proposition 1.1 implies that

dim(V_{G}(Ω^{n}(M) ⊗ Ω^{n}(M)^{∗} ⊗ N)) = dim(V_{G}(M) ∩ V_{G}(N)) < c.

In the quotient category Q_{c}, therefore, it follows that α/β = 0, as desired. □

Proposition 3.8. Suppose that 1 ≤ c < r. Then there are infinitely many isomorphism classes of indecomposable objects in Q_{c}.

Proof: Let V be an r-dimensional component of V_{G}(k). Then there is an elementary abelian subgroup E of order p^{r} such that V = res_{G,E}^{∗}(V_{E}(k)) is the image of V_{E}(k) under the map V_{E}(k) → V_{G}(k) induced by restriction.

Recall that V_{E}(k) is isomorphic to affine r-space and that the restriction map is finite-to-one on varieties. Thus if U is a c-dimensional subspace of V_{E}(k), then

X(U) = res_{G,E}^{∗−1}(res_{G,E}^{∗}(U))

is the union of a finite number of c-dimensional subspaces. Because k is infinite, V_{E}(k) is not the union of a finite number of c-dimensional subspaces. It follows that there is an infinite collection {U_{i} | i ∈ ℕ} of c-dimensional subspaces of V_{E}(k) such that U_{i} ⊆ X(U_{j}) if and only if i = j.

Set W_{i} = res_{G,E}^{∗}(U_{i}). Then {W_{i} | i ∈ ℕ} is an infinite collection of c-dimensional closed homogeneous subvarieties in V_{G}(k); moreover, if i ≠ j, then dim(W_{i} ∩W_{j}) < c. By Proposition 1.1(5) there is a kG-module M_{i} such that V_{G}(M_{i}) = W_{i}. Since M_{i} is a nonzero object of Q_{c}, it has a nonzero indecomposable summand N_{i} by Corollary 2.3, and V_{G}(N_{i}) = W_{i}. Lemma 3.7 implies that Hom_{Q_{c}}(N_{i}, N_{j}) = 0 for i ≠ j, so it follows that {N_{i} | i ∈ ℕ} is an infinite collection of pairwise nonisomorphic indecomposable objects in Q_{c}. This completes the proof. □

As the examples in Section 5 will show, the conclusion of Proposition 3.8 can fail when c = r.

§4. The Endomorphism Ring of the Trivial Module

The purpose of this section is to consider the structure of the endomorphism ring End_{Q_{r}}(k), where r is the p-rank of G. We begin by writing V_{G}(k) = W ∪V_{1} ∪⋯∪V_{t}, where V_{1}, …, V_{t} are all the distinct components of maximum dimension and W is the union of all the other components. Choose homogeneous elements ζ_{1}, …, ζ_{t} ∈ H^{∗}(G, k) such that

(∗)  W ∪ (⋃_{j≠i} V_{j}) ⊆ V_{G}(ζ_{i})  and  V_{G}(ζ_{i}) ∩ V_{i} ≠ V_{i}.

Lemma 4.1. There exist homogeneous elements ζ_{1}, …, ζ_{t} ∈ H^{∗}(G, k) satisfying (∗) such that deg ζ_{i} = deg ζ_{j} and ζ_{i}ζ_{j} = 0 for all i ≠ j.

Proof: Begin by choosing any homogeneous elements ζ_{1}, …, ζ_{t} so that (∗) holds. If i ≠ j, then V_{G}(ζ_{i}ζ_{j}) = V_{G}(ζ_{i})∪V_{G}(ζ_{j}) = V_{G}(k) so that ζ_{i}ζ_{j} is nilpotent. Let n_{ij} be an integer with (ζ_{i}ζ_{j})^{n_{ij}} = 0, and set N = max{n_{ij} | i ≠ j}. Then 0 = (ζ_{i}ζ_{j})^{N} = ζ_{i}^{N}ζ_{j}^{N} whenever i ≠ j, so the last condition in the statement of the lemma is satisfied if we replace ζ_{i} by ζ_{i}^{N} for all i. Now set n = lcm {deg ζ_{i} | i = 1, …, t}, and replace ζ_{i} by ζ_{i}^{n/ deg ζ_{i}} for all i. Then we also have deg ζ_{i} = deg ζ_{j} for all i and j, as desired. □

Lemma 4.2. Suppose that ζ_{1}, …, ζ_{t} are chosen as in Lemma 4.1. Set ζ = ζ_{1}+⋯+ζ_{t}, and let n = deg ζ. Then dim V_{G}(ζ) < dim V_{G}(k), and ζ̂: Ω^{n}(k) → k is a complexity isomorphism.

Proof: By (∗) it follows that for each i there is a maximal ideal M ∈ V_{i} such that ζ_{j} ∈ M if and only if j ≠ i. Hence ζ ∉ M and V_{G}(ζ) ∩ V_{i} ≠ V_{i} for all i. Since V_{1}, …, V_{t} are all of the components of V_{G}(k) of maximum dimension, dim V_{G}(ζ) < dim V_{G}(k). Moreover, L_{ζ} = Ker ζ̂ has variety V_{G}(L_{ζ}) = V_{G}(ζ), so ζ̂ is a complexity isomorphism, as desired. □

Theorem 3.5 provides an identification of End_{Q_{r}}(k) with the elements of degree zero in the localization of Êxt_{kG}^{∗}(k, k). As a consequence, the following result is immediate from Lemmas 4.1 and 4.2.

Lemma 4.3. The elements e_{1}, …, e_{t} ∈ End_{Q_{r}}(k) defined by e_{i} = ζ_{i}/ζ are orthogonal idempotents.

It should be pointed out, however, that because Q_{r} is generally not an abelian category, the existence of nontrivial idempotents in End_{Q_{r}}(k) does not imply that k decomposes as a direct sum. It does, of course, give a direct sum decomposition

End_{Q_{r}}(k) = e_{1} End_{Q_{r}}(k) ⊕ ⋯ ⊕ e_{t} End_{Q_{r}}(k).

Proposition 4.4. For all i the idempotent e_{i} is primitive, and e_{i} End_{Q_{r}}(k) is a local ring.

Proof: Let f ∈ e_{i} End_{Q_{r}}(k). Then f = e_{i} · (α/β), where α ∈ Êxt_{kG}^{∗}(k, k) and β ∈ H^{∗}(G, k) are homogeneous of the same degree and dim V_{G}(β) < dim V_{G}(k). If V_{G}(α) ∩ V_{i} = V_{i}, then V_{G}(ζ_{i}α) = V_{G}(ζ_{i}) ∪ V_{G}(α) = V_{G}(k). Hence ζ_{i}α is nilpotent and so is f = ζ_{i}α/ζβ.

Now suppose that V_{G}(α) ∩ V_{i} ≠ V_{i}. Let m and n be positive integers such that deg α^{m} = deg ζ^{n}. Let J = {j | V_{j} ∩ V_{G}(α) = V_{j}}, and set

α′ = α^{m} + ∑_{j∈J} ζ_{j}^{n}.

Then f^{m} = e_{i}·(α/β)^{m} = ζ_{i}α′/ζβ^{m} = e_{i}·(α′/β^{m}), and we will show that dim V_{G}(α′) < dim V_{G}(k) so that f is a unit.

Fix an integer ℓ with 1 ≤ ℓ ≤ t. If V_{ℓ} ∩ V_{G}(α) ≠ V_{ℓ}, then there is a maximal ideal M ∈ V_{ℓ} with α ∉ M. Since ℓ ∉ J, it follows by the properties of ζ_{j} that V_{ℓ} ⊆ V_{G}(ζ_{j}) for all j ∈ J. Hence ∑_{j∈J} ζ_{j}^{n} ∈ M and α′ ∉ M. On the other hand, if V_{ℓ} ∩ V_{G}(α) = V_{ℓ}, then ℓ ∈ J. Let M ∈ V_{ℓ} − V_{G}(ζ_{ℓ}). Then α ∈ M and ζ_{ℓ} ∉ M, but ζ_{j} ∈ M for all j ∈ J with j ≠ ℓ, so again it follows that α′ ∉ M. Thus in either case V_{ℓ} ∩ V_{G}(α′) ≠ V_{ℓ}. Because this condition holds for all ℓ, it follows that dim V_{G}(α′) < dim V_{G}(k), and f is a unit.

We have now seen that any f ∈ e_{i} End_{Q_{r}}(k) is either nilpotent or invertible. Hence e_{i} End_{Q_{r}}(k) is a local ring, and e_{i} is primitive. □

The following theorem is an immediate consequence of Proposition 4.4.

Theorem 4.5. Let t be the number of components of V_{G}(k) of maximum dimension. Then End_{Q_{r}}(k) is a direct sum of t local rings.

§5. Examples

In this section we present examples that demonstrate some of the behavior that can occur in the quotient categories. These examples show that the quotients have properties that are generally quite different from those of the original stable category. Specifically, we let k be an algebraically closed field of characteristic p > 0, and we consider the quotient category Q_{r} when G is a p-group of rank r.

Example 5.1.a. G is an elementary abelian 2-group of rank r.

In this case H^{∗}(G, k) is isomorphic to the polynomial ring R = k[ζ_{1}, …, ζ_{r}]. If ζ ∈ H^{∗}(G, k) is a nonzero homogeneous element, then V_{G}(ζ) ≠ V_{G}(k) because H^{∗}(G, k) is an integral domain. Since V_{G}(k) has only one component, it follows that dim V_{G}(ζ) < r, and S_{k} is the set of all nonzero homogeneous elements in H^{∗}(G, k). The map given by replacing ζ_{r} by 1 defines an isomorphism between the subring of elements of degree zero in S_{k}^{−1}R and the pure transcendental extension k(ζ_{1}, …, ζ_{r−1}) of k of degree r − 1. Hence End_{Q_{r}}(k) ≅ k(ζ_{1}, …, ζ_{r−1}). In particular, we see that the endomorphism ring of an indecomposable object can be infinite-dimensional over k.

When r = 2, we can say slightly more. If ζ ∈ H^{1}(G, k) is nonzero, then ζ̂ : Ω(k) → k is a complexity isomorphism, and it follows that Ω^{n}(k) ≅ k in Q_{2} for all n ∈ ℤ. Since all indecomposable modules of complexity two are of the form Ω^{n}(k) for some n, it follows that k is the unique indecomposable object in Q_{2}.

We remark in addition that Q_{2} has similar properties when G = A_{4} and p = 2: the trivial module is the unique indecomposable object in Q_{2}, and End_{Q_{2}}(k) is infinitedimensional over k.

Example 5.1.b. G is an elementary abelian p-group of rank r, and p is odd.

There are elements η_{1}, …, η_{r} of degree one and ζ_{1}, …, ζ_{r} of degree two such that

H^{∗}(G, k) ≅ k[ζ_{1}, …, ζ_{r}] ⊗ Λ(η_{1}, …, η_{r})

is the tensor product of a polynomial algebra and an exterior algebra. The multiplicative set S_{k} consists of all homogeneous elements not contained in the ideal generated by η_{1}, …, η_{r}, so the endomorphism ring of k is a local ring with a unique maximal ideal generated by elements of the form η_{i}η_{j}/ζ_{r}. The quotient by this maximal ideal is again a pure transcendental extension of degree r − 1.

The reader may notice, however, that for an arbitrary group the quotient of a component of End_{Q_{r}}(k) by its maximal ideal will not necessarily be a pure transcendental extension of k. Instead, suppose that V is the corresponding component of the variety V_{G}(k) and E is a maximal elementary abelian subgroup associated to V as in the Quillen Dimension Theorem (8.3.3 of [3]). Then the quotient will be an invariant subfield of a pure transcendental extension under the action of N_{G}(E)/C_{G}(E).

Example 5.2. G is the dihedral group D_{2^{n}} of order 2^{n}, n ≥ 3.

We begin with a brief summary of the indecomposable modules for the infinite dihedral group D_{∞} = ⟨x, y | x^{2} = y^{2} = 1⟩ over a field of characteristic two. The reader is referred to [8] for more detail.

Consider a, b, a^{−1}, and b^{−1} as the letters of a formal language, and let (a^{−1})^{−1} = a and (b^{−1})^{−1} = b. If ℓ is a letter, write ℓ^{∗} for either ℓ or ℓ^{−1}. A word C = ℓ_{1} ⋯ ℓ_{m} is given by a sequence of letters ℓ_{1}, …, ℓ_{m} subject to the condition that ℓ_{i} = a^{∗} implies ℓ_{i+1} = b^{∗} and ℓ_{i} = b^{∗} implies ℓ_{i+1} = a^{∗} for 1 ≤ i < m. The number m = |C| is called the length of C. Let W be the set of all words in a, b, a^{−1}, and b^{−1} together with two words of length zero, denoted 1_{a} and 1_{b}, with (1_{a})^{−1} = 1_{b} and (1_{b})^{−1} = 1_{a}. If C = ℓ_{1} ⋯ ℓ_{m} is a word, then its inverse is defined to be the word C^{−1} = ℓ_{m}^{−1} ⋯ ℓ_{1}^{−1}. Let ρ be the equivalence relation on W that identifies every word with its inverse, and let W_{1} = W/ρ.

If C is a word such that |C| is positive and even, then the m^{th} power C^{m} is the word obtained by juxtaposing m copies of the word C. Let W′ be the subset of W consisting of all words C such that |C| is positive and even, and C is not a power of a shorter word. Let ρ′ be the equivalence relation on W′ that identifies each word C = ℓ_{1} ⋯ ℓ_{m} with its inverse and with its images under cyclic permutations ℓ_{1} ⋯ ℓ_{m−1}ℓ_{m} ↦ ℓ_{m}ℓ_{1} ⋯ ℓ_{m−1}. Set W_{2} = W′/ρ′.

The words in W_{1} and W_{2} now give rise to two types of indecomposable modules for D_{∞}.

1. Modules of the first kind Let C = ℓ_{1} ⋯ ℓ_{m} be a word of length m in W. Let M(C) be an (m + 1)-

dimensional vector space with basis z_{0}, …, z_{m} on which a and b operate according to the following schema:

kz_{0}  kz_{1} u ℓ_{1}  u ℓ_{2} ⋯  kz_{m}. u ℓ_{m}

For example, if C = ab^{−1}aba^{−1}, then the schema is

kz_{0}  kz_{1} u a  w b kz_{2}  kz_{3} u a  kz_{4} u b  w a kz_{5}.

This diagram indicates that z_{0} maps to 0 under both a − 1 and b − 1; z_{1} maps to z_{0} under a − 1 and to z_{2} under b − 1; z_{2} maps to 0 under both a − 1 and b − 1; z_{3} maps to z_{2} under a − 1 and to 0 under b − 1; and so on. It is easy to check that M(C) ≅ M(C^{−1}).

2. Modules of the second kind Let V be a finite-dimensional vector space over k. An automorphism ϕ of V

is said to be indecomposable if its rational canonical form has only one block, and this block is associated with a power of an irreducible polynomial over k. Now let V = {(V, ϕ) | V is a finite-dimensional vector space over k and ϕ is an indecomposable automorphism of V }. If (C, V, ϕ) ∈ W′ × V with C = ℓ_{1} ⋯ ℓ_{m}, let M(C, V, ϕ) be the direct sum of m copies of V, say V_{0} ⊕ ⋯ ⊕ V_{m−1}. The group D_{∞} acts on M(C, V, ϕ) according to the schema

\begin{matrix}V_{0} u ℓ_{0} = 1_{V} & V_{1} u ℓ_{1} = ϕ & V_{2} uℓ_{2} = 1_{V} \\ V_{m−1} w ℓ_{m−1} = 1_{V} & ⋯ ℓ_{4} = 1_{V} w & V_{3}. uℓ_{3} = 1_{V} \end{matrix}

If C and C′ are words representing the same element of W_{2}, then M(C, V, ϕ) ≅ M(C′, V, ϕ).

The two kinds of modules described above provide a complete set of nonisomorphic indecomposable kD_{∞}-modules. Set q = 2^{n−2}. An indecomposable kD_{∞}-module is also a module for the quotient group G = D_{2^{n}} if and only if one of the following conditions holds:

(1) the module is of the first kind and the corresponding word does not contain (ab)^{q}, (ba)^{q}, or their inverses;

(2) the module is of the second kind and no power of the corresponding word contains (ab)^{q}, (ba)^{q}, or their inverses; or

(3) the module is the projective indecomposable module M((ab)^{q}(ba)^{−q}, k, 1_{k}).

Set A = (ab)^{q−1}a and B = (ba)^{q−1}b. Then all modules of the first kind except M(A) and M(B) have complexity two. The modules M(A) and M(B) are periodic, and so are all the modules of the second kind except the projective module kG itself. Thus every nonzero object in Q_{2} is isomorphic to a direct sum of modules of the first kind.

Lemma 5.2.1. Suppose that G = D_{2^{n}} is the dihedral group of order 2^{n} for n ≥ 3. Then k ⊕ k ≅ M(a) ⊕ M(b) in Q_{2}.

Proof: We will prove the lemma for the dihedral group D_{8} and leave the reader to make the slight changes necessary for D_{2^{n}} when n > 3.

Because M(a^{−1}b^{−1}a^{−1}) is periodic, it is easy to check that there is an isomorphism M(a) ≅ M(aba^{−1}b^{−1}a^{−1}) in Q_{2}. Similarly, M(b) ≅ M(bab^{−1}a^{−1}b^{−1}) in Q_{2}. Let f : M(aba^{−1}b^{−1}a^{−1}) → k ⊕ k and g : M(bab^{−1}a^{−1}b^{−1}) → k ⊕ k be epimorphisms. Then there is a short exact sequence

0  wU  wM(aba^{−1}b^{−1}a^{−1}) ⊕ M(bab^{−1}a^{−1}b^{−1})  w (f, g) k ⊕ k  w0,

and U is a module of the second kind. In particular, U is periodic so that in Q_{2} there are isomorphisms

k ⊕ k ≅ M(aba^{−1}b^{−1}a^{−1}) ⊕ M(bab^{−1}a^{−1}b^{−1}) ≅ M(a) ⊕ M(b).

This completes the proof. □

Proposition 5.2.2. If n ≥ 3 and G = D_{2^{n}} is the dihedral group of order 2^{n}, then Q_{2} has exactly three isomorphism classes of indecomposable objects. These classes are represented by the modules k, M(a), and M(b).

Proof: For simplicity we again give a proof for G = D_{8}, but the same proof works with minor modifications for any n ≥ 3.

Suppose, then, that G = D_{8}. We begin by showing that any non-periodic module of the first kind is isomorphic in Q_{2} to either k, M(a), or M(b). Let M be such a module, and let C be the corresponding word. If |C| ≤ 1, then it is trivial that M = M(C) is isomorphic to either k, M(a), or M(b).

Now let C be a word of length |C| > 1 such that M = M(C) is not periodic. The goal will be to show that M(C) is isomorphic in Q_{2} to M(D), where either |D| < |C| or D is a word of the form a ⋯ b^{−1}a^{−1}, a ⋯ b^{−1}, b ⋯ a^{−1}b^{−1}, a^{−1} ⋯ ba, a^{−1} ⋯ b, or b^{−1} ⋯ ab. Indeed, suppose that such a result holds. If |D| < |C|, then it follows by induction that M(C) ≅ M(D) is isomorphic in Q_{2} to either k, M(a), or M(b). If D = a ⋯ b^{−1}a^{−1}, let D_{1} and D_{2} be the subwords of D such that D = aD_{1} = D_{2}b^{−1}a^{−1}. These words determine submodules M(D_{1}) and M(D_{2}) of M(D), and the cokernels are given by M(D)/M(D_{1}) ≅ k and M(D)/M(D_{2}) ≅ M(a). Thus there are homomorphisms f : M(D) → M(a) and g : M(D) → M(a) with kernels M(D_{1}) and M(D_{2}) respectively. It is easy to check that there is a short exact sequence

0  wKer (f + g)  wM(D)  w f + g M(a)  w0

in which Ker (f + g) is a module of the second kind. In particular, Ker (f + g) is periodic, and it follows that M(D) ≅ M(a) in Q_{2}. Similar arguments show that if D = a ⋯ b^{−1}, then M(D) ≅ k in Q_{2}; and if D = b ⋯ a^{−1}b^{−1}, then M(D) ≅ M(b) in Q_{2}. In the remaining cases the result follows by dual arguments.

We now proceed to show that M(C) ≅ M(D) in Q_{2}, where D satisfies one of the above conditions. Because every word is equivalent to its inverse in W_{1}, the assumption that |C| > 1 implies that C is equivalent to a word of the form a ⋯ a, a ⋯ b, a ⋯ a^{−1}, a ⋯ b^{−1}, b ⋯ a, b ⋯ b, b ⋯ b^{−1}, a^{−1} ⋯ a, a^{−1} ⋯ b, or b^{−1} ⋯ b. Since M(C) is not periodic, it follows that C ≠ bab and C ≠ aba. First suppose that C = a ⋯ a. If in fact C = a ⋯ aba, then C must be of the form C = a ⋯ b^{−1}aba since C cannot contain the subword (ba)^{2}. Let D be the subword of C such that C = Db^{−1}aba. Because the module M(aba) is periodic, it is easy to see that M(C) ≅ M(D) in Q_{2}. Moreover, |D| < |C|, and the desired result follows in this case. Thus we may assume that C ≠ a ⋯ aba. Let D = Cba^{−1}b^{−1}a^{−1} = a ⋯ aba^{−1}b^{−1}a^{−1}. Because M(a^{−1}b^{−1}a^{−1}) is periodic, it follows that M(C) ≅ M(D) in Q_{2}, and D is a word of the desired type. Similar arguments apply if C has one of the remaining nine possible forms.

It now follows that there are at most three indecomposable objects in Q_{2}. The variety V_{G}(k) is a union of two planes intersecting in a line; the variety V_{G}(M(a)) is one of these planes, and V_{G}(M(b)) is the other. Because the varieties of M(a) and M(b) have a single component, it follows from Proposition 1.6 that M(a) and M(b) are indecomposable in Q_{2}. If k is not also indecomposable, then Proposition 1.6 implies that there is an object M′ such that k ≅ M(a) ⊕ M(b) ⊕ M′ in Q_{2}. If η is the function defined in Section 2, then Lemma 5.2.1 gives the equation η(M(a)) + η(M(b)) + 2η(M′) = 0. Thus we get a contradiction, and it follows that k is also indecomposable in Q_{2}. This completes the proof. □

Remarks:

1. Since k, M(a), and M(b) are nonisomorphic indecomposable objects of Q_{2}, the isomorphism k ⊕ k ≅ M(a) ⊕ M(b) shows that the Krull–Schmidt Theorem fails in this category.

2. The relation k⊕k ≅ M(a)⊕M(b) is actually quite curious. Since End_{Q_{2}}(k⊕k) is the ring of 2 × 2 matrices over End_{Q_{2}}(k) and End_{Q_{2}}(k) is a direct sum of two local rings by Theorem 4.5, one can check that End_{Q_{2}}(k ⊕k) is the direct sum of two copies of the ring of 2 × 2 matrices over a transcendental extension of k. On the other hand, Hom_{Q_{2}}(M(a), M(b)) = 0 and Hom_{Q_{2}}(M(b), M(a)) = 0 since M(a)⊗M(b) is periodic. Thus

End_{Q_{2}}(M(a) ⊕ M(b)) ≅ End_{Q_{2}}(M(a)) ⊕ End_{Q_{2}}(M(b)),

and it follows that End_{Q_{2}}(M(a)) ≅ End_{Q_{2}}(M(b)) must be the ring of 2 × 2 matrices over a transcendental extension of k. This fact can also be proven by an analysis of the results of [2]. Hence the cohomology rings of modules induced from maximal elementary abelian subgroups seem to play an important role in this subject. It is possible that the transfer maps from the maximal elementary abelian subgroups are also involved with the structure of the quotient categories.

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