STANDARD STRATIFICATIONS OF EI CATEGORIES AND ALPERIN’S WEIGHT CONJECTURE

PETER WEBB

Abstract. We characterize the finite EI categories whose representations are standardly stratified with respect to the natural preorder on the simple representations. The orbit category of a finite group with respect to any set of subgroups is always such a category. Taking the subgroups to be the p-subgroups of the group, we reformulate Alperin’s weight conjecture in terms of the standard and proper costandard representations of the orbit category. We do this using the properties of the Ringel dual construction and a theorem of Dlab, which have elsewhere been described for standardly stratified algebras where there is a partial order on the simple modules. We indicate that these results hold in the generality of an algebra whose simple modules are preordered, rather than partially ordered.

1. Introduction

An EI category is one in which every endomorphism is an isomorphism. In this paper we are particularly motivated by the examples of EI categories which are constructed from families of subgroups of a given group. These include the orbit category of a group with respect to a family of subgroups, the Frobenius category, the transporter category, and the Brauer category of a block. Groups themselves, as well as partially-ordered sets are also examples of EI categories.

A representation of a category over a commutative ring is simply a functor from the category to the category of modules for that ring. When the category is a group this is a representation in the usual sense, and if the category happens to be a poset we obtain a representation of the incidence algebra of the poset. We develop a particular aspect of the representation theory of EI categories, and this is a combination of the representation theories of groups and of posets.

The main application we give in this paper of the theory of representations of EI categories is to the local determination of properties of representations of finite groups, and in particular to Alperin’s Weight Conjecture. We reformulate this conjecture in terms of the standard and proper costandard modules for a standardly stratified algebra. A module category which is standardly stratified has similar properties to a highest weight category, but the notion is more general, and the aim in considering such structure is to simulate aspects of weight theory, such as appears with the representations of a complex semisimple Lie algebra. Before reaching the application to Alperin’s conjecture we first characterize the EI categories whose representations are standardly stratified with respect to a natural preorder on the simple modules. We do this in Section 2. It turns out that representations of the orbit category of a finite group with respect to a family of subgroups are always standardly stratified, and the stratification is related to the procedure used to compute the derived functors of the limit functor in [15] and [22].

2000 Mathematics Subject Classification. Primary 20C20; Secondary 16G10, 20J99.

In order to present our reformulations of Alperin’s conjecture, we need to invoke the development of the theory of standardly stratified algebras in a broader context than the one which appears in many accounts of the subject. The definition of a standardly stratified algebra given by Cline, Parshall and Scott [8] has as one of its ingredients a preordered set which parametrizes the simple modules. The key results which we will invoke are a theorem of Dlab and the properties of the Ringel dual algebra, and in [1, 2, 10, 17, 30] these were established under the hypothesis that the preordered set is actually a partiallyordered set. In [29] the Ringel dual algebra was introduced using a preordered set, and it was shown there that the dual algebra is again standardly-stratified, but it was not shown that a standardly-stratified algebra may be identified with its double dual, and neither was Dlab’s theorem considered. In fact all of these results hold in the generality that the simple modules are parametrized by a preordered set provided we are careful to set things up in the right way. Our purpose in Section 3 of this paper is to show how to do this. We have been informed since writing this paper that such a development has also been given in [13], which appeared in preprint form a few months earlier than the present paper.

In the last section of this paper we provide our reformulations of Alperin’s weight conjecture in the context of stratifications. We work with the representations of the orbit category of a finite group G with respect to its p-subgroups for some prime p and we consider the standard representations Δ_{λ}, the proper costandard representations \overline{∇}_{λ} and the canonical or partial tilting representations T_{λ} over an algebraically closed field of characteristic p. We show that Alperin’s weights correspond to the Δ_{λ} which are simple, and also to the \overline{∇}_{λ} which are injective. These in turn correspond by duality to the \overline{∇}_{λ} for the Ringel dual algebra which are T_{λ}. At the same time the simple representations of G are in correspondence with the Δ_{λ} which are T_{λ}, the Δ_{λ} which are injective, and also the Δ_{λ} for the Ringel dual algebra which are projective. Thus various reformulations of the conjecture that the number of weights equals the number of simple modules for G become apparent, and certain of these reformulations possess a symmetry which is lacking in some other forms of the conjecture. Thus Alperin’s weight conjecture is equivalent to the statement that the number of \overline{∇}_{λ} for the orbit category which are injective equals the number of Δ_{λ} for the Ringel dual algebra which are projective; and it is also equivalent to the statement that the number of Δ_{λ} equal to T_{λ} for the the orbit category equals the number of \overline{∇}_{λ} equal to T_{λ} for the Ringel dual algebra.

2. Some EI category algebras are standardly stratified

We start by describing some basic properties of representations of categories, and of EI categories in particular. Given a small category C and a field k (which could more generally be a commutative ring R with a 1), we may form the category algebra kC which is the k-vector space with the morphisms of C as a basis. A multiplication is defined on the basis elements by

αβ = \{ \begin{matrix}α ∘ β & if α and β can be composed, \\ 0 & otherwise. \end{matrix}

We will compose morphisms on the left and deal with left modules, so that here α ∘ β means do β first, then α. This is a construction which generalizes the notion of the group algebra of a group and incidence algebra of a poset, and the multiplication makes kC into an associative algebra, which has an identity element if and only if C has finitely many objects. In that case the identity is 1 = ∑_{x∈Ob C} 1_{x} and the elements 1_{x} form a set of pairwise mutually orthogonal idempotents in kC.

A representation of a category C is a functor F : C → k-mod. Extending the familiar property of the group algebra when C happens to be a group, we immediately see that representations of C may be identified with kC-modules, in that given a functor F : C → k-mod we obtain a kC-module ⨁_{x∈Ob C} F (x), and given a kC-module M we obtain a functor whose value at x ∈ C is 1_{x} · M. Evidently natural transformations of functors correspond to module homomorphisms under this identification.

We start with two general results, the first of which shows that in considering the representations of a category C, we can replace C by any full subcategory which contains at least one object from each isomorphism class.

Lemma 2.1. Let C and C′ be equivalent categories. Then the categories of representations of C and of C′ are also equivalent.

Proof. The equivalence of categories means there are functors A : C → C′ and B : C′ → C so that AB and BA are naturally isomorphic to the identity functors. Now if F : C → k-mod is a representation of C we obtain a representation F B of C′ and if G is a representation of C′ we obtain a representation GA of C. On composing the natural isomorphism of BA and the identity with F we obtain a natural isomorphism between the functor F ↦ F BA and the identity functor F ↦ F, and similarly the functor G ↦ GAB is isomorphic to G ↦ G. This demonstrates the equivalence of categories of representations. □

Our next result shows that the category algebras of categories with more than one isomorphism class of objects of the kind we will consider in the remainder of this paper are always non-trivially stratified in the sense of [8] . We need to know the definition of a stratifying ideal J in a finite-dimensional algebra A over a field, and this may be given in more than one way. One definition is that it is an ideal for which the inflation map induces an isomorphism Ext_{A/J}^{∗}(U, V) ≅ Ext_{A}^{∗}(U, V) for all A/J-modules U and V. In the proof of our result we find it easier to work with an equivalent definition which appears in [8].

Proposition 2.2. Let D be a full subcategory of a finite category C, and let e = ∑_{x∈D} 1_{x} ∈ kC where k is a field. Let E be the full subcategory of C whose objects are the objects of C not in D, and suppose that for all x ∈ D and y ∈ E we have Hom(x, y) = ∅. Then

(1) J = kCekC is a stratifying ideal in kC, and

(2) for all kE-modules U and V we have Ext_{kE}^{∗}(U, V) ≅ Ext_{kC}^{∗}(U, V), where U and V are regarded as kC-modules by inflation.

Proof. (1) We verify that the conditions of Definition 2.1.1 of [8] are satisfied. By construction J is idempotent, and we must verify that multiplication induces an isomorphism kCe ⊗_{ekCe} ekC → J and

Tor_{n}^{ekCe}(kCe, ekC) = 0 for all n > 0.

Now kCe = ekCe since both sides are the span in kC of all morphisms in D. This immediately implies that the Tor groups are zero because kCe is projective as an ekCe-module. We also have an isomorphism

kCe ⊗_{ekCe} ekC = ekCe ⊗_{ekCe} ekC
≅ ekC
= kCekC

induced by multiplication, the latter equality holding since both sides are the span in kC of the morphisms in C which have target in D.

(2) We may regard U and V as kC-modules via the homomorphism kC → kE which sends morphisms not in E to zero. Now the statement is equivalent to (1), as described in [8, 2.1.2]. □

If C is an EI category, that is, one in which all endomorphisms are isomorphisms, we see immediately that the set of isomorphism classes [x] of objects x in C forms a poset under the relation [x] ≤ [y] if and only if there is a morphism x → y. Thus if D is any full subcategory with the property that x ∈ D and [x] ≤ [y] imply y ∈ D, then D satisfies the hypothesis of the proposition. We may consider a chain of such subcategories ∅ = D_{0} ⊂ ⋯ ⊂ D_{n} = C in which each subcategory has one more isomorphism class of objects than the previous one, so that n is the number of isomorphism classes in C, and from this we obtain a stratification of kC of length n. For each kC-module M there is a corresponding filtration in which each factor is zero except on one isomorphism class of objects, where it is the value of M. This is exactly the filtration which has been used in the calculation of higher limits in [15] and [22].

Whereas category algebras of finite EI categories are always stratified, with a stratification of length equal to the number of isomorphism classes in the category, they are not always standardly stratified, and in the remainder of this section we characterize the circumstances in which this happens. We first recall the definition. Suppose that A is a finite-dimensional k-algebra whose simple modules are parametrized as S_{λ}, λ ∈ Λ where Λ is a preordered set (namely a set with a reflexive, transitive relation ≤). We write λ < μ (where λ, μ ∈ Λ) to mean λ ≤ μ but μ ≰ λ. Let P_{λ} be the projective cover of S_{λ}. According to [8, 2.2.1], the algebra A is standardly-stratified with respect to (Λ, ≤) if there exist modules Δ_{λ}, λ ∈ Λ, with the following properties:

(1) if the composition factor multiplicity [Δ_{λ} : S_{μ}] ≠ 0 then μ ≤ λ, and

(2) for each λ ∈ Λ there is a surjection P_{λ} → Δ_{λ} so that the kernel K_{λ} has a filtration with factors Δ_{μ} where μ > λ.

We readily see that if A is standardly-stratified then the Δ_{λ} are determined as the modules P_{λ}/K_{λ} where

K_{λ} = ∑_{{ϕ:Pμ→P_{λ} \atop μ>λ}} ϕ(P_{μ})

is the trace in P_{λ} of the projective modules P_{μ} with μ > λ.

We next recall the parametrization of the simple and projective representations of an EI category, which may be found in [9] and [20], and we summarize this description. The simple representations of an EI category C are in bijection with pairs (x, V) where x is an object of C taken up to isomorphism, and V is a simple k Aut(x)-module, again taken up to isomorphism. Indeed, given a pair (x, V) one readily constructs a simple functor whose support is the isomorphism class [x] of x and whose value on an object x′ isomorphic to x is V with the action of Aut(x′) on V obtained by transporting the action of Aut x via some fixed isomorphism x → x′. (Alternatively we may replace C by a skeletal subcategory by virtue of Lemma 2.1, in which case the simple functor associated to (x, V) is zero except on x, where it is V.) We will denote the simple functor so constructed by S_{x,V}. An elementary argument shows that these simple representations form a complete list of isomorphism types.

In the next result we describe the projective cover P_{x,V} of S_{x,V}.

Proposition 2.3. [9] Let C be a finite EI category and k a field. The projective cover P_{x,V} of S_{x,V} is a direct summand of the functor k[Hom(x, −)], which is projective. It is supported on {y ∈ Ob C | there is a morphism x → y}. On evaluation at x, P_{x,V} (x) is the projective cover of V as a k Aut(x)-module.

Proof. Since 1 = ∑_{x∈Ob C} 1_{x} is a sum of mutually orthogonal idempotents, kC · 1_{x} is a projective module for kC, and regarded as a functor this is k[Hom(x, −)] since on evaluation at an object y this is a vector space with the morphisms x → y as a basis, and the same is true of 1_{y}·kC·1_{x}. Let 1_{x} = e+f be a sum of orthogonal idempotents in k Aut(x) where k Aut(x)·e is the projective cover of V. Then kCe is a summand of k[Hom(x, −)], hence is projective, and it is indecomposable since it is generated by its value at x, where it is an indecomposable k Aut(x)- module, namely the projective cover of V. Since eS_{x,V} ≠ 0 we have constructed P_{x,V}. Its support is contained in the support of k[Hom(x, −)], which equals {y ∈ Ob C | there is a morphism x → y}. □

When A = kC is the category algebra of an EI category C, we take Λ to be the set of all pairs (x, V) where x ∈ Ob C and V is a simple k Aut(x)-module, taken up to isomorphism. There is a canonical preorder on Λ given by (x, V) ≤ (y, W) if and only if there exists a morphism x → y in C. In the remainder of this section we characterize the EI categories C for which kC is standardly stratified with respect to this preorder. For each λ = (x, V) we now define A-modules K_{λ} as above. Explicitly:

K_{x,V} = ∑_{{ϕ:P_{y,W} →P_{x,V} \atop (y,W)>(x,V )}} ϕ(P_{y,W}).

We define Δ_{x,V} = P_{x,V} /K_{x,V}.

Lemma 2.4. Let C be a finite EI category. Then

K_{x,V} (y) = \{ \begin{matrix}P_{x,V} (y) & if there is a non-isomorphism x → y, \\ 0 & otherwise, \end{matrix}

and

Δ_{x,V} (y) = \{ \begin{matrix}P_{V} & if x = y, \\ 0 & if x ≅ y̸ \end{matrix}

where P_{V} is the projective cover of V as an k Aut(x)-module.

Proof. The elements of Λ strictly greater than (x, V) are the (y, W) for which there is a non-isomorphism x → y. The corresponding indecomposable projective kC-modules P_{y,W} are precisely the indecomposable summands of the modules kC · 1_{y}, by Proposition 2.3. Thus the sum of images in P_{x,V} of homomorphisms from the P_{y,W} with (x, V) < (y, W) equals the sum of the images of the homomorphisms from the kC · 1_{y} for which there is a non-isomorphism x → y, and evaluated at such y this is P_{x,V} (y). Since kC · 1_{y} only has support on objects z for which there is a non-isomorphism y → z, it follows that K_{x,V} (y) = 0 if there is no non-isomorphism x → y. From Proposition 2.3 it now follows that Δ_{x,V} only has support on the isomorphism class of x, where it is P_{V}. □

We come now to the main result of this section.

Theorem 2.5. Let C be a finite EI category and let k be a field. Let Λ be the preordered set of pairs (x, V) which parametrizes the set of isomorphism classes of simple modules of the category algebra kC. Then kC is standardly stratified with respect to Λ if and only if for every morphism α : x → y in C the group Stab_{Aut(y)}(α) = {θ ∈ Aut(y) | θα = α} has order which is invertible in k.

Proof. We know that if kC is standardly stratified then the standard modules are the Δ_{x,V} defined earlier. Furthermore, the composition factors of Δ_{x,V} are precisely the S_{x,W} where W is a composition factor of P_{V}, and these always satisfy (x, W) ≤ (x, V). From this we see that kC is standardly stratified if and only if each module K_{x,V} has a filtration with factors of the form Δ_{y,W} where y > x.

We claim that K_{x,V} has a filtration with factors Δ_{y,W} where y > x if and only if each evaluation K_{x,V} (y) is projective as an k Aut(y)-module. This is because Δ_{y,W} is supported only on (the isomorphism class of) y, where it is the projective module P_{W}, and K_{x,V} is supported only on objects y with y > x. The filtration of K_{x,V} whose terms are the subfunctors generated by the values at objects which are initial sequences in a list of objects y_{1}, y_{2}, … where y_{i} > y_{j} implies i < j has the property that each factor is supported on a single isomorphism class. It will be a direct sum of Δ_{y,W} (and hence have a Δ-filtration) if and only if the evaluation on each object is projective. This is equivalent to requiring that each K_{x,V} (y) be projective, and so if this condition is satisfied K_{x,V} has a Δ-filtration as required. Conversely, if K_{x,V} has a Δ-filtration we see that for each y, K_{x,V} has a filtration with factors Δ_{y,W} (y) = P_{W}. Since these modules are projective, it is equivalent to require that K_{x,V} (y) be projective.

We conclude from this that kC is standardly stratified if and only if each evaluation P_{x,V} (y) is a projective k Aut(y)-module. By Proposition 2.3 each functor k[Hom(x, −)] is a direct sum of functors P_{x,V}, each occuring with non-zero multiplicity in the sum, and so every evaluation P_{x,V} (y) is projective if and only if every k[Hom(x, y)] is a projective k Aut(y)-module. Now k[Hom(x, y)] is a permutation module and it is projective if and only if every stabilizer in the action of Aut(y) on Hom(x, y) has order invertible in k (because in characteristic p, for example, it is projective if and only if it is projective on restriction to a Sylow p-subgroup, where it is free; see [4]). This completes the proof. □

Corollary 2.6. Let C be a finite EI category in which every morphism is an epimorphism and let k be a field. Then kC is standardly stratified with respect to the canonical preorder on Λ.

Proof. The stabilizer condition which appears in the statement of Theorem 2.5 is satisfied whenever C has the property that all morphisms are epimorphisms. In such a situation, if α : x → y is a morphism in C and θ_{1}, θ_{2} ∈ Aut(y) then θ_{1}α = θ_{2}α implies θ_{1} = θ_{2}, so that Stab_{Aut(y)}(α) = {1}, or in other words Aut(y) acts freely on Hom(x, y). □

Various categories constructed from the subgroups of a group satisfy the condition that all morphisms are epimorphism, and we mention some of these now. Let S be a set of subgroups of a group G. We define the transporter category T_{S} to have as its objects the members of S, and morphisms Hom(H, K) = N_{G}(H, K) = {g ∈ G | ^{g}H ⊆ K}. This latter set is the transporter of H into K, and when H = K it is simply the normalizer of H in G so that T_{S} is an EI-category. The composition of morphisms is group multiplication. It is clear that all morphisms in T_{S} are both epimorphisms and monomorphisms, since if h_{1}g = h_{2}g or gh_{1} = gh_{2} then h_{1} = h_{2}.

A related category is the orbit category O_{S} associated to S in which the objects are the coset spaces G/H where H ∈ S and the morphisms are the G-equivariant mappings. Because the image of each equivariant map G/H → G/K is a union of orbits, and G/K consists of only one orbit, each morphism is an epimorphism. In fact, it is well-known that all equivariant mappings G/H → G/K have the form α_{g} : xH ↦ xg^{−1}K where g ∈ N_{G}(H, K). The assignment g ↦ α_{g} specifies a surjective mapping N_{G}(H, K) → Hom_{O_{S}}(G/H, G/K) in which

{x ∈ N_{G}(H, K) | α_{x} = α_{g}} = Kg,

so that K\N_{G}(H, K) is in bijection with Hom_{O_{S}}(G/H, G/K). We see from this that the orbit category is obtained from the transporter category by replacing N_{G}(H, K) with K\N_{G}(H, K) as the morphism set from H to K. Orbit categories appear widely in the study of group actions on spaces. They are at the heart of Bredon’s notion of a coefficient system and in the construction of Bredon homology (see [6, 9, 24]). They are also important in the construction of approximations to classifying spaces of groups, and from the extensive literature on this topic we may select [15, 12, 7] to illustrate the application.

We also construct the Frobenius category or Quillen category F_{S} associated to the set of subgroups S (see [25, Sect. 47]). This is a category which plays a crucial role in the notion of a p-local finite group [7], and in the case when S consists of the elementary abelian p-subgroups it was used by Quillen in his stratification theorem for group cohomology. The category F_{S} has the elements of S as its objects, and Hom_{F_{S}} = N_{G}(H, K)/C_{G}(H). The morphisms may be identified with the set of group homomorphisms H → K which are of the form ‘conjugation by g’ for some g ∈ G. Such group homomorphisms are monomorphisms, and so all morphisms in F_{S} are monomorphisms. Thus all morphisms in the opposite category F_{S}^{op} are epimorphisms.

In view of this we obtain the following corollary.

Corollary 2.7. Over any field, representations of the orbit category O_{S}, the transporter category T_{S}, its opposite T_{S}^{op} and the opposite F_{S}^{op} of the Frobenius category are all standardly stratified with respect to the natural preorder on the simple modules specified by S_{x,V} ≤ S_{y,W} if and only if there exists a morphism x → y.

3. Dlab’s theorem and the Ringel dual

We suppose that A is a finite-dimensional algebra over a field k, whose simple modules are parametrized by a set Λ. The theory of standardly-stratified algebras described in (for example) [1, 2, 10, 16, 17, 30] was developed under the hypothesis that Λ is a poset. In fact the key assertions of this development hold when Λ is a preordered set, and we wish to indicate in this section that this is so. A development in this generality has also been given in [13] which appeared as a preprint a few months earlier than the present account and was unknown to me until after this paper was written. In [13] a different approach is taken to proving Dlab’s theorem.

We start by illustrating the kind of possibility that is allowed if Λ is a preordered set, but not a poset. Consider an algebra A with two simple modules labelled 1 and 2 whose regular representation is {1 \atop 2} 1 ⊕ {2 \atop 1} 2 (for example, the group algebra 𝔽_{3}S_{3}). We see that A is not standardly stratified with respect to any partial order on Λ = {1, 2}. For if 1 < 2 then Δ_{1} = 1, Δ_{2} = {2 \atop 1} and this does not give a stratification. Neither does the possibility 2 < 1, which is similar. Equally, A is not standardly stratified if Λ has the partial order in which 1 and 2 are not comparable. On the other hand we may use the trivial preorder in which 1 ≤ 2 and 2 ≤ 1. Now Δ_{1} = {1 \atop 2} 1 , Δ_{2} = {2 \atop 1} 2 and A is standardly stratified.

The trivial stratification just considered gives no extra information about an algebra, since every algebra has such a stratification, and it might be thought merely a matter of terminology whether one says an algebra is trivially stratified, or not stratified at all. However this trivial example underlies non-trivial stratifications of other algebras which are useful to us. For example an algebra with regular representation

A = _{1 {2 \atop 1} 3} ⊕ _{2 {1 \atop 2} 4}

is standardly stratified if 1 ≤ 2 ≤ 1 < 3 ≤ 4 ≤ 3, but not standardly stratified with any partial order on Λ = {1, 2, 3, 4}.

We now summarize the theory of the Ringel dual and Dlab’s theorem in the situation where Λ is a preordered set. The main task is to make definitions of the standard and costandard objects correctly as well as to impose a certain condition on Λ. After that, the arguments are just the same as the well-known ones which already appear in the literature. In presenting them here we claim no originality, except in so far as to observe that in some cases these arguments hold in greater generality than their original context. It seems valuable to have a statement of these results in the generality that Λ is a preordered set, since otherwise the question remains as to whether they do indeed hold.

Let A be a finite-dimensional algebra over a field k, whose simple modules are parametrized as S_{λ} where λ ∈ Λ, a preordered set. As before, if λ, μ ∈ Λ we will write λ < μ to mean λ ≤ μ and μ ≰ λ. We write λ ∼ μ if λ ≤ μ and μ ≤ λ. Let P_{λ} and I_{λ} be the projective cover and injective hull of S_{λ}. We define modules Δ_{λ}, \overline{Δ}_{λ}, ∇_{λ} and \overline{∇}_{λ} in this context as follows. We have already defined the standard module Δ_{λ} to be the largest quotient of P_{λ} all of whose composition factors are isomorphic to S_{μ} with μ ≤ λ. We define the proper standard module \overline{Δ}_{λ} to be the largest quotient of P_{λ} whose composition factors S_{μ} all satisfy μ < λ, except for a single S_{λ}. Similarly the costandard module ∇_{λ} is the largest submodule of I_{λ} with composition factors S_{μ} where μ ≤ λ and the proper costandard module \overline{∇}_{λ} is the largest submodule of I_{λ} with composition factors S_{μ} where μ < λ except for a single S_{λ}. We observe that if λ is minimal in Λ then \overline{Δ}_{λ} = \overline{∇}_{λ} = S_{λ} is simple, and that if λ is maximal in Λ then Δ_{λ} = P_{λ} is projective and ∇_{λ} = I_{λ} is injective.

We denote by F(Δ) the full subcategory of A-modules whose objects have a finite filtration in which each factor has the form Δ_{λ} for some λ ∈ Λ, and similarly we define F(\overline{Δ}), F(∇) and F(\overline{∇}). We see that if A is standardly-stratified in the sense of the definition in the last section then A ∈ F(Δ), but it is not apparent that the converse need be true in this generality. The problem is that it need not be the case that Ext groups between the Δs vanish appropriately. As an example consider (again) the algebra whose regular representation is _{A}A = 1^{{1 \atop 2}} ⊕ 2^{{2 \atop 1}} with two simple modules: 1 and 2. Taking this time the preorder in which 1 and 2 are not comparable, we see that Δ_{1} = 1 and Δ_{2} = 2 and that A ∈ F(Δ), but that A is not standardly stratified with this preorder.

To remedy this, we use the fact that there is associated to Λ a poset [Λ] whose elements are the equivalence classes [λ] of elements of λ under ∼ and where [λ] ≤ [μ] if and only if λ ≤ μ. We may place the elements of [Λ] in some order [Λ] = {ρ_{1}, ρ_{2}, …, ρ_{n}} so that ρ_{i} ≤ ρ_{j} implies i ≤ j and use this to define a new preorder on Λ: λ ≤′ μ if and only if λ ∈ ρ_{i}, μ ∈ ρ_{j} with i ≤ j. If A is standardly stratified with respect to (Λ, ≤) then it is also standardly-stratified with respect to (Λ, ≤′) with the same standard and proper standard modules (as we may easily see by an elementary argument), and so for many purposes we may work with (Λ, ≤′) instead of (Λ, ≤). A number of the results in this section will only be true under the hypothesis that ≤=≤′, which is equivalent to saying that the poset associated to (Λ, ≤) is a linear order, or that in every pair of elements of Λ the elements are comparable. This idea is present in [8, 2.2.3].

The standard and proper standard modules may also be defined in terms of idempotents. Frequently this is done elsewhere in the literature under the assumption that Λ is a poset, but it may also be done under the weaker assumption that Λ is a preordered set. To make the definitions work we suppose that the poset [Λ] = {ρ_{1}, ρ_{2}, …, ρ_{n}} associated to Λ is linearly ordered. For each λ ∈ Λ let e_{λ} ∈ A be a primitive idempotent corresponding to the simple module S_{λ}. For each equivalence class ρ_{i} of elements of Λ we define an idempotent E_{i} = ∑_{λ∈ρ_{i}} e_{λ}, so that E_{i} is the sum of as many primitive idempotents as there are elements in the equivalence class ρ_{i}. Let us put ϵ_{i} = ∑_{i≤j} E_{j}. Thus for λ ∈ ρ_{i} we have

Δ_{λ} ≅ (Ae_{λ})/(Aϵ_{i+1}(Rad A)e_{λ})

and

\overline{Δ}_{λ} ≅ (Ae_{λ})/(Aϵ_{i}(Rad A)e_{λ}).

The way to see this is to observe that Ae_{λ} ≅ P_{λ}, the projective cover of S_{λ}, so that (Rad A)e_{λ} is the unique maximal submodule of P_{λ} and Aϵ_{i+1}(Rad A)e_{λ} is the trace in (Rad A)e_{λ} of the projective modules P_{μ} where λ < μ. Similarly Aϵ_{i}(Rad A)e_{λ} is the trace in (Rad A)e_{λ} of the projective modules P_{μ} where λ ≤ μ.

We list some basic properties of these modules.

Proposition 3.1. Let A be a finite-dimensional k-algebra whose simple modules are parametrized (up to isomorphism) by a preordered set Λ for which the associated poset [Λ] is linearly ordered. The modules we have defined satisfy the following properties.

(1) Hom_{A}(Δ_{λ}, ∇_{μ}) ≠ 0 implies λ ∼ μ.

(2) Hom_{A}(Δ_{λ}, \overline{∇}_{μ}) ≠ 0 implies λ = μ, and also Hom_{A}(\overline{Δ}_{μ}, ∇_{λ}) ≠ 0 implies λ = μ.

(3) Hom_{A}(\overline{Δ}_{λ}, \overline{∇}_{μ}) ≠ 0 implies λ = μ.

(4) If μ ≤ λ then Ext_{A}^{1}(Δ_{λ}, S_{μ}) = 0 and Ext_{A}^{1}(S_{μ}, ∇_{λ}) = 0.

(5) Ext_{A}^{1}(Δ_{λ}, Δ_{μ}) ≠ 0 implies λ < μ, and also Ext_{A}^{1}(∇_{μ}, ∇_{λ}) ≠ 0 implies λ < μ.

(6) Ext_{A}^{1}(Δ_{λ}, \overline{Δ}_{μ}) ≠ 0 implies λ < μ, and also Ext_{A}^{1}(\overline{∇}_{μ}, ∇_{λ}) ≠ 0 implies λ < μ.

(7) Ext_{A}^{1}(\overline{Δ}_{λ}, Δ_{μ}) ≠ 0 implies λ ≤ μ, and also Ext_{A}^{1}(∇_{μ}, \overline{∇}_{λ}) ≠ 0 implies λ ≤ μ.

(8) Ext_{A}^{1}(\overline{Δ}_{λ}, \overline{Δ}_{μ}) ≠ 0 implies λ ≤ μ, and also Ext_{A}^{1}(\overline{∇}_{μ}, \overline{∇}_{λ}) ≠ 0 implies λ ≤ μ.

(9) Ext_{A}^{1}(Δ_{λ}, ∇_{μ}) = 0 for all λ, μ ∈ Λ.

(10) Ext_{A}^{1}(Δ_{λ}, \overline{∇}_{μ}) = 0 and Ext_{A}^{1}(\overline{Δ}_{λ}, ∇_{μ}) = 0 for all λ, μ ∈ Λ.

(11) Ext_{A}^{1}(\overline{Δ}_{λ}, \overline{∇}_{μ}) ≠ 0 implies λ ∼ μ.

Proof. In each case where there are two statements they are equivalent to each other by the duality between modules for A and its opposite algebra. We only prove the first statement. All of the arguments are routine, being similar to the ones which prove 1.2 and 1.3 of [11] or 2.2.2 and 2.2.8 of [8], for example.

(1) If there is a non-zero homomorphism ϕ : Δ_{λ} → ∇_{μ} we deduce that S_{λ} is a composition factor of ∇_{μ} and S_{μ} is a composition factor of Δ_{λ}. Thus λ ≤ μ and μ ≤ λ so λ ∼ μ.

(2) If there is a non-zero homomorphism ϕ : Δ_{λ} → \overline{∇}_{μ} then μ ≤ λ as in (1), but now if λ ≠ μ then λ < μ since S_{λ} is a composition factor of \overline{∇}_{μ}. This would imply λ < μ ≤ λ, which is not possible.

(3) If there is a non-zero homomorphism ϕ : \overline{Δ}_{λ} → \overline{∇}_{μ} then either λ = μ or else λ < μ and μ < λ, which is not possible.

(4) These follow from the fact that Δ_{λ} is the largest quotient of P_{λ} with composition factors ≤ λ, and that ∇_{λ} is the largest submodule of I_{λ} with composition factors ≤ λ. It may also be proved by the argument of (5).

(5) We use the short exact sequence 0 → K_{λ} → P_{λ} → Δ_{λ} → 0 to compute Ext. The semisimple top of K_{λ} only has composition factors S_{ν} with λ < ν so Hom_{A}(K_{λ}, Δ_{μ}) ≠ 0 implies Δ_{μ} has such a composition factor, and λ < ν ≤ μ.

(6) This is similar to (5).

(7) We use the short exact sequence 0 → \overline{K}_{λ} → P_{λ} → \overline{Δ}_{λ} → 0 to compute Ext, where the composition factors of the semisimple top of \overline{K}_{λ} are S_{ν} with λ ≤ ν. Thus Hom_{A}(\overline{K}_{λ}, Δ_{μ}) ≠ 0 implies Δ_{μ} has such a composition factor, and λ ≤ ν ≤ μ.

(8) This is similar to (7).

(9) The argument for this is given in 2.2.8 of [8].

(10) This and (9) are perhaps the most interesting arguments. We use the short exact sequence 0 → K_{λ} → P_{λ} → Δ_{λ} → 0 as before to compute Ext, and because Hom_{A}(K_{λ}, \overline{∇}_{μ}) ≠ 0 implies λ < μ we deduce that if Ext_{A}^{1}(Δ_{λ}, \overline{∇}_{μ}) ≠ 0 then λ < μ.

We may also use the injective presentation 0 → \overline{∇}_{μ} → I_{μ} → \overline{C}_{μ} → 0, where the socle of \overline{C}_{μ} has composition factors S_{ν} with μ ≤ ν, so that Hom_{A}(Δ_{λ}, \overline{C}_{μ}) ≠ 0 implies μ ≤ ν ≤ λ so μ ≤ λ. Because we cannot simultaneously have λ < μ and μ ≤ λ we deduce that Ext_{A}^{1}(Δ_{λ}, \overline{∇}_{μ}) = 0.

(11) Here Hom_{A}(\overline{K}_{λ}, \overline{∇}_{μ}) ≠ 0 implies λ ≤ μ and Hom_{A}(\overline{Δ}_{λ}, \overline{C}_{μ}) ≠ 0 implies μ ≤ λ. □

Corollary 3.2. Let A be a finite-dimensional k-algebra whose simple modules are parametrized (up to isomorphism) by a preordered set Λ for which the associated poset [Λ] = {ρ_{1}, ρ_{2}, …, ρ_{n}} is linearly ordered, and let the idempotents ϵ_{i} be defined as above.

(1) Let M be a finite-dimensional A-module. Then M lies in F(Δ) if and only if for every i the module (Aϵ_{i}M)/(Aϵ_{i+1}M) is a direct sum of modules Δ_{λ} with λ ∈ ρ_{i}.

(2) F(Δ) is closed under taking direct summands.

(3) A is standardly-stratified if and only if A ∈ F(Δ), or equivalently, every finite-dimensional projective A module lies in F(Δ).

Proof. (1) The argument is implicit in [8] Section 2.2. Assuming M ∈ F(Δ), by (5) of Proposition 3.1 M has a filtration in which the factors occur in an order compatible with the linear order on [Λ]. Such a filtration is a refinement of the trace filtration with terms Aϵ_{i}M and each factor is a direct sum of modules Δ_{λ} again by (5) of Proposition 3.1. The converse implication is easy.

(2) This is the usual argument, as in [23] Section 3 or [11] 1.4.

(3) From the definition we have that if A is standardly stratified then A ∈ F(Δ). Conversely if A ∈ F(Δ) then each indecomposable projective P lies in F(Δ) and by part (1) the filtration with terms Aϵ_{i}P has Δ factors. Since P is indecomposable projective the top factor must be Δ_{λ} for some λ, and the rest of the factors must be Δ_{μ} with μ > λ by the property of this filtration. □

We also have the following:

Corollary 3.3. Let A be a finite-dimensional k-algebra whose simple modules are parametrized by a preordered set Λ for which the associated poset [Λ] is linearly ordered.

(1) F(Δ) and F(\overline{Δ}) are closed under kernels of epimorphisms.

(2) F(∇) and F(\overline{∇}) are closed under cokernels of monomorphisms.

Proof. Parts (1) and (2) are dual to each other and only one of them needs to be proved. The statement about F(Δ) follows as in 1.5 of [11] and the statement about F(\overline{Δ}) follows as in 3.2 of [1]. □

Dlab showed [10] under the hypothesis that Λ itself is linearly-ordered that the left regular representation _{A}A lies in F(Δ) if and only if the right regular representation A_{A} lies in F(\overline{Δ}) (where now the \overline{Δ}s are right modules), or what is the same on considering ordinary duality with respect to the field k, that all finitely-generated injective left Amodules lie in F(\overline{∇}). A similar thing works when Λ is a preordered set. We state a result which is a combination of the statements which appear in [1, Theorem 3.1] and [17, Theorem 3]. Rather than work with left Δs and right \overline{Δ}s (as is done elsewhere) we choose to work entirely with left modules, so that instead of the right modules \overline{Δ} we work with their vector space duals, which are the left modules \overline{∇}.

Theorem 3.4. (Dlab’s Theorem) Let A be a finite-dimensional kalgebra whose simple modules are parametrized (up to isomorphism) by a preordered set Λ for which the associated poset [Λ] is linearly ordered. The following statements are all equivalent to the statement that A is standardly stratified.

(Δ1) F(Δ) contains all finitely-generated projective modules. (\overline{∇}1) F(\overline{∇}) contains all finitely-generated injective modules.

(2) Ext_{A}^{2}(Δ_{λ}, \overline{∇}_{μ}) = 0 for all λ, μ ∈ Λ. (Δ3) F(Δ) = \{X | Ext_{A}^{1}(X, \overline{∇}_{μ}) = 0\} for all μ ∈ Λ. (\overline{∇}3) F(\overline{∇}) = \{X | Ext_{A}^{1}(Δ_{μ}, X) = 0\} for all μ ∈ Λ.

These conditions are in turn equivalent to the following seemingly stronger conditions:

(2′) Ext_{A}^{k}(Δ_{λ}, \overline{∇}_{μ}) = 0 for all λ, μ ∈ Λ and all k ≥ 1. (Δ3′) F(Δ) = \{X | Ext_{A}^{k}(X, \overline{∇}_{μ}) = 0 for all μ ∈ Λ and all k ≥ 1\}. (\overline{∇}3′) F(\overline{∇}) = \{X | Ext_{A}^{k}(Δ_{λ}, X) = 0 for all λ ∈ Λ and all k ≥ 1\}.

Proof. We have observed in part (3) of Corollary 3.2 that condition (Δ1) is equivalent to the condition that A is standardly stratified. The proofs of the equivalences which appear in [1] and [17] go through verbatim with the modules Δ and \overline{∇} which we have defined here, since these modules satisfy the same formal properties as the ones considered in [1] and [17].

Certain implications are immediate, as in [1] and [17]. These are (Δ3′) ⇒ (2′) ⇒ (2), (\overline{∇}3′) ⇒ (2′), (Δ3) ⇒ (Δ1) and (\overline{∇}3) ⇒ (\overline{∇}1).

We prove (Δ1) ⇒ (2′) by an inductive argument involving dimensionshifting, as is done at the bottom of page 101 of [17], using part 10 of Proposition 3.1 and part 1 of Corollary 3.3; (\overline{∇}1) ⇒ (2′) is proved similarly.

(Δ3) ⇒ (Δ3′) and (\overline{∇}3) ⇒ (\overline{∇}3′) are also proved in this fashion, as is indicated on page 102 of [17].

The harder parts of the proof are (2) ⇒ (Δ3) which proceeds as on pages 102 and 103 of [17], and (2) ⇒ (\overline{∇}3), which proceeds as on pages 5 and 6 of [1]. □

By analogy with [2, Theorem 1.6] we now have:

Theorem 3.5. Let A be a standardly-stratified algebra with respect to a preordered set Λ for which the associated poset [Λ] is linearly ordered. Then

(1) F(Δ) is a functorially finite and resolving subcategory of Amodules.

(2) F(\overline{∇}) is a covariantly finite and coresolving subcategory of Amodules.

Proof. We see from Corollary 3.3 and conditions (Δ1) and (\overline{∇}1) of Dlab’s Theorem that F(Δ) is resolving and F(\overline{∇}) is coresolving. The fact that F(Δ) is functorially finite follows from [23] using part (5) of Proposition 3.1. The fact that F(\overline{∇}) is covariantly finite follows from (1), condition (\overline{∇}3) of Dlab’s theorem and [5, 3.3]. □

The Ringel dual of a standardly-stratified algebra was constructed in [2] (under the hypothesis that Λ is a poset) and it was also constructed independently in [29, Theorem 9.1] (without this hypothesis), where as part of a more general construction it was shown that the Ringel dual of a standardly-stratified algebra is again standardly-stratifed. The approach used in [29] was to classify the indecomposable Ext-injective objects in F(Δ) as the ‘Ext-injective hulls’ T_{λ} of the Δ_{λ}, without using the property that they are precisely the indecomposable objects in F(Δ) ∩ F(\overline{∇}).

In [29, Theorem 9.1] we gave a sufficient condition for the Ringel dual algebra to be standardly stratified, but unfortunately there was a significant typographical error in the statement of the theorem, and we take the opportunity to point this out. In that theorem the modules which are here called Δ_{λ} were called Θ(λ) and the hypothesis Hom(Θ(λ), Θ(ρ)) = 0 unless λ > ρ was made. It should have read ‘Hom(Θ(λ), Θ(ρ)) = 0 unless λ ≥ ρ’, and the proof used this latter condition. We comment also that in [29, Theorem 9.10] the Θ(λ) were indexed by the opposite preorder to the one used here.

In fact, the usual development of the properties of the Ringel dual, as described for example in [2], holds in the situation where Λ is a preordered set, and we now summarize it. The Ringel dual is the algebra B = End_{A}(T) where T = ⨁_{λ∈Λ} T_{λ}. We adopt the convention consistent with [29], but opposite to a frequently used convention, that homomorphisms are applied and composed on the left. Thus starting with a left A-module T, we regard T also as a left B-module. There are four contravariant functors we consider, namely F_{A} = Hom_{A}(−, T) : A-mod → B-mod, G_{A} = D Hom_{A}(T, −) : A-mod → B-mod where D(X) = Hom_{k}(X, k), and also F_{B} = Hom_{B}(−, T), G_{B} = D Hom_{B}(T, −) which go in the opposite direction to F_{A} and G_{A}. At this point we introduce the notation of placing a superscript A on a symbol that denotes an A-module, and a superscript B on a symbol that denotes a B-module. The exception to this is T, which is both an A-module and a B-module, and appears without a superscript.

Theorem 3.6. Let A be a finite-dimensional k-algebra whose simple modules are parametrized (up to isomorphism) by a preordered set Λ for which the associated poset [Λ] is linearly ordered. The isomorphism types of indecomposable objects of F(Δ^{A}) ∩ F(\overline{∇}^{A}) are parametrized as T_{λ} where λ ∈ Λ. For each λ there are short exact sequences

\begin{matrix}0 → Δ_{λ}^{A} → T_{λ} → X_{λ}^{A} → 0 \\ 0 → Y_{λ}^{A} → T_{λ} → \overline{∇}_{λ}^{A} → 0 \end{matrix}

where X_{λ}^{A} has a filtration with factors Δ_{μ}^{A} where μ < λ and Y_{λ}^{A} has a filtration with factors \overline{∇}_{μ}^{A} where μ ≤ λ. (In fact T_{λ}^{A} is the Extinjective hull in F(Δ^{A}) of Δ_{λ}^{A} (in the sense of [29]).) The module T = ⨁_{λ∈Λ} T_{λ}^{A} is a tilting module for A. The dual algebra B = End_{A}(T) is again standardly stratified, with preordered set Λ^{op}, standard modules Δ_{λ}^{B} = F_{A}(Δ_{λ}^{A}) and proper costandard modules \overline{∇}_{λ}^{B} = G_{A}(\overline{∇}_{λ}^{A}). Up to isomorphism the indecomposable summands of T as a B-module are precisely the indecomposable modules in F(Δ^{B}) ∩ F(\overline{∇}^{B}), T is a tilting module for B, and A ≅ End_{B}(T) which is Morita equivalent to the dual algebra of B. The functors F_{A} and F_{B} provide a duality between F(Δ^{A}) and F(Δ^{B}) which interchanges P_{λ}^{A} with T_{λ}^{B} and T_{λ}^{A} with P_{λ}^{B}. The functors G_{A} and G_{B} provide a duality between F(\overline{∇}^{A}) and F(\overline{∇}^{B}) which interchanges I_{λ}^{A} with T_{λ}^{B} and T_{λ}^{A} with I_{λ}^{B}.

Proof. We will describe a way to approach this which at many points is similar to the line of development in [2]. For a number of arguments we choose to quote from [29], but other references are possible.

The modules in F(Δ^{A}) ∩ F(\overline{∇}^{A}) are the Ext-injective objects of F(Δ^{A}) by Dlab’s Theorem, and these were classified in [29, 8.6] as the Ext-injective hulls of the Δ_{λ}^{A}. This provides the first of the short exact sequences in the statement of the theorem and it follows from the construction there that X_{λ}^{A} has a filtration with factors Δ_{μ}^{A} where μ < λ. It follows from [29, Theorem 9.1] that B is standardly-stratified with preordered set Λ^{op} (according to the convention we are using here) and with standard modules Δ_{λ}^{B} = F (Δ_{λ}^{A}).

We next show that T is a tilting module, for which we must show that T has finite projective dimension, Ext_{A}^{i}(T, T) = 0 for all i > 0, and that the regular representation _{A}A has a finite resolution by modules which are summands of direct sums of T. In fact every object of F(Δ^{A}) has both a finite resolution by projective modules and also a finite resolution by Ext-injective modules, from the properties of Extprojective and Ext-injective hulls developed in [29] (for example), and the Ext-injective hulls are always summands of a direct sum of copies of T. The fact that Ext_{A}^{i}(T, T) = 0 for all i > 0 is immediate from (Δ3′) or (\overline{∇}3′) of Dlab’s theorem.

Because T is a tilting module the canonical map A → End_{B}(T) is an isomorphism, and so the isomorphism types of projective A-modules biject with the isomorphism types of summands of T as a B-module, and these are parametrized by Λ. Since the indecomposable objects of F(Δ^{B})∩F(\overline{∇}^{B}) are also parametrized by Λ, we will know that, as a Bmodule, T is the sum of all of them (perhaps taken with multiplicity) provided we can show that T is Ext-injective in F(Δ^{B}). For this it suffices to show that Ext_{B}^{1}(Δ_{λ}^{B}, T) = 0 for all λ ∈ Λ. Take an Extinjective resolution

0 → Δ_{λ}^{A} → T_{0}^{A} → ⋯ → T_{n}^{A} → 0

where the A-modules T_{i}^{A} are direct sums of summands of T. Applying the functor F_{A}, which is exact on F(Δ^{A}) and sends summands of T to projectives, we obtain a projective resolution

0 ← Δ_{λ}^{B} ← P_{0}^{B} ← ⋯ ← P_{n}^{B} ← 0

and compute Ext_{B}^{∗}(Δ_{λ}^{B}, T) as the homology H_{∗}(Hom_{B}(P_{•}^{B}, T)). Since Hom_{B}(−, T) = F_{B} is an equivalence between the projective B-modules on the one hand and the sums of summands of T on the other, we see that Hom_{B}(P_{•}^{B}, T) is the original resolution of Δ_{λ}^{A}, which is acyclic above degree 0. This shows that Ext_{B}^{1}(Δ_{λ}^{B}, T) = 0, and hence A ≅ End_{B}(T) is indeed the dual algebra of B (up to Morita equivalence). It follows also that F_{B}(Δ_{λ}^{B}) = Δ_{λ}^{A}.

We prove that F_{A}, F_{B} are inverse dualities between F(Δ^{A}) and F(Δ^{B}). This is proven for example in [8, 3.8.2], but to save the reader the trouble of chasing references we present an argument. We show that both F_{A} and F_{B} are full and faithful. In the case of F_{A}, it is full and faithful on the full subcategory of F(Δ^{A}) whose modules are the direct sums of summands of T, by the Fitting correspondence. Because every module in F(Δ^{A}) has an Ext-injective resolution by such modules, F_{A} is full and faithful on F(Δ^{A}). The argument for F_{B} is similar, and so F_{A}, F_{B} are inverse dualities between F(Δ^{A}) and F(Δ^{B}) since they are exact and exchange Δ_{λ}^{A} with Δ_{λ}^{B}.

It follows from tilting theory (see [21] Theorem 1.16 taken together with the duality isomorphism which appears on page 114 of [21], or [14]) that the functors G_{A} and G_{B} provide a duality between F(\overline{∇}^{A}) and F(\overline{∇}^{B}) and by the Fitting correspondence these functors interchange indecomposable summands of T and indecomposable injective modules. More specifically, consider the indecomposable summand T_{λ}^{A} of T which has a Δ^{A}-filtration with Δ_{λ}^{A} at the bottom, and let e_{λ} ∈ B be the idempotent which is the composite of projection and inclusion T → T_{λ}^{A} → T. Then F_{A}(T_{λ}^{A}) = Hom_{A}(T_{λ}^{A}, T) ≅ Be_{λ} and Hom_{A}(T, T_{λ}^{A}) ≅ e_{λ}B are the indecomposable projective left and right B-modules corresponding to λ, from which we see that G_{A}(T_{λ}^{A}) = I_{λ}^{B} is the injective left B-module parametrized by λ. Hence also G_{A}(I_{λ}^{A}) = T_{λ}^{B}.

Because the \overline{∇}_{λ}^{A} and \overline{∇}_{λ}^{B} are identified in F(\overline{∇}^{A}) and F(\overline{∇}^{B}) as the objects which do not have proper filtrations by any other objects in these categories, it follows that G_{A} and G_{B} interchanges them, so that G_{A}(\overline{∇}_{λ}^{A}) = \overline{∇}_{ρ}^{B} for some ρ depending on λ. We show that λ = ρ.

To do this, we show that in any sequence

0 → Y^{A} → T_{λ}^{A} → \overline{∇}_{ρ}^{A} → 0

where Y ∈ F(\overline{∇}^{A}) we must have λ = ρ. It will follow from this on applying the functor G_{A} that G_{A}(\overline{∇}_{λ}^{A}) is a submodule of I_{λ}^{B}, and hence must be \overline{∇}_{λ}^{B}. We start by observing that this same procedure shows that in any such sequence the right-hand term must be the same \overline{∇}_{ρ}^{A}, because its image under G_{A} is always \overline{∇}_{λ}^{B}, being a submodule of I_{λ}^{B}. Thus all factors \overline{∇}_{ν}^{A} in a \overline{∇}^{A}-filtration of Y^{A} must satisfy ν ≤ ρ by Proposition 3.1 part (8). Since Δ_{λ}^{A} is a submodule of T_{λ}^{A}, for some such ν we have Hom_{A}(Δ_{λ}^{A}, \overline{∇}_{ν}^{A}) ≠ 0 which implies λ = ν by Proposition 3.1 part (2), so λ ≤ ρ. At the same time, all factors Δ_{σ} of T_{λ} in a Δfiltration have either σ < λ or σ = λ by Proposition 3.1 part (5) and there exists such a σ for which Hom_{A}(Δ_{σ}, \overline{∇}_{ρ}) ≠ 0. This implies that σ = ρ by Proposition 3.1 part (2). Thus either ρ < λ, which is not possible by our previous conclusion, or ρ = λ, which is what we were aiming to prove. □

We see that the T_{λ}^{A} have a stronger property than the one stated in the last theorem, namely that if Δ_{ρ}^{A} → T_{λ}^{A} is any monomorphism then necessarily ρ = λ and the quotient has a Δ^{A}-filtration, and also that if T_{λ}^{A} → \overline{∇}_{ρ}^{A} is any epimorphism then again λ = ρ and the kernel has a \overline{∇}^{A}-filtration. This follows because on applying F_{A} we obtain an epimorphism of B-modules P_{λ}^{B} → Δ_{ρ}^{B} which forces λ to equal ρ and the kernel to have a Δ^{B}-filtration, whereupon on applying F_{B} we obtain that the cokernel of Δ_{λ}^{A} → T_{λ}^{A} has a Δ^{A}-filtration. The argument for T_{λ}^{A} → \overline{∇}_{ρ}^{A} is similar using G_{A} and G_{B}.

There are further properties of these categories which we could mention and whose proofs we leave to the reader. Thus it follows from the fact that any module in F(Δ^{A}) has a finite resolution by summands of direct sums of T, taken together with condition (\overline{∇}3) of Dlab’s theorem, that F(\overline{∇}^{A}) may be characterized as \{X | Ext_{A}^{1}(T, X) = 0\}. There is also a BGG reciprocity statement, which follows from (2) of Proposition 3.1.

4. Representations of the orbit category and Alperin’s weight conjecture

In this section we focus on the orbit category O_{S} of a finite group G with respect to the set S of p-subgroups of G where p is the characteristic of the algebraically closed field k and write A = kO_{S}. The orbit category has as its objects the transitive G-sets G/H where H ∈ S and Aut_{O_{S}}(G/H) ≅ N_{G}(H)/H, so that the simple kO_{S}-modules are parametrized by pairs (G/H, V) where H ∈ S is taken up to conjugacy and V is a simple representation of N_{G}(H)/H taken up to isomorphism. For brevity we will write such a pair as (H, V), and write S_{H,V}^{A}, P_{H,V}^{A}, Δ_{H,V}^{A} for the corresponding simple, projective and standard A-modules, etc. When we consider these modules for the Ringel dual B of A, they will acquire a superscript B.

In [3] Alperin defines a weight of G to be a pair (H, V) where H is a p-subgroup of G taken up to conjugacy and V is a simple projective k[N_{G}(H)/H]-module, taken up to isomorphism. Alperin’s weight conjecture states that the number of weights for G at the prime p equals the number of isomorphism types of irreducible kG-modules.

We see that the weights are a subset of the preordered set which parametrizes the simple representations of O_{S}. We will identify by categorical properties subsets of the F(Δ) and F(\overline{∇}) which biject with the simple kG-modules, and with the weights. In particular we identify by categorical properties the Δ_{H,V}^{A} when H = 1 and when (H, V) is a weight. For completeness, we also identify the Δ_{H,V}^{A} when H is a Sylow p-subgroup of G. We conclude with reformulations of Alperin’s conjecture.

In the following results the canonical modules to which we refer are the indecomposable modules T_{H,V}^{A} in F(Δ^{A}) ∩ F(\overline{∇}^{A}) and T_{H,V}^{B} in F(Δ^{B}) ∩ F(\overline{∇}^{B}).

Theorem 4.1. Let A = kO_{S} where O_{S} is the orbit category of G with respect to its p-subgroups, and let B be the Ringel dual of A. The following are equivalent.

(1) Δ_{H,V}^{A} = S_{H,V}^{A} is a simple kO_{S}-module,

(2) \overline{∇}_{H,V}^{A} = I_{H,V}^{A} is injective,

(3) \overline{∇}_{H,V}^{B} = T_{H,V}^{B} is a canonical B-module.

(4) (H, V) is a weight.

Proof. To show that (1) ⇔ (4) observe that Δ_{H,V}^{A} is the projective cover P_{V} of V as a module for k[N_{G}(H)/H], concentrated at the object G/H of O_{S}, and it is simple if and only if the projective cover is simple, which occurs if and only if V is projective.

The equivalence (2) ⇔ (3) is formal because of the duality between F(\overline{∇}^{A}) and F(\overline{∇}^{B}).

It remains to show (2) ⇔ (4). We use the fact that I_{H,V}^{A} is isomorphic to Hom_{k}(P_{H,V}^{A^{op}}, k), so that I_{H,V}^{A} (H) = P_{V}. Assuming (2) we have

I_{H,V}^{A} (H) = P_{V} = \overline{∇}_{H,V}^{A} (V) = V,

so that V is projective and (H, V) is a weight. Conversely if (H, V) is a weight then I_{H,V}^{A} (H) = V = \overline{∇}_{H,V}^{A} (H). Apart from S_{H,V}^{A}, all the other composition factors S_{K,W}^{A} of I_{H,V}^{A} have K <_{G} H and so (K, W) is strictly smaller than (H, V). We see from the definition of \overline{∇}_{H,V}^{A} that \overline{∇}_{H,V}^{A} = I_{H,V}^{A}. □

In the proof of the next result we quote some of the theory of Mackey functors, as much as a convenience as for any other reason. As a guide to theory of Mackey functors we may refer to [28].

Theorem 4.2. Let A = kO_{S} where O_{S} is the orbit category of G with respect to its p-subgroups, and let B be the Ringel dual of A. The following are equivalent.

(1) Δ_{H,V}^{A} = T_{H,V}^{A} is a canonical kO_{S}-module,

(2) Δ_{H,V}^{A} = I_{H,V}^{A} is an injective A-module,

(3) Δ_{H,V}^{B} = P_{H,V}^{B} is a projective B-module.

(4) H = 1.

Proof. The equivalence (1) ⇔ (3) is formal because of the duality between F(Δ^{A}) and F(Δ^{B}).

Condition (1) is equivalent to the statement that Δ_{H,V}^{A} is Ext-injective in F(Δ^{A}), because the indecomposable Ext-injectives are precisely the T_{K,W}^{A}, and T_{H,V}^{A} is the Ext-injective hull of Δ_{H,V}^{A} (in the terminology of [29]). Since injectives are Ext-injective, we obtain the implication (2) ⇒ (1).

We next prove (4) ⇒ (2), namely that if H = 1 then Δ_{H,V}^{A} is injective. We may construct injectives using the duality between covariant functors and contravariant functors on O_{S}. Writing M(x)^{∗} = Hom_{k}(M(x), k), given a covariant functor M we obtain a contravariant functor M^{∗}, and conversely, and it is evident that projectives and injectives are interchanged by this duality. Since k Hom(−, x) is a projective contravariant functor (by an analogous result to Proposition 2.3), the functor k Hom(−, x)^{∗} is an injective covariant functor. Taking x to be the object G/1 in O_{S} we obtain an injective functor which is the direct sum of the Δ_{1,V}^{A} corresponding to the decomposition of the regular representation kG into projectives. Since every Δ_{1,V}^{A} appears here, they are all injective.

We prove finally that (1) ⇒ (4) and will do it by showing that if H ≠ 1 is a p-subgroup and V is a simple k[N_{G}(H)/H]-module then there is a non-split short exact sequence of kO_{S}-modules

0 → Δ_{H,V}^{A} → M_{1} → M_{2} → 0

in which all terms lie in F(Δ^{A}). This will show that Δ_{H,V}^{A} is not Extinjective in F(Δ^{A}) and hence not T_{H,V}^{A}.

We will construct M_{1} in stages. Let X be the projective cover of V as a k[N_{G}(H)/H]-module. When we regard X as a k[N_{G}(H)]-module it has vertex H and a Green correspondent U, say, as a kG-module (see [4] for background). Consider the kO_{S}-module F = F Q_{U} which assigns to each subgroup J the fixed quotient F Q_{U}(J) = k ⊗_{kJ} U. Now F (1) = U is indecomposable and the functorial morphisms F (1) → F (H) (i.e. the corestriction maps) are all surjective, so F is generated as a kO_{S}- module by F (1). It follows that F is indecomposable as a kO_{S}-module, since if F = F_{1}⊕F_{2} then one summand, say F_{1}, must have F_{1}(1) = F (1) and hence F_{1} = F, because F is generated by its value at 1.

Next we claim that the value of F at H has X as a summand. This is because the restriction U ↓_{N_{G}(H)}^{G} has X as a summand by the properties of Green correspondence, and this survives to the fixed quotient under H. Thus Δ_{H,V}^{A} is a quotient of a submodule of F, because it is supported only at H.

Now let L → F be the projective cover of F as a kO_{S}-module, so that L is also generated by its value at 1, and L ≅ P_{1,V_{1}}^{A} ⊕ ⋯ ⊕ P_{1,V_{n}}^{A} as a direct sum of indecomposable projectives. For each subgroup Q we let ⟨L(Q)⟩ be the subfunctor of L generated by its value at Q. Now

∑_{Q≥H}⟨L(Q)⟩/ ∑_{Q>H}⟨L(Q)⟩ ≅ ⨁ Δ_{H,V_{i}}^{A}

has Δ_{H,V}^{A} as an image, and so some indecomposable projective functor P_{1,W}^{A} has Δ_{H,V}^{A} appearing in a Δ^{A}-filtration. The quotient by the term in this filtration of P_{1,W}^{A} which leaves Δ_{H,V}^{A} in the socle of the quotient is an indecomposable module M in F(Δ^{A}) (indecomposable because its semisimple quotient is simple) which, provided H ≠ 1, gives rise to a non-split extension 0 → Δ_{H,V}^{A} → M_{1} → M_{2} → 0 with all terms in F(Δ^{A}). It follows if H ≠ 1 that Δ_{H,V}^{A} is not Ext-injective. □

Although we will not use it in reformulating Alperin’s conjecture, we include the next result for completeness.

Theorem 4.3. Let A = kO_{S} where O_{S} is the orbit category of G with respect to its p-subgroups, and let B be the Ringel dual of A. The following are equivalent.

(1) Δ_{H,V}^{A} = P_{H,V}^{A} is a projective kO_{S}-module,

(2) Δ_{H,V}^{A} is Ext-projective in F(Δ^{A}),

(3) Δ_{H,V}^{B} = T_{H,V}^{B} is a canonical B-module,

(4) H is a Sylow p-subgroup of G.

Proof. Some of the implications are immediate. We have (4) ⇒ (1) because (H, V) is a minimal element of Λ. Also, (1) ⇒ (2) is trivial.

(2) ⇔ (3) follows from the duality between F(Δ^{A}) and F(Δ^{B}). (2) ⇒ (1) holds because the projectives lie in F(Δ^{A}), so they lie

among the Ext-projective objects, and the number of indecomposable Ext-projectives equals the number of indecomposable projectives (for example, by [29, 8.6] where it is shown that the indecomposable Extprojectives are the Ext-projective covers of the Δs).

It remains to demonstrate (1) ⇒ (4), which we do by showing that if H is not a Sylow p-subgroup then Δ_{H,V}^{A} is not projective. So let H be a p-subgroup of G and let X be the projective cover of V as a k[N_{G}(H)/H]-module. In the notation used in [27] and [29], consider the Mackey functor (Inf_{N_{G}(H)/H}^{N_{G}(H)} F Q_{X}) ↑_{N_{G}(H)}^{G}, namely the inflation to N_{G}(H) of the fixed quotient functor F Q_{X}, induced to G. In [29] such Mackey functors played a key role as factors in certain filtrations, and were denoted there Δ_{H,X}, but to avoid confusion we do not use this notation here, because it is not the same as our present usage. In [29, 3.4] we show that this Mackey functor is generated by its value at H. This implies that it is still generated by its value at H if we consider only the operations which form the covariant part of the Mackey functor (see e.g. [26, 2.1]), and hence if we let M be the kO_{S}-module which is the restriction to O_{S} of this Mackey functor then M is generated by its value at H. As in [29, 3.1] we have M(H) = X, which is indecomposable, and so we deduce that M is indecomposable as an kO_{S}-module (since if M = M_{1} ⊕ M_{2} then, say, M_{1}(H) = M(H) and hence M_{1} = M).

Consider now the short exact sequence of kO_{S}-modules

0 → K → M → Δ_{H,V}^{A} → 0

where K is the submodule of M generated by the values of M at groups strictly larger than H. Suppose that H is not a Sylow p-subgroup of G. Then there is a non-identity p-subgroup J/H of N_{G}(H)/H and it has a non-zero fixed quotient X_{J}. This implies that M(J) ≠ 0 and hence that K ≠ 0 (because K(J) ≠ 0). Since M is indecomposable the short exact sequence is not split, and so Δ_{H,V}^{A} is not projective. □

We immediately obtain several reformulations of Alperin’s weight conjecture by substituting for the numbers of simple kG-modules or weights the other numbers indicated by these theorems. The reformulations are appealing because they are expressed in terms of objects defined by categorical properties of a single category, kO_{S}-mod, together with knowledge of the preorder on Λ. The structure of this category has the potential to provide a reason why the numbers are equal since up to Morita equivalence the group algebra may be recovered from kO_{S}-mod and Λ using Theorem 4.2. Our hope is that the theory of stratifications may shed light on Alperin’s conjecture.

Some of our reformulations of Alperin’s conjecture reveal a symmetry between the two numbers which it asserts to be equal, and it is particularly apparent if we allow ourselves to express them in terms of the Ringel dual. We single out now two of the reformulations which have this symmetry.

Corollary 4.4. Let A = kO_{S} where O_{S} is the orbit category of G with respect to its p-subgroups, and let B be the Ringel dual of A. The assertion of Alperin’s weight conjecture for the group G is equivalent to each of the following two statements.

(1) The number of λ = (H, V) for which Δ_{λ}^{A} = T_{λ}^{A} is equal to the number of λ for which \overline{∇}_{λ}^{B} = T_{λ}^{B}.

λ (1, 1) (1, 2) (C_{2}, 1)
simple S_{λ}^{A}  a  b  c
projective P_{λ}^{A}  {a \atop a c}  {b \atop c}  c
injective I_{λ}^{A}  {a \atop a}  b {a b \atop c}
Δ_{λ}^{A}  {a \atop a}  b  c
\overline{∇}_{λ}^{A}  a  b {a b \atop c}
T_{λ}^{A}  {a \atop a}  b {a b \atop a c}

Table 1. Modules for A = 𝔽_{2}O_{S} where S is the set of 2-subgroups of S_{3}.

(2) The number of λ = (H, V) for which Δ_{λ}^{B} = P_{λ}^{B} is equal to the number of λ for which \overline{∇}_{λ}^{A} = I_{λ}^{A}.

We illustrate these assertions with a small example, which nevertheless demonstrates the key features. We consider the orbit category O_{S} for the symmetric group G = S_{3}, taking S to be the 2-subgroups of S_{3}. Thus S contains the identity subgroup and three subgroups of order 2 and O_{S} has four objects. Since the three subgroups of order 2 are all conjugate, the corresponding objects of O_{S} are all isomorphic, and we denote one of these objects by G/C_{2}. By Lemma 2.1 we can delete the other two isomorphic objects without changing the Morita type of the category algebra, and we do this. The automorphism groups of the objects are Aut(G/1) ≅ G and Aut(G/C_{2}) = 1, so that taking k = 𝔽_{2} (a splitting field!) there are three simple representations of O_{S} parametrized by pairs (G/1, 1) = (1, 1), (G/1, 2) = (1, 2) and (G/C_{2}, 1) = (C_{2}, 1), where we denote representations of the automorphism groups by their dimensions. To simplify the notation we will write the simple kO_{S}-modules as a = S_{1,1}^{A}, b = S_{1,2}^{A} and c = S_{C_{2},1}^{A}.

The relevant representations of A = kO_{S} and its Ringel dual, B, appear in Tables 1 and 2. We see that the weights (1, 2) and (C_{2}, 1) determine the columns of the table for A in which the Δ_{λ}^{A} are simple, and also in which the \overline{∇}_{λ}^{A} are injective. In the table for B these are the columns in which \overline{∇}_{λ}^{B} = T_{λ}^{B}. At the same time the columns indexed by (1, 1) and (1, 2) are those in which Δ_{λ}^{A} = T_{λ}^{A} or equivalently Δ_{λ}^{A} = I_{λ}^{A} for A, and P_{λ}^{B} = Δ_{λ}^{B} for B. For completeness we describe all the indecomposable 𝔽_{2}O_{S}-modules. The ones in F(Δ^{A}) and F(\overline{∇}^{A}) appear

λ (1, 1) (1, 2) (C_{2}, 1)
simple S_{λ}^{B}  α  β  γ
projective P_{λ}^{B}  α_{{γ \atop α}}  β _{{γ \atop α ; γ_{α} β}}
injective I_{λ}^{B}  {γ \atop α} ; {γ \atop α}  {γ \atop β}  γ^{{γ \atop α}}
Δ_{λ}^{B}  α_{{γ \atop α}}  β  γ
\overline{∇}_{λ}^{B}  {γ \atop α}  {γ \atop β}  γ
T_{λ}^{B}  {γ \atop α} ; {γ \atop α}  {γ \atop β}  γ

Table 2. Modules for B = the Ringel dual of A.

F(Δ^{A}) : \begin{aligned}& ⋄ {b \atop c} & & & {a \atop a} ⋄ \\ & ↗ & ↘ & ↗ \\ ⋄ c & & & {a b \atop a c} ⋄ \\ & ↘ & ↗ & ↘ \\ & ⋄ {a \atop a c} & & & b ⋄ \end{aligned}

F(\overline{∇}^{A}) : \begin{aligned}⋄ & {a \atop a} & ⋄ \\ & & ↗ & ↘ \\ & ⋄ a b_{a c} & & a \\ & ↗ & ↘ & ↗ \\ a & & & {a b \atop c} ⋄ \\ & & & ↘ \\ & & & ⋄ b ⋄ \end{aligned}

Table 3. Modules for A = 𝔽_{2}O_{S} with a Δ^{A}-filtration and with a \overline{∇}^{A}-filtration.

in Table 3. In addition to the modules displayed in these diagrams there is one further indecomposable 𝔽_{2}O_{S}-module, which is {a \atop c}. In these pictures we place a symbol ⋄ to the left of a module when it is Extprojective, and to the right of a module when it is Ext-injective.

We conclude with some remarks. The first is that there is a blockby-block refinement of Alperin’s conjecture which may be described as follows. Given a block idempotent e^{2} = e ∈ Z(kG), for each psubgroup H ≤ G the Brauer morphism provides a central idempotent Br_{H}(e) ∈ Z(k[C_{G}(H)]) ⊆ k[N_{G}(H)] (see [25]). Regarding each simple k[N_{G}(H)/H]-module V as a k[N_{G}(H)]-module we may say that V belongs to e if Br_{H}(e) · V ≠ 0 (in which case Br_{H}(e) acts as the identity on V and Br_{H}(e) · V = V). The block-by-block version of Alperin’s weight conjecture states that for each block of kG, the number of simple kG-modules belonging to the block equals the number of weights belonging to the block.

We see from the example with S_{3} at p = 2 that kO_{S} need not respect the blocks of kG since kS_{3} has two blocks, but kO_{S} has only one block. It thus appears at first sight that the reformulations of Alperin’s conjecture using kO_{S} might not be well adapted to block-by-block versions.

In fact, we may produce such reformulations in the following way. The category algebra kO_{S} contains an idempotent 1_{G/H} for each object G/H of O_{S} These idempotents are orthogonal and 1_{G/H} · kO_{S} · 1_{G/H} ≅ k[N_{G}(H)/H]. Given a block idempotent e of kG we regard each Br_{H}(e) as an element of 1_{G/H} · kO_{S} · 1_{G/H} and let E = ∑_{H∈S/G} Br_{H}(e). Now E is idempotent, and E · kO_{S} · E is an algebra whose simple modules are the S_{H,V} where V belongs to e, and which is standardly stratified, with structure obtained by applying the functor M → E · M to the stratification of kO_{S}. We leave it to the interested reader to complete the argument that the analogues of 4.1 and 4.2 hold for E · kO_{S} · E. Thus, for example, H = 1 and V belongs to e if and only if Δ_{H,V}^{EAE} = T_{H,V}^{EAE}; and (H, V) is a weight belonging to e if and only if \overline{∇}_{H,V}^{EAE} = I_{H,V}^{EAE}.

We remark that the property required of kO_{S} which Corollary 4.4 states to be equivalent to Alperin’s conjecture is not satisfied by standardly stratified algebras in general, and it is quite easy to find examples of this. One such example is given on page 153 of [2] where there is exhibited a standardly stratified algebra in which the number of λ for which Δ_{λ} = T_{λ} is 1 but the number of λ for which \overline{∇}_{λ} = I_{λ} is 0. This means that to establish Alperin’s conjecture by using these reformulations, further specific properties of the orbit category algebra kO_{S} must be used. If we knew how to identify the right properties then it would no doubt be possible to prove Alperin’s conjecture. Equally, without knowing a proof it is hard to see what the appropriate properties might be.

The many reformulations of the conjecture which have appeared over the years can be viewed as attempts to identify the mechanism which underlies it, and the present work is no different in this respect. Like some other attempts we have focused on the structure of the orbit category, and we mention [19, 24] as recent contributions which also exploit this structure. By comparison, another recent approach of Linckelmann [18] uses the structure of the Frobenius category. This has the advantage that it behaves better than the orbit category when it comes to getting block-by-block versions of the conjecture, but it has the disadvantage that the Frobenius category loses some information about p-subgroups, which it appears to be necessary to restore by considering an appropriate extension category. At the moment it is hard to know which category to favour. It seems quite possible that once the conceptual underpinnings of Alperin’s conjecture have been properly worked out and understood, many of these approaches to the conjecture will become viable.

Acknowledgement

I wish to thank the Bernoulli Center of the EPFL, Lausanne, for its hospitality and support while part of this work was undertaken.

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 E-mail address: webb@math.umn.edu

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA