Exact Category of Modules of Constant Jordan type

Jon F. Carlson^{1} and Eric M. Friedlander^{2}

1 Department of Mathematics, University of Georgia

  jfc@math.uga.edu

2 Department of Mathematics, Northwestern University, Evanston, IL 60208

  eric@math.northwestern.edu

To Yuri Manin with admiration

Summary. For a finite group scheme G, we continue our investigation of those finite dimensional kG-modules which are of constant Jordan type. We introduce a Quillen exact category structure C(kG) on these modules and investigate K_{0}(C(kG)). We study which Jordan types can be realized as the Jordan types of (virtual) modules of constant Jordan type. We also briefly consider thickenings of C(kG) inside the triangulated category stmod(kG).

1 Introduction

Together with Julia Pevtsova, the authors introduced in [6] an intriguing class of modules for a finite group G (or, more generally, for an arbitrary finite group scheme), the kG-modules of constant Jordan type. This class includes projective modules and endotrivial modules. It is closed under taking direct sums, direct summands, k-linear duals, and tensor products. We have several methods for constructing modules of constant Jordan type, typically using cohomological techniques. In addition, Andrei Suslin has introduced several interesting constructions which associate modules of constant Jordan type to an arbitrary finite dimensional kG-module in the special case that G = ℤ/pℤ × ℤ/pℤ.

What strikes us as remarkable is how challenging is the problem of classifying modules of constant Jordan type even for relatively simple finite group schemes. In this paper we address two other aspects of the theory that also present formidable challenges. The first is the realization problem of determining which Jordan types can actually occur for modules of constant Jordan type. The second question concerns stratification of the entire module category by modules of constant Jordan type.

To consider realization, we give the class of kG-modules of constant Jordan type the structure of a Quillen exact category C(kG) using “locally split short exact sequences.” This structure suggests itself naturally once kG-modules are treated from the point of view of π-points as in [10], a point of view necessary to even define modules of constant Jordan type. With respect to this exact category structure, the Grothendieck group K_{0}(C(kG)) arises as a natural invariant. There are natural Jordan type functions JType, \overline{JType} defined on K_{0}(C(kG)) which are useful to formulate questions of realizability of (virtual) modules of constant Jordan type. The reader will find several results concerning the surjectivity of these functions.

A seemingly very difficult goal is the classification of kG-modules of constant Jordan type, or at least the determination of K_{0}(C(kG)). In this paper, we provide a calculation of K_{0}(C(kG)) for two very simple examples: the Klein four group and the first infinitesimal kernel of SL_{2}.

The category C(kG) possesses many closure properties. However, the complexity of this category is reflected in the observation that an extension of modules of constant Jordan type need not be of constant Jordan type. We conclude this paper by a brief consideration of a stratification of the stable module category stmod(kG) by “thickenings” of C(kG).

We are very grateful to Julia Pevtsova and Andrei Suslin for many discussions. This paper is part of a longer term project which will reflect their ideas and constructions.

2 The exact category C(kG)

As shown in [6], there is a surprising array of kG-modules of constant Jordan type. One evident way to construct new examples out of old is to use locally split extensions (see, for example, Proposition 2.4). In order to focus on examples which seem more essential, we introduce in Definition 2.3 the Quillen exact category C(kG) of modules of constant Jordan type whose admissible short exact sequences are those which are locally split.

We begin by recalling the definition of a π-point of a finite group scheme over k. This is a construction that is necessary to formulate the concept of modules of constant Jordan type.

Definition 2.1. Let G be a finite group scheme with group algebra kG (the linear dual of the coordinate algebra k[G]). A π-point of G is a map of Kalgebras α_{K} : K[t]/t^{p} → KG_{K} which is left flat and which factors through some abelian unipotent subgroup scheme U_{K} ⊂ G_{K}; here K is an arbitrary field extension of k and G_{K} is the base extension of G along K/k.

Two π-points α_{K}, β_{L} of G are said to be equivalent (denoted α_{K} ∼ β_{L}) provided that for every finite dimensional kG-module M the K[t]/t^{p}-module α_{K}^{∗}(M_{K}) is projective if and only if the L[t]/t^{p}-module β_{L}^{∗}(M_{L}) is projective.

In [10], it is shown that the set of equivalence classes of π-points of G admits a scheme structure which is defined in terms of the representation theory of G and which is denoted Π(G). Moreover, it is verified that this scheme is isomorphic to the projectivization of the affine scheme of H^{•}(G, k),

Π(G) ≅ P roj H^{•}(G, k).

Here, H^{•}(G, k) is the finitely generated commutative k-algebra defined to be the cohomology algebra H^{∗}(G, k) if p = char(k) equals 2 and to be the subalgebra of H^{∗}(G, k) generated by homogeneous classes of even degree if p > 2.

We next introduce admissible monomorphisms and admissible epimorphisms, formulated in terms of π-points.

Definition 2.2. Let G be a finite group scheme. A short exact sequence of kG-modules 0 → M_{1} → M_{2} → M_{3} → 0 is said to be locally split if its pullback via any π-point α_{K} : K[t]/t^{p} → KG is split as a short exact sequence of K[t]/t^{p}-modules. We shall frequently refer to such a locally split short exact sequence as an admissible sequence.

Moreover, we say that a monomorphism f : M_{1} → M_{2} of kG-modules is an admissible monomorphism if it can be completed to a locally split short exact sequence. Similarly, an epimorphism g : M_{2} → M_{3} is said to be an admissible epimorphism if it can be completed to a locally split short exact sequence.

We typically identify an admissible monomorphism with the inclusion of the image of the injective map f : M → N.

The objects of our study are kG-modules of constant Jordan type as defined below (and introduced in [6]). We recall that a finite dimensional K[t]/t^{p} module M of dimension N is isomorphic to

a_{p}[p] + ⋯ + a_{i}[i] ⋯ + a_{1}[1], ∑_{i} a_{i} · i = N,

where [i] = K[t]/t^{i} is the indecomposable K[t]/t^{p}-module of dimension i. We refer to the p-tuple (a_{p}, …, a_{1}) as the Jordan type of M and designate this Jordan type by JType(M); the (p − 1)-tuple (a_{p−1}, …, a_{1}) will be called the stable Jordan type of M.

Definition 2.3. A finite dimensional module M for a finite group scheme G is said to be of constant Jordan type if the Jordan type of α_{K}^{∗}(M_{K}) is independent of the choice of the π-point α_{K} of G.

In the following proposition, we see that the class of modules of constant Jordan type is closed under locally split extensions.

Proposition 2.4. Let G be a finite group scheme and let E denote a short exact sequence 0 → M_{1} → M_{2} → M_{3} → 0 of finite dimensional kG-modules.

1. Assume E is locally split. Then M_{1} and M_{3} are of constant Jordan type if and only if M_{2} is of constant Jordan type.

2. If M_{1} and M_{3} are modules of constant Jordan type, then E is locally split if and only if α_{K}^{∗}(M_{2,K}) ≃ α_{K}^{∗}(M_{1,K}) ⊕ α_{K}^{∗}(M_{3,K}) for some representative α_{K} of each generic point of Π(G).

Proof. Observe that a short exact sequence of finite dimensional K[t]/t^{p}- modules 0 → N_{1} → N_{2} → N_{3} → 0 is split if and only if JType(N_{2}) = JType(N_{1}) + JType(N_{3}). Thus, if E is locally split and if M_{1}, M_{3} have constant Jordan type, then M_{2} does as well.

If E is locally split and M_{2} has constant Jordan type, then the proof that both M_{1} and M_{3} have constant Jordan type is verified using the same argument as that of [6, 3.7] using [6, 3.5]; the point is that the Jordan type of M_{1} at some representative of a generic point of Π(G) must be greater or equal to the Jordan type of M_{1} at some representative of any specialization. The same applies to M_{2} and M_{3}. As shown in [11, 4.2], the Jordan type of any finite dimensional kG-module at a generic point of Π(G) is independent of the choice of π-point representing that generic point.

If 0 → V_{1} → V_{2} → V_{3} → 0 is a short exact sequence of k[t]/t^{p}-modules, then the Jordan type of V_{2} must be greater or equal to the sum of the Jordan types of V_{1} and V_{3}. Thus, the observation of the preceding paragraph verifies the following: assume that E is a short exact sequence of kG-modules with M_{1}, M_{3} of constant Jordan type; if M_{3} has the minimal possible Jordan type (namely the sum of the Jordan types of M_{1} and M_{3}) at each generic π-point of G, then M_{3} must have constant Jordan type. ⊓ ⊔

As the following proposition asserts, admissible monomorphisms and admissible epimorphisms as in Definition 2.2 are associated to structures of exact categories (in the sense of Quillen [17]) on the category mod(kG) of finite dimensional kG-modules and the full subcategory C(kG) of modules of constant Jordan type.

Proposition 2.5. The collection \underline{E} of locally split short exact sequences of finite dimensional kG-modules constitutes a class of admissible sequences providing mod(kG) with the structure of an exact category in the sense of Quillen (cf. [17]).

Similarly, the class \underline{E}_{C} of locally split short exact sequences of kG-modules of constant Jordan type also constitutes a class of admissible sequences, thereby providing C(kG) with the structure of exact subcategory of mod(kG).

Proof. According to Quillen, to verify that \underline{E} provides mod(kG) with the structure of an exact category we must verify three properties. The first property consists of the conditions that any short exact sequence isomorphic to one in \underline{E} is itself in \underline{E}; that \underline{E} contains all split short exact sequences; and that if 0 → M′ →^{i} M →^{j} M′′ → 0 is a short exact sequence in \underline{E}, then i : M′ → M is a kernel for j : M → M′′ and j : M → M′′ is a cokernel for i : M′ → M. These conditions are essentially immediate.

The second property consists of the conditions that the class of admissible monomorphisms (i.e., monomorphisms occurring in an exact sequence of \underline{E}) is closed under composition and closed under push-out with respect to any map of mod(kG); similarly that the class of admissible epimorphisms is closed under composition and pull-back. This follows from the observation that these properties hold for split exact sequences, thus also for those exact sequences split at every π-point.

The third property asserts that any map f : M′ → M of kG-modules with the property that there exists some map g : M → Q of kG-modules such that the composition g ∘f : M′ → M → Q is an admissible monomorphism is itself an admissible monomorphism; and the analogous statement for admissible epimorphisms. This is clear, for any splitting of the composition α_{K}^{∗}(g ∘ f) : α_{K}^{∗}(M′_{K}) → α_{K}^{∗}(M_{K}) → α_{K}^{∗}(Q_{K}) gives a splitting of α_{K}^{∗}(f) : α_{K}^{∗}(M′_{K}) → α_{K}^{∗}(M_{K}).

In view of Proposition 2.4, the preceding discussion for \underline{E} applies equally to the class \underline{E}_{C} of locally split short exact sequences of C(kG). ⊓ ⊔

The following proposition makes the evident points that not every short exact sequence of modules of constant Jordan type is locally split, that some non-split short exact sequences are locally split, and that a non-locally split extension of modules of constant Jordan type might not have a middle term which is of constant Jordan type.

Proposition 2.6. Let E be an elementary abelian p-group and let I ⊂ kE be the augmentation ideal. Then the short exact sequence of kE-modules of constant Jordan type

\xymatrix{0 \ar[r] & I^{i}/I^{j} \ar[r] & I^{i}/I^{ℓ} \ar[r] & I^{j}/I^{ℓ} \ar[r] & 0}

is not locally split for any i < j < ℓ ≤ p.

If the rank of E is at least 2, then a non-trivial negative Tate cohomology class ξ ∈ Ĥ^{n}(G, k) determines a locally split (but not split) short exact sequence of the form

0 → k → E → Ω^{n−1}(k) → 0.

In contrast, if the rank of E is at least 2 and if 0 ≠ ζ ∈ H^{1}(E, k), then the associated short exact sequence 0 → k → M → k → 0 of kE-modules is such that M does not have constant Jordan type.

Proof. By [6, 2.1], each I^{i}/I^{ℓ} is of constant Jordan type. The fact that the sequence 0 → I^{i}/I^{j} → I^{i}/I^{ℓ} → I^{j}/I^{ℓ} → 0 is not locally split follows from Proposition 2.4 and an easy computation of Jordan types. The second assertion is a special case of [6, 6.3].

Finally, the extension of k by k determined by ζ ∈ H^{1}(E, k) is split when pulled-back via the π-point α_{K} if and only if α_{K}^{∗}(ζ_{K}) = 0 ∈ H^{1}(K[t]/t^{p}, K).

The subset of Π(E) consisting of those [α_{K}] satisfying α_{K}^{∗}(ζ_{K}) = 0 is a hyperplane of Π(E) ≅ ℙ^{r−1} where r is the rank of E. Consequently, the Jordan type of α_{K}^{∗}(M_{K}) is 2[1] on a hyperplane of Π(E) and is [2] otherwise.

⊓ ⊔

3 The Grothendieck group K_{0}(C(kG))

For any exact category E specified by a class \underline{E} of admissible exact sequences, we denote by K_{0}(E) the Grothendieck group given as the quotient of the free abelian group of isomorphism classes of objects of E modulo the relations generated by [M_{1}] − [M_{2}] + [M_{3}] whenever 0 → M_{1} → M_{2} → M_{3} → 0 is an admissible sequence in \underline{E}. In this section, we begin a consideration of K_{0}(C(kG)), the Grothendieck group of modules of constant Jordan type (with respect to locally split short exact sequences).

Following Quillen [17], we could further consider K_{i}(C(kG)) for i > 0. Granted the current state of our understanding, the challenge of investigating K_{0}(C(kG)) is sufficiently daunting that we postpone any consideration of K_{i}(C(kG)), i > 0.

Proposition 3.1. For any finite group scheme G, there are natural embeddings of exact categories

\xymatrix{P(G) \ar[r] & C(kG), & C(kG) \ar[r] & mod(kG).}

The first is from the exact category P(G) of finitely generated projective kGmodules into the category C(kG) of modules of constant Jordan type; the second is from C(kG) into the category mod(kG) of all finitely generated kGmodules. These embeddings induce homomorphisms

\xymatrix{K_{0}(kG) ≡ K_{0}(P(G)) \ar[r] & K_{0}(C(kG)) \ar[r] & K_{0}(mod(kG)).}

Moreover, these homomorphisms are contravariantly functorial with respect to a closed immersion i : H → G of finite group schemes.

Proof. Since every short exact sequence of projective modules is split, we conclude that the full embedding P(G) → C(kG) of the category of finite dimensional projective kG-modules into the category of kG-modules of constant Jordan type (with admissible short exact sequences being the locally split short exact sequences) is an embedding of exact categories. The embedding C(kG) ⊆ mod(kG) is clearly an embedding of exact categories.

If i : H → G is a closed embedding, then kG is projective as a kH-module (cf. [15, 8.16]) so that any projective kG-module restricts to a projective kHmodule. By [6, 1.9], restriction of a kG-module of constant Jordan type to kH is again of constant Jordan type. Since restriction is exact and preserves locally split sequences (because every π-point of H when composed with i becomes a π-point of G), we obtain the asserted naturality with respect to i : H → G. ⊓ ⊔ Remark 3.2. If a module M ∈ C(kG) admits an admissible filtration (meaning the inclusion maps are admissible monomorphisms)

M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{n} = M,

then [M] = ∑_{i=1}^{n}[M_{i}/M_{i−1}] ∈ K_{0}(C(kG)).

As an immediate corollary of Proposition 3.1, we have the following.

Corollary 3.3. If G is a finite group, then K_{0}(P(G)) → K_{0}(C(kG)) is injective. More generally, if the Cartan matrix for the finite group scheme G is non-degenerate, then K_{0}(P(G)) → K_{0}(C(kG)) is injective.

Proof. A basis for K_{0}(P(G)) is given by the classes of the indecomposable projective modules, while a basis K_{0}(mod(kG)) is given by the classes of the irreducible modules. The Cartan matrix for kG represents the natural map K_{0}(P(G)) to K_{0}(mod(kG)) with respect to these bases. By Proposition 3.1, this map factors through K_{0}(C(kG)). This proves the second statement. A theorem of Brauer (cf. [3, I.5.7.2]) says that if G is a finite group then the Cartan matrix for kG is non-singular. ⊓ ⊔

We consider a few elementary examples.

Example 3.4. Let G be the cyclic group ℤ/p. Then

K_{0}(C(kℤ/p)) ≃ ℤ^{p}.

The map K_{0}(P(G)) → K_{0}(C(kℤ/p)) is identified with the map ℤ → ℤ^{p}, a ↦ (a, 0, …, 0). The map K_{0}(C(kℤ/p)) → K_{0}(mod(kℤ/p)) is identified with the map ℤ^{p} → ℤ, (a_{p}, …, a_{1}) ↦ ∑_{i} ia_{i}.

Proposition 2.4 and the universal property of the Grothendieck group K_{0}(C(kG)) immediately imply the following proposition.

Proposition 3.5. Sending a kG-module M of constant Jordan type to the Jordan type of α_{K}^{∗}(M_{K}) and to the stable Jordan type of α_{K}^{∗}(M_{K}) for any π-point α_{K} of G determines homomorphisms

\xymatrix{JType : K_{0}(C(kG)) \ar[r] & ℤ^{p}, & \overline{JType} : K_{0}(C(kG)) \ar[r] & ℤ^{p−1}.}

(1)

We view an element of K_{0}(C(kG)) as the class of a virtual kG-module of constant Jordan type. In the next section, we shall investigate to what extent the homomorphisms JType, \overline{JType} are surjective; in other words, the realizability of Jordan types by virtual kG-modules of constant Jordan type.

The function JType is not injective: for example, a module of constant Jordan type and its k-linear dual have the same Jordan type. The example of ℤ/2 × ℤ/2 given in Proposition 3.6 provides a more explicit example of non-injectivity of JType.

We can achieve the next example because we know exactly what are the indecomposable kE-modules for the Klein-four group E = ℤ/2 × ℤ/2 (cf. [2], [13]).

Proposition 3.6. For any field k of characteristic 2, the group algebra kE of the Klein four group E = ℤ/2 × ℤ/2 satisfies

K_{0}(C(kE)) ≃ ℤ^{3}.

Proof. We recall from [6, 6.2] that any kE-module of constant Jordan type is of the form kE^{e} ⨁(⊕_{i}Ω^{n_{i}}(k)). One easily checks there is an admissible (i.e., locally split, short) exact sequence of kE-modules of the following form:

\xymatrix{0 \ar[r] & Ω^{2}(k) \ar[r] & Ω^{1}(k) ⊕ Ω^{1}(k) \ar[r] & k \ar[r] & 0}  (2)

Hence, [Ω^{2}(k)] = 2[Ω^{1}(k)]−[k] in K_{0}(C(kE)). Consecutive applications of the Heller shift to the sequence (2) thus imply that K_{0}(C(kE)) is generated by the classes of the three kE-modules: kE, k, Ω^{1}(k).

We define a function σ on the class of modules of constant Jordan type by sending M ≃ kE^{e} ⨁(⊕_{i}Ω^{n_{i}}(k)) to σ(M) = ∑_{i} n_{i}. We proceed to show that σ is additive on admissible sequences, and, hence, induces a homomorphism σ : K_{0}(C(kE)) → ℤ.

Let ξ : 0 → M → L → N → 0 be an admissible sequence of kE-modules of constant Jordan type, and assume N = N_{1} ⊕ N_{2}. Since Ext_{E}^{1}(N, M) = Ext_{E}^{1}(N_{1}, M) ⊕ Ext_{E}^{1}(N_{2}, M), we have ξ = ξ_{1} + ξ_{2} where ξ_{i} ∈ Ext_{E}^{1}(N_{i}, M). Moreover, both ξ_{1}, ξ_{2} are admissible, and the additivity of σ on ξ_{1}, ξ_{2} implies additivity of σ on ξ. Hence, we may assume that N is an indecomposable nonprojective module of constant Jordan type, that is, N ≃ Ω^{n}(k). Similarly, we may assume M ≃ Ω^{m}(k).

Thus, we may assume that ξ has the form

\xymatrix{0 \ar[r] & Ω^{m}(k) \ar[r] & kE^{e} ⨁(⊕_{i}Ω^{n_{i}}(k)) \ar[r] & Ω^{n}(k) \ar[r] & 0}

By [6, 6.9], dim Ω^{2a}(k) = 4a+1; hence, dim Ω^{2a+1}(k) = 4(a+1)+1 (using the short exact sequence 0 → Ω^{2a+1}(k) → P_{2a} → Ω^{2a}(k) → 0). Comparing dimensions, we conclude that σ is additive on an admissible sequence ξ if and only if σ is additive on Ω^{1}(ξ). (We are implicitly using [6, 1.8] asserting that M has constant Jordan type if and only if Ω^{1}(M) has constant Jordan type.) Thus, we may further assume that n = 0. Hence, ξ has the form

\xymatrix{0 \ar[r] & Ω^{m}(k) \ar[r] & kE^{e} ⨁(⊕_{i}Ω^{a_{i}}(k)) \ar[r] & k \ar[r] & 0}

Since ξ is split on restriction to any π-point, there exists i such that the restriction Ω^{a_{i}}(k) → k is not the zero map. Hence, the map Ω^{a_{i}}(k) → k is surjective. Since the left end of the sequence ξ is projective free, we conclude that e = 0. Because the Jordan type of the middle term is the sum of the Jordan type of the ends, the middle term must have exactly two summands. Hence, the sequence has the form

\xymatrix{0 \ar[r] & Ω^{m}(k) \ar[r] & Ω^{a}(k) ⊕ Ω^{b}(k) \ar[r] & k \ar[r] & 0}

Because ξ represents an element in H^{1}(G, Ω^{m}(k)) ≃ Ĥ^{1−m}(G, k) that vanishes on restriction along any π-point, 1 − m < 0 by [6, 6.3]; in other words, m is positive. Likewise, a and b are not negative. That is, the map Ω^{a}(k) → k represents an element in Ĥ^{a}(G, k) which does not vanish on restriction to some π-point. This can only happen if a ≥ 0. The same argument shows that b ≥ 0.

Comparing dimensions (see [6, 6.9]), we get 2m + 1 + 1 = dim Ω^{m}(k) + dim k = dim Ω^{a}(k) + dim Ω^{b}(k) = 2a + 1 + 2b + 1. Hence, m + 0 = a + b, and we conclude that σ is additive.

We consider the map

Ψ = (JType, σ) : K_{0}(C(kE)) ⟶ ℤ^{2} ⊕ ℤ.

This is well defined by Proposition 3.5 and the observation that σ is additive as shown above. To prove the proposition, it suffices to show that Ψ has image a subgroup of finite index inside ℤ^{2} ⊕ ℤ. This follows from the observation that the vectors Ψ(k) = (0, 1, 0), Ψ(Ω^{1}(k)) = (1, 1, 1), and Ψ(kE) = (2, 0, 0) are linearly independent over ℚ. ⊓ ⊔

The preceding proof shows that the admissible sequences for the Klein four group have a generating set consisting of sequences of the form

\xymatrix{0 \ar[r] & Ω^{m+n+a}(k) \ar[r] & Ω^{m+a}(k) ⊕ Ω^{n+a}(k) ⊕ kE^{e} \ar[r] & Ω^{a}(k) \ar[r] & 0,}

(3)

where m, n > 0. Here,

2e = |m + n + a| + |a| − |m + a| − |n + a|;

hence, e = 0 unless a is negative and m + n + a is positive. This fact is of use in the discussion to follow.

There are examples of group schemes for which the Cartan matrix is singular so that we can not apply Corollary 3.3 in these examples. Perhaps the simplest is the first Frobenius kernel G = 𝔊_{1} of the algebraic group 𝔊 = SL_{2} with p > 2. In this case, kG is isomorphic to the restricted enveloping algebra of the restricted p-Lie algebra sl_{2} . It is known (cf. [12, 2.4]) that kG has (p + 1)/2 blocks one of which has only a single projective irreducible module. Each of the other (p − 1)/2 blocks has two non-isomorphic irreducible modules. Now suppose that B is one of these blocks. It has irreducible modules S and T. The projective covers Q_{S} of S and Q_{T} of T have the forms

\xymatrix{ &  & S \ar[dl] \ar[dr] &  &  &  & T \ar[dl] \ar[dr] \\ Q_{S} = & T \ar[dr] &  & T , \ar[dl] & Q_{T} = & S \ar[dr] &  & S \ar[dl] \\  &  & S &  &  &  & T.}

In particular, for any module M in B, Rad^{3}(M) = 0, and moreover, if Rad^{2}(M) ≠ {0}, then M contains a projective direct summand. That is, if M is an indecomposable non-projective B-module then Rad^{2}(M) = {0}.

We see from the above that the Cartan matrix of kG is a p × p matrix consisting of (p+1)/2 block matrices along the diagonal, (p−1)/2 of which are the Cartan matrices of blocks B of the group algebra as above. The Cartan matrix of such a block is a 2 × 2 matrices with 2 in every entry. The other block matrix of the Cartan matrix of kG is a 1 × 1 identity matrix. Hence the natural map from K_{0}(P(G)) to K_{0}(mod(kG)) is not injective. However, as we see below, K_{0}(P(G)) still injects into K_{0}(C(kG)).

Proposition 3.7. Let G = 𝔊_{1} where 𝔊 = SL_{2} and assume that p ≥ 3. Then

K_{0}(C(kG)) ≃ ℤ^{3p−2}.

Moreover, the map K_{0}(P(G)) → K_{0}(C(kG)) is injective.

Proof. We begin by recalling that rational SL_{2}-modules are kG-modules of constant Jordan type by [6, 2.5]; in particular, every simple kG-module is a module of constant Jordan type.

As stated above, there are (p − 1)/2 blocks B_{1}, …, B_{(p−1)/2} as above. In addition there is another block B′ containing only a single indecomposable module which is projective. Therefore,

K_{0}(C(kG)) ≃ K_{0}(C(B′)) ⊕ ∑_{i=1}^{(p−1)/2} K_{0}(C(B_{i})).

We know that K_{0}(C(kB′)) ≃ ℤ. Consequently, it suffices to prove that K_{0}(C(B_{i})) ≃ ℤ^{6} for each i.

We consider B = B_{i} with simple modules S and T. We may assume that S and T have stable constant Jordan types 1[i] and 1[p − i] respectively. The indecomposable B-modules can be classified using standard methods similar to those of [18] or the diagrammatic methods of [5]. Every non-simple, nonprojective indecomposable B-module M has the property that Soc(M) = Rad(M) is a direct sum of t copies of one of the simple modules S or T. The quotient M/ Rad(M) is a direct sum of r copies of the other simple T or S. Moreover, we must have that r is one of t−1, t, or t+1, that is, |t−r| ≤ 1. In the case that r = t, it is evident that the module M has periodic cohomology.

Or, at least, it is clear that the dimensions of Ω^{n}(M) are bounded for all n. This means that the module M must have proper non-trivial support variety in Π(G), as the annihilator of its cohomology must be non-trivial. Consequently, such a module M can not have constant Jordan type. Thus, we conclude that |t − r| = 1, and that every indecomposable module of constant Jordan type is a syzygy of an irreducible module (cf. (7), (8) and (9), below). This last fact can also be deduced from recent results of Benson [4] on algebras with radical cube zero.

Using the diagrammatic methods, sketched below (as can also be done in the situation of Proposition 3.6), we first verify that the non-trivial admissible sequences are generated by sequences of the form

0 → Ω^{2(m+n)+a}(𝔛) → Ω^{2m+a}(𝔛) ⊕ Ω^{2n+a}(𝔛) ⊕ P → Ω^{a}(𝔛) → 0 (4)

and

0 → Ω^{2(m+n−1)+a}(𝔛) → Ω^{2m−1+a}(𝔜) ⊕ Ω^{2n−1+a}(𝔜) ⊕ Q → Ω^{a}(𝔛) → 0 (5)

and

0 → Ω^{2(m+n)−1+a}(𝔜) → Ω^{2m+a}(𝔛) ⊕ Ω^{2n−1+a}(𝔜) ⊕ R → Ω^{a}(𝔛) → 0 (6)

where 𝔛 and 𝔜 are either S or T and 𝔜 is not the same as 𝔛. Here a can be any integer and m, n > 0. The modules P, Q and R are projective modules which are required for the exactness. In any of these sequences, the projective module P, Q or R is a eumber of copies of the projective cover of a simple module in the socle of the left-most term of the sequence, or in the top of the right-most term of the sequence. In what follows it might be helpful to note that, for dimension reasons, the projective module P in sequence (4) is zero except in the cases that 2(m + n) + a is positive and a is negative. Likewise, Q in sequence (5) is zero except when 2(m + n − 1) + a is positive and a is negative. And a similar thing happens for sequence (6). To be very specific, P in (4) is a sum of copies of Q_{T} if 𝔛 = S and a is even or if 𝔛 = T and a is odd. Otherwise, it is a sum of copies of Q_{S}. The module Q in (5) and R in (6) are sum of copies of Q_{T} in the cases that 𝔛 = S, 𝔜 = T and a is even and 𝔛 = T, 𝔜 = S and a is odd. Otherwise, they are sum of copies of Q_{S}. The verification that these sequences, (4), (5) and (6), generate the collection of admissible sequences prodeeds as follows.

As in the proof of Proposition 3.6, we are looking for sequences representing elements of Ext_{kG}^{1}(−, −) which vanish on restriction along any π-point. Because Ext_{kG}^{1} distributes over direct sums, and any such distribution still satisfies the vanishing condition, we can restrict our consideration to sequences which have indecomposable end terms. Consequently, the right end term can be considered to be Ω^{a}(𝔛) for 𝔛 either S or T. Suppose that 𝔛 is S. We can translate the sequence by Ω^{−a} so that the right term is isomorphic to S. Note also that because the two end terms each have only one single nonprojective Jordan block on restriction to any Π-point, there are at exactly two nonprojective direct summands in the middle term of the sequence.

Now we consider the diagrams for the syzygies of S amd T. For example,

\xymatrix{Ω^{−1}(S) = &  & S \ar[dl] \ar[dr] &  & Ω^{−1}(T) = &  & T \ar[dl] \ar[dr] \\  & T &  & T , &  & S &  & S}  (7)

\xymatrix{Ω(S) = & T \ar[dr] &  & T , \ar[dl] & Ω(T) = & S \ar[dr] &  & S \ar[dl] \\  &  & S &  &  &  & T.}  (8)

\xymatrix{Ω^{2}(S) = & S \ar[dr] &  & S \ar[dl] \ar[dr] &  & S , \ar[dl] & Ω^{2}(T) = & T \ar[dr] &  & T \ar[dl] \ar[dr] &  & T \ar[dl] \\  &  & T &  & T &  &  &  & S &  & S.}

(9)

The diagram for an arbitrary syzygy of either S or T is merely an elongation of these diagrams.

If we have a locally split sequence whose right hand term is S (or T) and if Ω^{m}(𝔛) occurs in the middle term (with 𝔛 either S or T), then m must be nonnegative. The reason is that otherwise (m < 0) the Jordan type of the kernel of such a map has too many non projective blocks at any Π-point. In addition, such a map could not be right split at any π-point, because the socle of the kernel would have more irreducible constituents than the head.

To finish the verification that the non-trivial admissible sequences are generated by sequences of the form (4), (5), or (6), we suppose that there is a locally split sequence whose right hand term is S. The middle term must consist of two terms of the form Ω^{m}(𝔛) and Ω^{n}(𝔜), with m, n > 0 (if either m or n is 0, then the sequence splits). Because both of these terms must map surjectively to S, we must have that 𝔛 ≃ S if m is even, and 𝔛 ≃ T if m is odd. The same happens for 𝔜 and n. Finally, we notice that the left term of the sequence is determined entirely by its composition factors. A complete analysis, which we leave to the reader, reveals that the sequence must look like one of (4), (5), or (6).

Consider the following two collections of indecomposable B-modules:

U_{1} = {Ω^{m}(S), m even} ∪ {Ω^{n}(T), n odd} ∪ {Q_{T} }

and

U_{2} = {Ω^{m}(S), m odd} ∪ {Ω^{n}(T), n even} ∪ {Q_{S}}.

Observe that each of the sequences (4), (5) or (6) for any values of a, m, n, involves either modules all of which are isomorphic to elements of U_{1} or modules isomorphic to elements of U_{2}.

Let E(B) be the exact category of all B-modules of constant Jordan type with the admissible sequences being the split exact sequences. Thus, K_{0}(E(B)) is the Green ring ℤ[B] of B and is a free ℤ-module on the classes of elements in the union of U_{1} and U_{2}. Moreover factoring out the relators coming from the sequences (4) and (5) defines K_{0}(C(B)) as a quotient of ℤ[B]. As in the proof of Proposition 3.6, we conclude that the images of classes in U_{1} (respectively, U_{2}) in K_{0}(C(B)) are generated by the images of the classes [S], [Ω^{1}(T)] and [Q_{T}] (resp., [T], [Ω^{1}(S)] and [Q_{S}]).

Now let M_{1} be the subgroup of K_{0}(E(B)) generated by the classes of modules in U_{1}, and let M_{2} be the subgroup generated by the classes of modules in U_{2}. Then K_{0}(E(B)) ≃ M_{1} ⊕ M_{2}. Let K be the subgroup generated by the relators determined by sequences (4) (5) and (6). That is, K is the kernel of the homomorphism of K_{0}(E(B)) onto K_{0}(C(B)). Notice that K = K_{1} ⊕ K_{2} where each K_{i} = M_{i} ∩ K is generated by the relators determined by those sequences of the form (4) (5) and (6) which only involve modules from U_{i}. Consequently, we have that

K_{0}(C(B)) ≃ K_{0}(E(B))/K ≃ M_{1}/K_{1} ⊕ M_{2}/K_{2}

Therefore, to complete the proof of the proposition, it suffices to exhibit homomorphisms M_{1}/K_{1} → ℤ^{3} and M_{2}/K_{2} → ℤ^{3} whose images are cofinite. We do this by showing that there are isomorphism θ_{i} : M_{i}/K_{i} ≃ K_{0}(C(k′E)) for i = 1, 2 and appealing to Proposition 3.6; here, E = ℤ/2×ℤ/2 is the Klein four group and k′ is a field of characteristic 2.

For this purpose we define maps γ_{1} : M_{1} → K_{0}(E(k′E)) and γ_{2} : M_{2} → K_{0}(E(k′E)) as follows:

γ_{1}([Ω^{n}(S)]) = [Ω^{n}(k′)] (for n even), γ_{2}([Ω^{n}(S)]) = [Ω^{n}(k′)] (for n odd),

γ_{1}([Ω^{n}(T)]) = [Ω^{n}(k′)] (for n odd), γ_{2}([Ω^{n}(T)]) = [Ω^{n}(k′)] (for n even),

γ_{1}([Q_{T}]) = [k′E], γ_{2}([Q_{S}]) = [k′E].

The reader should remember that γ_{i} is only defined on the classes of modules in U_{i}. Moreover, each γ_{i} is clearly surjective.

The important thing to note is that γ_{1} takes the relators coming from those sequences (4), (5) and (6) involving only the modules of U_{1} bijectively to the relators coming from the sequences (3), which are then zero in K_{0}(E(k′E)). To see this we need only replace the module 𝔛, 𝔜 in sequences (4), (5) and (6) by k′ and the projective modules P and Q by the appropriate sum of copies of k′E.

Moreover, relators coming from the sequences (3) generate the kernel of the natural quotient map from K_{0}(E(k′E)) to K_{0}(C(k′E)). Consequently, γ_{1} induces an isomorphism θ_{1} from M_{1}/K_{1} to K_{0}(C(k′E)). Likewise γ_{2} induces in isomorphism θ_{2} : M_{2}/K_{2} ≃ K_{0}(C(k′E)). ⊓ ⊔

4 Realization of Jordan types

In this section, we initiate the investigation of the images of the Jordan type functions, JType, \overline{JType}, introduced in Proposition 3.5. Our first proposition establishes the surjectivity of JType for an elementary abelian p-group of arbitrary rank. The reader is cautioned that this establishes realizability of Jordan types by virtual modules of constant Jordan type. Even for the rank 2 elementary abelian p-group E = ℤ/p × ℤ/p with p > 3, we do not know whether there is a kE-module M of constant Jordan type with \overline{JType}(M) = [2], whereas Proposition 4.1 shows that we can realize the Jordan type [2] in a virtual module.

Proposition 4.1. The map

\xymatrix{JType : K_{0}(C(kE)) \ar[r] & ℤ^{p}}

of Proposition 3.5 is surjective provided that E is an elementary abelian pgroup.

Proof. It suffices to verify that M = kE/I^{i} has constant Jordan type of the form 1[i] + a_{i−1}[i − 1] ⋯ + a_{1}[1] for each i, 1 ≤ i ≤ p, where I is the augmentation ideal of kE. The fact that M has constant Jordan type is verified in [6, 2.1]. The fact that a_{j} = 0 for j > i follows from the fact that I^{i} · M = 0. The fact that a_{i} = 1 follows from the observation that the generator of M is not annihilated by (g − 1)^{i} ∈ I for any generator g of E whereas any element of I/I^{i} is annihilated by (g − 1)^{i} for every generator g of E. ⊓ ⊔

The Jordan type of a direct sum of k[t]/t^{p}-modules is the sum of the Jordan types. The stable Jordan type of the Heller shift of the k[t]/t^{p}-module ∑_{i=1}^{p−1} a_{i}[i] equals ∑_{i=1}^{p−1} a_{p−i}[i]. The Jordan type of a tensor product is given by the following proposition.

Proposition 4.2. (cf. [6, 10.2]) Let [i] be an indecomposable k[t]/t^{p}-module of dimension i for 1 ≤ i ≤ p. Then if j ≥ i, we have that

[i]⊗[j] = \{ \begin{matrix}[j − i + 1] + [j − i + 3] + … + [j + i − 3] + [j + i − 1] & if j + i ≤ p \\ [j − i + 1] + … + [2p − 1 − i − j] + (j + i − p)[p] & if j + i > p. \end{matrix}

The following proposition strongly restricts the possible images of the stable Jordan type function \overline{JType} : K_{0}(C(kG)) → ℤ^{p−1} of Proposition 3.5. We say that a subset S ⊂ ℤ^{p−1} is closed under Heller shifts if {a_{1}, …, a_{p−1}} is in S whenever σ = {a_{p−1}, … a_{1}} ∈ S. Similarly, we say that S ⊂ ℤ^{p−1} is closed under direct sums (respectively, tensor products) if σ + τ ∈ S (resp. σ ⊗ τ ∈ S) whenever σ, τ ∈ S. Here, σ ⊗ τ is defined by the formula of Proposition 4.2 if both σ, τ have a single non-zero entry and is defined more generally by imposing biadditivity.

Proposition 4.3. Let S ⊂ ℤ^{p−1} be a set of stable Jordan types. Moreover,

1. If S has the form \overline{JType}(C(kG)) for some finite group scheme G, then S is closed under Heller shifts, direct sums, and tensor products.

2. If S = {m[1] + n[p − 1]; m, n ∈ ℤ}, then S is closed under Heller shifts, direct sums, and tensor products.

3. If [1], [2] ∈ S and S is closed under Heller shifts, direct sums, and tensor products, then S = ℤ^{p−1}.

4. Similarly, if [1], [3] ∈ S with p > 2 and S is closed under Heller shifts, direct sums, and tensor products, then S = ℤ^{p−1}.

5. On the other hand, the assumption that [1], [4] ∈ S and S is closed under Heller shifts, direct sums,and tensor products does not imply that S = ℤ^{p−1} for p = 11.

Proof. Statement (1) is a consequence of [6, 1.8].

Statement (2) easily follows from the fact that [p − 1] ⊗ [p − 1] = [1] + (p − 2)[p].

To prove (3), we observe that [i] ⊗ [2] = [i + 1] + [i − 1] for i, 2 ≤ i < p. Thus, [1], [2] ⊗ [2] ∈ S implies that [3] ∈ S. Proceeding by induction, we see that [i], [i − 1], [i] ⊗ [2] ∈ S implies that [i + 1] ∈ S.

For (4), we may assume that p > 5. Then we obtain [3]⊗[3] = [1]+[3]+[5] which implies that [5] ∈ S; [5]⊗[3] = [3]+[5]+[7], so that [7] ∈ S. Continuing, we conclude that [2i − 1] ∈ S, 1 ≤ [2i − 1] ≤ p − 2. Applying Heller shifts to [2i − 1], we conclude the stable type [p − 2i + 1] ∈ S for 1 ≤ [2i − 1] ≤ p − 2, so that [j] ∈ S for all j, 1 ≤ j ≤ [p − 1].

Finally, let p = 11 so that [4] ⊗ [4] = [1] + [3] + [5] + [7] and p − 4 = 7. Moreover, [7] ⊗ [4] = [7] + [9] + [11]. We conclude that S contains the span of {[1], [4], [7], [10], [3] + [5], [8] + [6], [7] + [9]}. On the other hand, further tensor products with these classes never yield a Jordan type ∑ a_{i}[i] with a_{3} ≠ a_{5}, a_{6} ≠ a_{8} or a_{7} ≠ a_{9}. ⊓ ⊔

The applicability of Proposition 4.3 to the question of realizability of stable Jordan types is reflected in the following proposition and its corollary.

Proposition 4.4. Suppose that G is a finite group which has a normal abelian Sylow p-subgroup. There exists a kG-module with constant stable Jordan type [2] + n[1] for some n. Moreover, the stable Jordan type function

\xymatrix{\overline{JType} : K_{0}(C(kG)) \ar[r] & ℤ^{p−1}}

is surjective.

Proof. Let E ⊆ G be the subgroup consisting of all elements of order dividing p. This is the unique maximal elementary abelian subgroup of G. Let M = k_{E}^{↑G} be the induced module from the trivial module on E. Notice that because E is normal in G, E acts trivially on M and hence M is a module of constant Jordan type s[1] where s is the index of E in G.

For any t > 0 we have that

Ext_{kG}^{t}(k, M) ≅ H^{t}(G, k_{E}^{↑G}) ≅ H^{t}(E, k).

by the Eckmann-Shapiro Lemma. In particular, Ext_{kG}^{1}(k, M) ≅ H^{1}(E, k) has a k-basis consisting of elements γ_{1}, …, γ_{r}, where r is the rank of E. The elements have the property that for any π-point α_{K} : K[t]/(t^{p}) → KG the restriction of some α_{K}^{∗}(γ_{i}) is not zero for some i. It follows that the tuple ζ = (γ_{1}, …, γ_{r}) is an element of Ext_{kG}^{1}((k_{E}^{↑G})^{r}, k) which does not vanish when restricted along any π-point. Then ζ represents a sequence

\xymatrix{ζ :  0 \ar[r] & (k_{E}^{↑G})^{r} \ar[r] & B \ar[r] & k \ar[r] & 0}

which is not split on restriction along any π-point. As the first term in the sequence has constant Jordan type rs[1], the middle term of the sequence must have constant Jordan type [2] + (rs − 1)[1].

The surjectivity of \overline{JType} now follows from Proposition 4.3(2). ⊓ ⊔

Corollary 4.5. Let G be a finite unipotent abelian group scheme. Then the stable Jordan type function

\xymatrix{\overline{JType} : K_{0}(C(kG)) \ar[r] & ℤ^{p−1}}

is surjective.

Proof. Because G is a unipotent abelian group scheme, its group algebra kG is isomorphic as an algebra (though not as a Hopf algebra) to the group algebra of a finite abelian p-group. Hence, the previous proposition applies, since kG has a module of constant Jordan type [2]+n[1] as constructed in the proof. ⊓ ⊔

We next briefly consider the question of realizability of stable Jordan types by indecomposable kG-modules by utilizing the Auslander-Reiten theory of almost split sequences (cf. [1]). In the following proposition, τ : stmod(kG) → stmod(kG) denotes the Auslander-Reiten translation. This is a functor on the stable category and if G is a finite group or if kG is a symmetric algebra, then τ(M) = Ω^{2}(M) for any finite dimensional kG-module M.

Proposition 4.6. Let Θ be a connected component of the Auslander-Reiten quiver which has tree class A_{∞}. Let K̂ be the subgroup of K_{0}(C(kG)) generated by the classes of the modules in Θ. Let X_{0} denote a kG-module in Θ that becomes an initial node of the tree of Θ once projectives are deleted. Then, K̂ is generated by the classes {[τ^{n}(X_{0})] |n ∈ ℤ} and, if Θ contains a projective module P, by the class [P] of that projective module.

Proof. It suffices to show that the class [M] of any indecomposable module M in Θ can be written as a linear combination of the classes of elements on the bottom row of the stable part of Θ (see diagram that follows). This is obvious if M lies on the bottom row of Θ. So assume that M is not on the bottom row. Moreover, to prove this for M it suffices to do so for τ^{n}(M) for some n ∈ ℤ. This is because the functor τ is additive.

The stable part of the component Θ has the form

[figure: …; …; X_{2}; τ^{−1}(X_{2}); …; X_{1}; τ^{−1}(X_{1}); …; X_{0}; τ^{−1}(X_{0}); τ^{−2}(X_{0})]

Without loss of generality, we can assume that M = X_{n+1} for some n. In the case that n = 0, the almost split sequence for M has the form

\xymatrix{0 \ar[r] & X_{0} \ar[r] & X_{1} ⊕ ϵ \ar[r] & τ^{−1}(X_{0}) \ar[r] & 0}

where ϵ is either the zero module or the projective module P. This is an admissible sequence by [6]. Hence we have that [M] = [X_{1}] = [X_{0}] + [τ^{−1}(X_{0})] − [ϵ] in K_{0}(C(kG)).

If n > 0, then there is an almost split sequence having the form

\xymatrix{0 \ar[r] & X_{n} \ar[r] & X_{n+1} ⊕ τ^{−1}(X_{n−1}) \ar[r] & τ^{−1}(X_{n}) \ar[r] & 0.}

Again this sequence is admissible, and we have that [X_{n}] + [τ^{−1}(X_{n})] = [X_{n+1}] + [τ^{−1}(X_{n−1})]. The proposition now follows by induction. ⊓ ⊔

As an example of the applicability of the following corollary, recall that a finite p-group G has a single block and this block has wild representation type provided Π(G) has dimension at least 1 and G is not a dihedral, quaternion or semi-dihedral 2-group. The proof of this corollary is essentially a verbatim repetition of the proof of [6, 8.8] granted Proposition 4.6.

Corollary 4.7. Let G be a finite group and assume that k is algebraically closed. Assume that there exists an indecomposable kG-module of constant Jordan type with stable Jordan type \underline{a} = ∑_{i=1}^{p−1} a_{i}[i] which lies in a block of wild representation type. Then there exists an indecomposable kG-module of constant Jordan type with stable Jordan type n\underline{a} for any n > 0.

Proof. By Erdmann’s Theorem [7], the connected component of the Auslander-Reiten quiver of such a module has tree class A_{∞}. In the notation of the proof of Proposition 4.6, if X_{0} has stable constant Jordan type \underline{a}, then because τ commutes with restrictions along π-points, so also does τ^{n}(X_{0}). Thus, by the relations developed in the proof, X_{1} has stable constant Jordan type 2\underline{a}, and inductively, X_{n} has stable constant Jordan type (n + 1)\underline{a}. ⊓ ⊔

For an arbitrary finite group scheme, Corollary 4.7 would appear to remain valid in view of work of R. Farnsteiner [8], [9].

We next give a cohomological criterion for the realizability of all stable Jordan types by virtual modules of constant Jordan type.

Theorem 4.8. Let G be a finite group scheme with the property that there exist odd dimensional classes ζ_{1} ∈ H^{2d_{1}−1}(G, k), …, ζ_{m} ∈ H^{2d_{m}−1}(G, k) such that ∩_{i=1}^{m}V (β(ζ_{i})) = 0, where β : H^{odd}(G, k) → H^{ev}(G, k) is the Bockstein cohomological operation of degree +1. Let

L_{ζ_{1},...,ζ_{m}} = ker\{ \xymatrix{∑ ζ̃_{i} : ∑_{i=1}^{m} Ω^{2d_{i}−1}(k) \ar[r] & k} \}.

If p > 2, then L_{ζ_{1},...,ζ_{m}} is a module of constant Jordan type whose stable Jordan type has the form (m − 1)[p − 1] + 1[p − 2]. Consequently, for such G,

\xymatrix{\overline{JType} : K_{0}(C(kG)) \ar[r] & ℤ^{p−1}}

is surjective.

Proof. Because ∩_{i=1}^{m}V (β(ζ_{i})) = 0, there does not exist an equivalence class of π-points [α_{K}] ∈ Π(G) with α_{K}^{∗}(β(ζ_{i,K})) = 0 ∈ H^{2d_{i}}(K[t]/t^{p}, K) for all i, 1 ≤ i ≤ m. Hence, for each α_{K} of G there exists some i such that α_{K}^{∗}(ζ_{i,K}) ≠ 0 ∈ H^{2d_{i}−1}(K[t]/t^{p}, K). Hence, Proposition [6, 6.7] implies that L_{ζ_{1},...,ζ_{m}} is of constant Jordan type with stable Jordan type (m − 1)[p − 1] + 1[p − 2].

The second assertion follows from the first by Proposition 4.3(2). ⊓ ⊔

We proceed to verify (in Proposition 4.11) below that the condition of Theorem 4.8 is satisfied by many finite groups. To do so, we use the following theorem of D. Quillen which asserts that H^{∗}(GL(n, 𝔽_{ℓ}), k) maps isomorphically onto the invariants of the cohomology of a direct product of cyclic groups. Specifically, we have the following.

Theorem 4.9. (D. Quillen [16, §8]) Assume p > 2, let 𝔽_{ℓ} be a finite field with (ℓ, p) = 1 and let r be the least integer such that p divides the order of the units 𝔽_{ℓ^{r}}^{∗} of 𝔽_{ℓ^{r}}. Let π = Gal(𝔽_{ℓ^{r}}/𝔽_{ℓ}). Then the restriction map

H^{∗}(GL(n, 𝔽_{ℓ}), k) → H^{∗}((𝔽_{ℓ^{r}}^{∗})^{×m}, k)^{π^{×m}⋊Σ_{m}}

is an isomorphism, where n = mr+e, 0 ≤ e < r. In particular, H^{∗}(GL(n, 𝔽_{ℓ}), k)_{red} is a polynomial algebra on generators x_{i} ∈ H^{2ri}(GL(n, 𝔽_{ℓ}), k)), 1 ≤ i ≤ m.

The following is an easy consequence of Quillen’s theorem.

Proposition 4.10. We retain the hypotheses and notation of Theorem 4.9. Assume furthermore that p^{2} does not divide the order of the units of 𝔽_{ℓ^{r}}. Then the cohomology H^{∗}((𝔽_{ℓ^{r}}^{∗})^{×m}, k)^{π^{×m}} is an exterior algebra on generators t_{1}, …, t_{m} in degrees 2r − 1 tensor a symmetric algebra on the Bocksteins of the t_{i}’s, u_{1} = β(t_{1}), …, u_{m} = β(t_{m}) in degrees 2r.

Set ω_{i}^{s} to be the class

∑_{j_{1}<...<j_{i}} t_{j_{s}} · u_{j_{1}} ⋯ u_{j_{s−1}} · u_{j_{s+1}} ⋯ u_{j_{i}} ∈ H^{2ri−1}((𝔽_{ℓ^{r}}^{∗})^{×m}, k)^{π^{×m}}

for any s, 1 ≤ s ≤ i, and set ζ_{i} to be the class

∑_{1≤s≤i} ω_{i}^{s} ∈ (H^{2ri−1}((𝔽_{ℓ^{r}}^{∗})^{×m}, k)^{π^{×m}})^{Σ_{m}} ≅ H^{2ri}(GL(n, 𝔽_{ℓ}), k).

Then the Bockstein applied to ζ_{i} equals i times x_{i},

β(ζ_{i}) = i · x_{i} ∈ H^{2ri}(GL(n, 𝔽_{ℓ}), k).

Proof. The Bockstein of each ω_{i}^{s} equals ∑_{j_{1}<...<j_{i}} u_{j_{1}} ⋯ u_{j_{r}} ∈ H^{2ri}(T (n, 𝔽_{ℓ}), k), for any s, where T (n, 𝔽_{ℓ}) is the torus which we can take to consist of the diagonal matrices. This element is equal to the restriction of x_{i} ∈ H^{2ri}(GL(n, 𝔽_{ℓ}), k). Thus, i · x_{i} and β(ζ_{i}) ∈ H^{2ri}(GL(n, 𝔽_{ℓ}), k) both restrict to i · ∑_{j_{1}<...j_{i}} u_{j_{1}} ⋯ u_{j_{i}} ∈ H^{2ri}(T (n, 𝔽_{ℓ}), k). Since the restriction map H^{2ri}(GL(n, 𝔽_{ℓ}), k) → H^{2ri}(T (n, 𝔽_{ℓ}), k) is injective, the proposition follows.

⊓ ⊔

Proposition 4.11. Assume p > 2, let 𝔽_{ℓ} be a finite field with (ℓ, p) = 1 and let r be the least integer such that p divides the order of the units 𝔽_{ℓ^{r}}^{∗} of 𝔽_{ℓ^{r}}. Let π = Gal(𝔽_{ℓ^{r}}/𝔽_{ℓ}). Assume further that p^{2} does not divide the order of the units 𝔽_{ℓ^{r}}^{∗} of 𝔽_{ℓ^{r}}. If n = mr + e, 0 ≤ e < r and if m < p, then any finite group G admitting an embedding G ⊂ GL(n, 𝔽_{ℓ}) satisfies the hypothesis of Theorem 4.8.

In particular, for any such finite group G,

\xymatrix{\overline{JType} : K_{0}(C(kG)) \ar[r] & ℤ^{p−1}}

is surjective.

Proof. By Theorem 4.8, we may construct a kGL(n, 𝔽_{ℓ})-module L_{ζ_{1},...,ζ_{m}} (see 4.8) of constant Jordan type with stable Jordan type m[p − 1] + 1[p − 2]. The restriction of L_{ζ_{1},...,ζ_{m}} to kG remains a module of constant Jordan type with the same stable Jordan type. Thus, the corollary follows by applying Proposition 4.3(2). ⊓ ⊔

5 Stratification by C(kG)

The modules of constant Jordan type do not form a triangulated subcategory of the stable module category stmod(kG). On the other hand, C(kG) enjoys many good properties: as recalled earlier (from [6]), C(kG) is closed under direct sums, tensor products, k-linear duals, retracts, and Heller shifts. In stmod(kG), M ↦ Ω^{−1}(M) is the translation functor on the triangulated category stmod(kG). Hence, C(kG) is a full subcategory of stmod(kG) closed under translations.

In this section, we briefly consider “thickenings” of C(kG) ⊂ stmod(kG), adopting terminology of [14]. The question here is what modules can be assembled by successive extensions of modules of constant Jordan type. This is motivated partly by the observation that if a kG-module M is an extension of two modules of constant Jordan type then there is lower bound on the Jordan type (and perhaps even a minimal Jordan type) that can occur at any π-point, namely, the sum of Jordan types of the two extending modules. A well known class of examples are the modules L_{ζ} for ζ ∈ H^{2m}(G, k) for some m > 0. Such a module is defined by a sequence

\xymatrix{0 \ar[r] & L_{ζ} \ar[r] & Ω^{2m}(k) \ar[r]^{ζ̃} & k \ar[r] & 0}

where the map ζ̃ represents the cohomology class ζ. For any π-point α_{K} of G, α_{K}^{∗}(L_{ζ,K}) has Jordan type s[p] for some s > 0 if α_{K}^{∗}(ζ_{K}) ≠ 0 and Jordan type s[p] + [p − 1] + [1] if α_{K}^{∗}(ζ_{K}) = 0.

Definition 5.1. Set thick^{1}(C) equal to C(kG). For n > 1, set thick^{n}(C) equal to the smallest full subcategory of stmod(kG) which is closed under retracts and which contains all finite dimensional kG-modules M fitting in a distinguished triangle M′ → M → N → Ω^{−1}(M′) with M′ ∈ thick^{n−1}(C) and N a kG-module of constant Jordan type (i.e., M ∈ thick^{1}(C)).

Furthermore, define

Thick(C) ≡ ∪_{n} thick^{n}(C) ⊂ stmod(kG),

the smallest thick subcategory of C(kG) containing C(kG).

Any finite dimensional kG-module has a finite filtration with associated graded module a direct sum of irreducible modules (obtained by successively considering socles of quotient modules). Since the only isomorphism class of irreducible modules for a finite p-group is that of the trivial module and since every irreducible k(SL_{2})_{1}-module has constant Jordan type by [6], we immediately conclude the following proposition.

Proposition 5.2. Let G be a finite group scheme with the property that every irreducible kG-module has constant Jordan type. Then

Thick(C) = stmod(kG).

In particular, if G is either a finite p-group or the first infinitesimal kernel (SL_{2})_{1} of SL_{2}, then Thick(C) = stmod(kG).

The socle filtration employed in the proof of the above filtration might not be a good measure of the “level” of a kG-module as we shall see in Proposition 5.9.

Recall that a block B of kG is said to have defect group a given p-subgroup P ⊂ G if P is the smallest subgroup of G such that every module in B is a direct summand of a kG-module obtained as the induced module of a kP - module. As shown in [11, 4.12], this implies that any kG-module in B has support in i_{∗}(Π(P)) ⊂ Π(G), where i : Π(P) → Π(G) is induced by the inclusion P ⊂ G.

Proposition 5.3. Let G be a finite group and let B be a block of kG with defect group P ≠ {1}. If i_{∗} : Π(P) → Π(G) is not surjective, then no non-projective module in B has constant Jordan type.

Moreover, if B is such a block, then no non-projective module in B is in Thick(C).

Proof. If M is a kG-module in B, then the support variety Π(G)_{M} is contained in i_{∗}(Π(P)) ⊂ Π(G), a proper subvariety of Π(G). Consequently, M does not have constant Jordan type. Thus the only modules of constant Jordan type in B are the projective modules. If a B-module were in Thick(C), then it would have to be an extension of projective B-modules and hence projective. ⊓ ⊔

Remark 5.4. The hypothesis of Proposition 5.3 is satisfied in numerous examples. For example the alternating group A_{9} on nine letters and the first Janko group J_{1} have such blocks in characteristic 2. The Mathieu group M_{12} has such a block in characteristic 3.

We conclude this investigation of modules of constant Jordan type by returning to the special case G = E an elementary abelian p-group. We utilize a class of modules of constant Jordan type discovered by Andrei Suslin, those with the “constant image property.” The widespread prevalence of such modules reveals that C(kE) must necessarily be large and complicated.

We consider an elementary abelian p-group E of rank r, and we consistently use the notation kE = k[x_{1}, …, x_{r}]/(x_{1}^{p}, …, x_{r}^{p}). We denote by I_{E} the augmentation ideal of kE, and by \overline{k} some choice of algebraic closure of k.

The following definition is similar to those definitions found in [11, §1]

Definition 5.5. A finite dimensional kE-module M is said to have the constant image property if for any 0 ≠ w_{α} = α_{1}x_{1} + ⋯ + α_{r}x_{r} ∈ \overline{k}E,

w_{α}M\overline{k} = Rad M\overline{k}.

Example 5.6. The module

M = Rad^{(r−1)(p−1)}(kE) ≡ I_{E}^{(r−1)(p−1)}

has the constant image property.

To verify this, first observe that M\overline{k} = Rad^{(r−1)(p−1)}(\overline{k}E), so that we may assume that k is algebraically closed. Every non-zero monomial of degree (r − 1)(p − 1) + 1 in k[x_{1}, …, x_{r}] has the form x_{1}^{a_{1}} ⋯ x_{r}^{a_{r}} for 0 ≤ a_{i} < p for all i. But as a_{1} + ⋯ + a_{r} = (r − 1)(p − 1) + 1, it must be that a_{1} > 0; i.e., I_{E}^{(r−1)(p−1)+1} = x_{1}I_{E}^{(r−1)(p−1)}. Hence,

Rad M ≡ I_{E}M = Rad^{(r−1)(p−1)+1}(kE) = x_{1}M

and we have verified the condition of Definition 5.5 for w_{α} = x_{1}. This is sufficient, for given any w_{α} ≠ 0 there is an automorphism of kE which takes w_{α} to x_{1} and takes I_{E}^{n} to I_{E}^{n} for all n.

Remark 5.7. As the reader can easily verify, the direct sum of modules with the constant image property and any quotient of a module with constant image property again have the constant image property. Thus, starting with Example 5.6, we obtain many additional modules with constant image property. Moreover, a simple induction argument implies that if the kEmodule M has the constant image property, then for any n > 0 and any 0 ≠ w_{α} = α_{1}x_{1} + ⋯ + α_{r}x_{r} ∈ \overline{k}E,

w_{α}^{n}M\overline{k} = Rad^{n} M\overline{k}.

Proposition 5.8. Suppose that M is a kE-module with the constant image property. Then M has constant Jordan type.

Proof. For any α = (α_{1}, …, α_{r}) ≠ \underline{0} ∈ \overline{k}^{r}, the Jordan type of w_{α} on M\overline{k} is equivalent to the data consisting of the sequence of dimensions

Dim M\overline{k}, Dim w_{α}M\overline{k}, Dim w_{α}^{2}M\overline{k}, …, Dim w_{α}^{p−1}M\overline{k}.

Thus, Remark 5.7 implies that each of the w_{α} have the same Jordan type on M\overline{k}. According to the original definition of maximal Jordan type [11, 1.4], some w_{α} has maximal Jordan type for M. Thus, the complement of the nonmaximal variety of M contains all \overline{k}-rational points of 𝔸^{r}, and hence the non-maximal variety of M is empty. This is equivalent to the assertion that M has constant Jordan type. ⊓ ⊔

Proposition 5.9. Suppose that E is an elementary abelian p-group of rank r. Then

thick^{2r}(C) = stmod(kE).

Proof. We first assume that Rad^{p}(M) = I_{E}^{p}M = 0 and proceed to verify that M is in thick^{2}(C). Namely, let Q = kE^{t} be the injective hull of M. Then we have an injection φ : M ⟶ kE^{t}. Because I_{E}^{p}M = 0, we must have that φ(M) ⊆ I_{E}^{(r−1)(p−1)}kE^{t} = (I_{E}^{(r−1)(p−1)})^{t}. This yields the exact sequence

\xymatrix{0 \ar[r] & M \ar[r] & (I_{E}^{(r−1)(p−1)})^{t} \ar[r] & N \ar[r] & 0}

where N is the quotient. Hence in stmod(kG) there is a distinguished triangle

\xymatrix{M \ar[r] & (I_{E}^{(r−1)(p−1)})^{t} \ar[r] & N \ar[r] & Ω^{−1}(M) \ar[r] & …}

Consequently, applying Proposition 5.8, we conclude that M ∈ thick^{2}(C). In general, the modules

M/Rad^{p}(M), Rad^{p}(M)/ Rad^{2p}(M), …, Rad^{(r−1)p}(M)/ Rad^{rp}(M)

each belong to thick^{2}(C), because they are all annihilated by I_{E}^{p}. Because I_{E}^{rp}(M) = 0, this readily implies that M is in thick^{2r}(C) as asserted. ⊓ ⊔

References

1. M. Auslander, I. Reiten, and S. Smalo, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36, 1995.

2. V. Basev, Representations of the group ℤ_{2} × ℤ_{2} in a field of characteristic 2, (Russian) Dokl. Akad. Nauk. USSR 141 (1961), 1015- 1018.

3. D. Benson, Representations and Cohomology, vols. I and II, Cambridge University Press, 1991.

4. D. Benson, Resolutions over symmetric algebras with radical cube zero, (preprint).

5. D. J. Benson and J. F. Carlson, Diagrammatic methods for group representations and cohomology, Comm. in Algebra, 15(1987), 53-121.

6. J. Carlson, E. Friedlander, and J. Pevtsova, Modules of constant Jordan type, To appear in J. fur die reine und ang. Math. (2007).

7. K. Erdmann, On Auslander-Reiten components of group algebras, J. Pure Appl. Algebra 104 (1995), 149-160.

8. R. Farnsteiner, On the Auslander-Reiten quiver of an infinitesimal group, Nagoya Math. J. 160 (2000), 103-121.

9. R. Farnsteiner, Support spaces and Auslander-Reiten components, to appear in Composit. Math.

10. E. Friedlander and J. Pevtsova, Π-supports for modules for finite group schemes, To appear in the Duke Math J.

11. E. Friedlander, J. Pevtsova, and A. Suslin, Maximal and generic Jordan types, To appear in Inventiones Math.

12. E. Friedlander and B. Parshall, Modular representation theory of Lie algebras, American J. Math 110 (1988), 1055-1094.

13. A. Heller, I Reiner, Indecomposable representations, Illinois J. Math 5 (1961), 314-323.

14. S. Iyengar, Levels in triangulated categories, Lecture given at Leeds.

15. J. C. Jantzen, Representations of Algebraic Groups, Academic Press (1987).

16. D. Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Annals of Math 96 (1972), 552-586.

17. D. Quillen, Higher K-theory, I; in Algebraic K-theory I, SLN 341 (1973), 77 –139.

18. C. M. Ringel, The indecomposable representations of the dihedral 2-groups, Math. Ann. 214 (1975), 19–34.