MODULES OF CONSTANT JORDAN TYPE

JON F. CARLSON^{∗}, ERIC M. FRIEDLANDER^{∗} AND JULIA PEVTSOVA^{∗}

Abstract. We introduce the class of modules of constant Jordan type for a finite group scheme G over a field k of characteristic p > 0. This class is closed under taking direct sums, tensor products, duals, Heller shifts and direct summands, and includes endotrivial modules. It contains all modules in an Auslander-Reiten component which has at least one module in the class. Highly non-trivial examples are constructed using cohomological techniques. We offer conjectures suggesting that there are strong conditions on a partition to be the Jordan type associated to a module of constant Jordan type.

Contents

0. Introduction 1 1. Constant Jordan Type 3 2. Examples of modules of constant Jordan type 7 3. Specialization, Γ(G)_{M}, and constant Jordan type 9 4. Behavior with respect to tensor products 13 5. Endotrivial modules 20 6. Constructing modules of constant Jordan type 24 7. Constraint on ranks 34 8. Auslander Reiten Components 36 9. Questions and Conjectures 42 10. APPENDIX: Decomposition of tensor products of k[t]/t^{p}-modules 45 References 46

0. Introduction

In [18] and [20], the second and third authors have introduced a seemingly naive approach to the study of representations of finite groups and related structures on vector spaces over a field k of characteristic p > 0. The basic idea is to restrict representations to certain subalgebras (“π-points”) isomorphic to the group algebra of ℤ/pℤ, for we completely understand the representation theory of the algebra kℤ/pℤ in terms of partitions (or “Jordan types”). The simplicity of this approach enables the authors to consider representation theory in a very general context (of a finite group scheme G over an arbitrary field of characteristic p > 0) and prove both global results about the stable module category and explicit results for specific examples. The naivety of the approach is somewhat misleading for underlying many theorems are somewhat difficult cohomological results, especially results giving finite generation and detection modulo nilpotence on subalgebras of special form.

2000 Mathematics Subject Classification. 16G10, 20C20, 20G10.

∗ partially supported by the NSF .

In a recent paper [21], the second and third authors in collaboration with Andrei Suslin have adopted this naive point of view to formulate and investigate new invariants for such representations. The authors introduce “maximal” and “generic” Jordan types for a given representation whose existence even in the very special case of the finite group ℤ/pℤ × ℤ/pℤ is highly non-trivial.

Indeed, this special example G = ℤ/pℤ × ℤ/pℤ (for p > 2) is challenging from a representation-theoretic point of view for its group algebra has wild representation type. With such “wildness” in mind, it is natural to investigate classes of representations of G with special properties. That is the purpose of this present paper, in which we investigate modules of constant Jordan type. Although the formulation of this concept requires the approach of “π-points” and our study utilizes many of the techniques of the papers mentioned above, the resulting class of modules appears to be a most natural one to study for those considering the modular representation theory of finite groups, p-restricted Lie algebras, and other finite group schemes. We remark here on two aspects of this class of modules of constant Jordan type: it includes the much-studied class of endotrivial modules and also includes many other modules even in the special case of an elementary abelian p-group; the classification of such modules of constant Jordan type appears to be very difficult, sufficiently difficult that even in the special case G = ℤ/pℤ × ℤ/pℤ we can only speculate on what Jordan types are realized.

In Definition 1.5, we introduce the concept of a kG-module M of constant Jordan type, a finite dimensional module with the property that α_{K}^{∗}(M_{K}) has Jordan type independent of the π-point \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} with K/k an arbitrary field extension. As verified in Theorem 5.6, a kG-module is an endotrivial module if and only if it has constant Jordan type of a very special form. For certain explicit finite group schemes, various examples of modules of constant Jordan type can be constructed directly as we show in §2. Much of our effort in the first half of this paper is dedicated to showing that the class of modules of constant Jordan type is closed under various natural operations: Heller shifts (Proposition 1.8), direct sums (Proposition 1.8), direct summands (Theorem 3.7), linear duals (Proposition 5.2), and tensor products (Corollary 4.3).

To establish these results, we continue the study initiated in [21] of the condition that a π-point α_{K} be maximal for a given kG-module M. This analysis should be of interest for the study of general kG-modules. In §3, we formulate the natural relationship of strict specialization of π-points (closely related to the relationship of specialization of equivalence classes of π-points considered in [20]). In §4, we investigate the surprisingly subtle behavior of the condition of maximality of πpoints with respect to the tensor product of two given kG-modules. The relevance of maximality of π-points for a kG-module M is emphasized by Proposition 3.6 which asserts that the kG-module M has constant Jordan type if and only if the non-maximal support variety of M, Γ(G)_{M}, is empty.

In the second half of this paper, we give several methods of constructing modules of constant Jordan type. One is provided in Proposition 6.1 and another in Theorem 6.6 (as well as Proposition 6.7). Much preliminary effort is required for us to establish in Theorem 6.13 that our second method provides examples which can not be realized by the first. Indeed, the example provided by this theorem shows how subtle is the behavior of the class of modules of constant Jordan type with respect to extensions. A third method of construction using the Auslander-Reiten theory of almost split sequences is detailed in §8: Theorem 8.7 establishes that any module in the same Auslander-Reiten component as a module of constant Jordan type is also of constant Jordan type, whereas Theorem 8.8 constructs kG-modules of constant Jordan type n[1] + (proj) provided that G satisfies a mild cohomological property.

As one indication of the combinatorics involved in the existence of modules of constant Jordan type, we show in Theorem 7.1 that our techniques give a new proof of a special case of Macaulay’s Generalized Principal Ideal Theorem.

We have made little progress in classifying those partitions which are realizable as the Jordan type of modules of constant Jordan type. For example, for a rank 2 elementary abelian p-group E with p > 3, we conjecture but can not prove that no partition of type n[p] + 1[2] is the Jordan type of kE-module of constant Jordan type. In §9, we mention numerous questions and conjectures on the constraints of a Jordan type associated to a module of constant Jordan type.

Throughout this paper, k will denote an arbitrary field of finite characteristic and p > 0 will denote the characteristic of k. Without explicit mention to the contrary, kG-modules are assumed to be finitely generated. We let M_{n}(k) denote the algebra of n × n matrices over k.

We thank Valery Alexeev, David Eisenbud, and Sasha Premet for help with Theorem 7.1. We also thank Karin Erdmann for drawing our attention to possible connections with the Auslander-Reiten theory. We are particularly grateful to the referee for many constructive comments and suggestions.

1. Constant Jordan Type

In this first section, we introduce modules of constant Jordan type and investigate some of the basic properties of this class of modules.

Recall that a finite group scheme G (over k) is a group scheme over k whose coordinate algebra k[G] is finite dimensional over k. We denote the linear dual of k[G] by kG and call this the group algebra of G. A (rational) G-module is a comodule for k[G] or equivalently a kG-module. If K/k is a field extension, then we denote by G_{K} the base change of the k-group scheme G to the K-group scheme G_{K} = G ×_{Spec k} Spec K. We observe that the group algebra KG_{K} of G_{K} equals KG = K ⊗_{k} kG.

We remind the reader that the isomorphism class of a finite dimensional k[t]/t^{p}- module M of dimension n (over k) is given by a partition of n into blocks of size ≤ p. Equivalently, if we let \xymatrix{ρ_{M} : k[t]/t^{p} \ar[r] & M_{n}(k)} be the representation associated to M, then the isomorphism type of M is specified by the conjugacy class of the element ρ_{M}(t) ∈ M_{n}(k) whose p-th power is 0. We shall often denote the isomorphism type of M by a_{p}[p] + ⋯ + a_{1}[1], where a_{i} denotes the number of blocks of size i in the partition of n associated to M.

We call the isomorphism type of a finite dimensional k[t]/t^{p}-module M the Jordan type of M. For any finite dimensional k[t]/t^{p}-module M, the stable Jordan type of M is the “stable equivalence” class of Jordan types, where two Jordan types a_{p}[p] + ⋯ + a_{1}[1] and b_{p}[p] + ⋯ + b_{1}[1] are stably equivalent if a_{i} = b_{i}, for all i < p.

We may view a Jordan type a_{p}[p] + ⋯ + a_{1}[1] as a partition of n = ∑_{i=1}^{p} ia_{i}. We shall compare Jordan types using the dominance order. Let \underline{n} = [n_{1} ≤ n_{2} ≤ … ≤ n_{k}], \underline{m} = [m_{1} ≤ m_{2} ≤ … ≤ m_{k}] be two partitions of N. Then \underline{n} dominates \underline{m}, written \underline{n} ≥ \underline{m}, iff

∑_{i=j}^{k} n_{i} ≥ ∑_{i=j}^{k} m_{i}.

for all j, 1 ≤ j ≤ k. For Jordan types \underline{a} = a_{p}[p]+⋯+a_{1}[1] and \underline{b} = b_{p}[p]+⋯+b_{1}[1], we have \underline{a} ≥ \underline{b} in the dominance order if and only if the following equivalent condition is satisfied:

∑_{i=j+1}^{p} a_{i}(i − j) ≥ ∑_{i=j+1}^{p} b_{i}(i − j)

for any j, 1 ≤ j ≤ p. Note that ∑_{i=j+1}^{p} a_{i}(i − j) is simply the rank of the j^{th} power of the matrix with Jordan type \underline{a}.

Remark 1.1. Let M, N be k[t]/t^{p}-modules of dimension n given by

\xymatrix{ρ_{M}, ρ_{N} : k[t]/t^{p} \ar[r] & M_{n}(k).}

Then the Jordan type \underline{a} of M is greater or equal to (respectively, greater than) the Jordan type of \underline{b} of N if and only if for every j, 1 ≤ j < p, the rank of ρ_{M}^{j} is greater or equal to the the rank of ρ_{N}^{j} (resp., and strictly greater for some j).

Definition 1.2. A π-point for a finite group scheme G is a left flat map of K-algebras \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} , for some field extension K/k, which factors through the group algebra KC_{K} ⊂ KG_{K} of some unipotent abelian subgroup scheme C_{K} ⊂ G_{K}. If M is a finite dimensional kG-module, the Jordan type of the π-point α_{K} on M is the isomorphism class of the K[t]/t^{p}-module α_{K}^{∗}(M_{K}) (where M_{K} = K ⊗_{k} M). We emphasize here that α_{K}^{∗}(M_{K}) denotes the restriction of M_{K} to a K[t]/t^{p}-module along the map α_{K}. We say that the Jordan type of α_{K}^{∗}(M_{K}) is the Jordan type of α_{K} on M.

Definition 1.3. Let \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} , \xymatrix{β_{L} : L[t]/t^{p} \ar[r] & LG} be π-points of G. Then α_{K} is said to be a specialization of β_{L} (written β_{L} ↓ α_{K}) if for every finite dimensional kG-module M the K[t]/t^{p}-module α_{K}^{∗}(M_{K}) is projective whenever the L[t]/t^{p}-module β_{L}^{∗}(M_{L}) is projective. We say that α_{K}, β_{L} are equivalent and write α_{K} ∼ β_{L} provided that α_{K} ↓ β_{L} and β_{L} ↓ α_{K}.

The following theorem summarizes the close relationship between the set of equivalence classes of π-points of G and the cohomology H^{•}(G, k). Here, H^{•}(G, k) = H^{∗}(G, k), the cohomology algebra of G provided that p = 2; for p > 2, H^{•}(G, k) ⊂ H^{∗}(G, k) denotes the commutative subalgebra of even dimensional classes.

Theorem 1.4. ([20, 3.6]) The set of equivalence classes of π-points of a finite group scheme G, written Π(G), admits a scheme structure determined by the stable module category, stmod(kG). With this structure, Π(G) is isomorphic to the scheme P roj H^{•}(G, k).

In particular, the closed subsets of Π(G) are of the form Π(G)_{M} where M is a finite dimensional kG-module and Π(G)_{M} is the subset of those equivalence classes of π-points \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} such that α_{K}^{∗}(M_{K}) is not projective.

We now introduce modules of constant Jordan type, whose study is the primary object of interest in this paper.

Definition 1.5. The finite dimensional kG-module M is said to be of constant Jordan type if the Jordan type of α_{K}^{∗}(M_{K}) is independent of the choice of π-point

\xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} .

Remark 1.6. The Jordan type of α_{K}^{∗}(M_{K}) for a finite dimensional kG-module M at a π-point \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} typically depends not only upon the equivalence class [α_{K}] ∈ Π(G) but also upon the representative of this equivalence class. However, there are some exceptions. The central conclusion of [21] is that in either of the following two situations, the Jordan type of α_{K}^{∗}(M_{K}) does not change if we replace α_{K} by some β_{L} with α_{K} ∼ β_{L}:

(1) If [α_{K}] ∈ Π(G) is a generic point; otherwise said, if α_{K} is a generic π-point.

(2) If for the given finite dimensional kG-module M, there does not exist any π-point β_{L} such that the Jordan type of β_{L}^{∗}(M_{L}) is strictly greater than the Jordan type of α_{K}^{∗}(M_{K}). In this situation we say that α_{K} has maximal Jordan type on M.

We recall [21, 5.1] that the non-maximal support variety, Γ(G)_{M} ⊂ Π(G) associated to a finite dimensional kG-module M is defined to be the (closed) subspace of those points x ∈ Π(G) with the property that for some (and thus any) representative α_{K} of x the Jordan type of α_{K}^{∗}(M_{K}) is not maximal for M, or equivalently, α_{K} does not have maximal Jordan type on M.

Remark 1.6(2) immediately leads us to the following equivalent formulation of the property of constant Jordan type.

Proposition 1.7. A finite dimensional kG-module is of constant Jordan type a_{p}[p]+ ⋯ + a_{1}[1] if and only if for each equivalence class [α_{K}] ∈ Π(G) there exists some representative \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} with the property α_{K}^{∗}(M_{K}) has type a_{p}[p] + ⋯ + a_{1}[1].

Since any π-point \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} has the property that α_{K}^{∗} commutes with direct sums and (modulo projectives) Heller shifts, we conclude the following.

Proposition 1.8. Let G be an arbitrary finite group scheme.

• The trivial kG-module k has constant Jordan type.

• A finite dimensional projective kG-module has constant Jordan type. If kG is not semi-simple, then the Jordan type of a kG-projective module P is equal to \frac{dim_{k} P}{p} [p].

• If Ω^{i}(k) denotes the i-th Heller shift of k for some i ∈ ℤ, then Ω^{i}(k) has constant Jordan type equal to n[p] + 1[1] for some n ≥ 0 if i is even and equal to m[p] + 1[p − 1] for some m ≥ 0 if i is odd.

• If M has constant Jordan type, then Ω^{i}(M) also has constant Jordan type for any i ∈ ℤ

• If M, M′ are kG-modules of constant Jordan type, then M ⊕ M′ also has constant Jordan type.

The preceding proposition will be supplemented in subsequent sections by propositions asserting that other familiar operations on modules of constant Jordan type yield modules of constant Jordan type: taking a direct summand (by Theorem 3.7), taking the tensor product (by Corollary 4.3), and taking Hom_{k}(−, −) (by 5.4).

We make explicit the following elementary functoriality property.

Proposition 1.9. If \xymatrix{f : H \ar[r] & G} is a flat map of finite group schemes and if M is a kG-module of constant Jordan type, then f^{∗}(M) is a kH-module of the same constant Jordan type.

Proof. If \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KH} is a π-point of H, then f ∘ α_{K} is a π-point of G and the Jordan type of α_{K}^{∗}((f^{∗}M))_{K}) equals that of (f ∘ α_{K})^{∗}(M_{K}). □

Remark 1.10. To verify whether or not a given finite dimensional kG-module M has constant Jordan type it suffices to check that the Jordan type of α_{K}^{∗}(M_{K}) does not vary as [α_{K}] ∈ Π(G) ranges over closed points of Π(G). Thus, it suffices to consider α_{K} with K/k finite. In particular, if k is algebraically closed, it suffices to consider k-rational points of Π(G). The collection of these points is denoted by P (G), and was investigated extensively in [18].

Because a kG-module M has constant Jordan type if and only if its base change M_{K} has constant Jordan type as a KG-module for any field extension K/k, one could replace k by its algebraic closure and consider only k-rational points for the algebraically closed field k.

2. Examples of modules of constant Jordan type

We give examples of modules of constant Jordan type in special situations. Perhaps it is worth remarking that these examples are quite different from endotrivial modules considered in §5.

Proposition 2.1. Let E be an elementary abelian p-group and let I ⊂ kE be the augmentation ideal of the group algebra kE. Then, for any n ≥ m, I^{m}/I^{n} is a kE-module of constant Jordan type.

Proof. Let \xymatrix{β : k[t]/t^{p} \ar[r] & kE} ≃ k[t_{1}, …, t_{r}]/(t_{1}^{p}, …, t_{r}^{p}) be the π-point defined by

β(t) = t_{1} and let \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KE} be an arbitrary π-point. Since α_{K} is flat and α_{K}(t) has p-th power 0, α_{K}(t) is a polynomial in t_{1}, …, t_{r} with constant term 0 and non-vanishing linear term ([18, 2.2,2.6]). Consequently, we may choose an automorphism \xymatrix{θ_{α} : KE \ar[r] & KE} which sends t_{1} to α_{K}(t), so that α_{K} = θ_{α} ∘ β. The automorphism θ_{α} necessarily sends any power I^{m} of the augmentation ideal isomorphically onto itself, so that

θ_{α}^{−1} : α_{K}^{∗}(I^{m}/I^{n}) = (θ_{α} ∘ β)^{∗}((I^{m}/I^{n})_{K}) ≃ (β^{∗}(I^{m}/I^{n}))_{K}.

In other words, the Jordan type of α_{K}^{∗}(I^{m}/I^{n}) does not depend upon the choice of π-point α_{K}. □

Example 2.2. As an elementary, specific case of Proposition 2.1, we consider kE/I^{2} where E is an elementary abelian p-group of rank r. This is a kE-module of dimension r + 1 and can be represented explicitly as follows. Give the kE-module structure on k^{r+1} by defining the generators {t_{1}, t_{2}, …, t_{r}} to act by multiplication by {e_{1,2}, e_{1,3}, …, e_{1,r+1}}, pairwise commuting elementary matrices of size (r+1)×(r+1) with p-th power 0. The constant Jordan type of the module kE/I^{2} is 1[2]+(r−1)[1].

Remark 2.3. By Proposition 5.2, the dual (kE/I^{2})^{#} of kE/I^{2} is also a module of constant Jordan type of the same Jordan type as kE/I^{2}. In the special case of r = 2, (kE/I^{2}, (kE/I^{2})^{#}) constitute the example produced years ago by Jens Jantzen of two non-isomorphic modules with the same “local Jordan type.”

Example 2.4. We give a somewhat more interesting example to show how the residue characteristic p plays a role. Consider the elementary abelian p-group E of rank 2, so that kE = k[x, y]/(x^{p}, y^{p}). Define the kE-module W of dimension 13 generated by v_{1}, v_{2}, v_{3}, v_{4} and spanned as a k-vector space by

{v_{1}, v_{2}, v_{3}, v_{4}, x(v_{1}), x(v_{2}), x(v_{3}), x(v_{4}), x^{2}(v_{1}), x^{2}(v_{2}), x^{2}(v_{3}), y(v_{1}), yx(v_{1})}

with

x(v_{i}) = y(v_{i+1}), y^{2}(v_{1}) = x^{2}(v_{4}) = x^{3}(v_{i}) = 0.

We represent this module with the following diagram

(1)  [figure: •^{⟨v_{1}⟩}; y; x; •^{⟨v_{2}⟩}; y; x; •^{⟨v_{3}⟩}; y; x; •^{⟨v_{4}⟩}; y; x; •; x; •; y; x; •; y; x; •; y; x; •; y; •; •; •; •]

The vertices correspond to k-linear space generators, and the arrows indicate the action of the generators x and y of the group algebra. A vertex with no out-coming arrows corresponds to a trivial 1-dimensional submodule.

If p = 5, then W ≃ I^{3}/I^{6}, and W has constant Jordan type 3[3] + 2[2]. More generally, the Jordan type of the kE-module I^{p−2}/I^{p+1} is (p − 2)[3] + 2[2].

If p > 5, then W does not have constant Jordan type. Namely, the Jordan type for both x and y on W is 3[3] + 2[2], whereas the Jordan type of x + y is 4[3] + 1[1]. The Jordan blocks of size 3 for the action of x + y are generated by v_{1}, v_{2}, v_{3} and v_{4} and the trivial block is generated by x(v_{1} − v_{2} + v_{3} − v_{4}) + y(v_{4}).

Proposition 2.5. Let sl_{2} denote the p-restricted Lie algebra of 2×2 matrices of trace 0 and let u(sl_{2}) denote the restricted enveloping algebra of sl_{2}, the group algebra of the height 1 infinitesimal group scheme G = SL_{2(1)}. If the finite dimensional u(sl_{2})- module M is the restriction of a rational SL_{2}-module, then M has constant Jordan type. On the other hand, other u(sl_{2})-modules typically do not have constant Jordan type.

Proof. Let \xymatrix{ρ : SL_{2} \ar[r] & GL_{m}} determine a rational SL_{2}-module M of dimension m. Then the induced map of p-restricted Lie algebras \xymatrix{dρ : sl_{2} \ar[r] & gl_{m}} defines the associated u(sl_{2})-module structure on M. For any field extension K/k and any x ∈ SL_{2}(K), the rational SL_{2,K}-module M_{K}^{ρ(x)} given by Ad(ρ(x)) ∘ ρ is isomorphic to M_{K} and thus the associated u(sl_{2,K})-module M_{K}^{ρ(x)} given by d(Ad(ρ(x) ∘ ρ) is isomorphic to M_{K}.

Recall that the space of k-rational points of Π(G) can be identified with the space of k-rational lines of the nilpotent variety N (sl_{2}). Indeed, each equivalence class of π-points is represented by some \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & u(sl_{2,K})} sending t to a nilpotent matrix of sl_{2}(K). Moreover, the Jordan type of a given finite dimensional u(sl_{2})- module does not depend upon the choice of such a representative of a given point of Π(G) by [21, 3.1]. The action of SL_{2}(K) on Π(G)_{K} sending α_{K} to dρ(Ad(x))∘α_{K} = Ad(ρ(x)) ∘ α_{K} corresponds to the natural action of SL_{2} on N (sl_{2}). Consequently, the transitivity of this action together with Proposition 1.7 implies that the rational SL_{2}-module M has constant Jordan type as a u(sl_{2})-module.

On the other hand, any u(sl_{2})-module M whose Π-support space Π(G)_{M} is a non-empty proper subset of the 1-dimensional variety Π(G) has Jordan type of M at a generic π-point of Π(G) equal to \frac{m}{p}[p] (where m = dim M) and strictly smaller Jordan type at any π-point representing a point in Π(G)_{M}. Such M abound, since every finite subset of Π(G) is of the form Π(G)_{M} for some finite dimensional kGmodule M. For example, let b ⊂ sl_{2} be the Lie subalgebra of lower triangular matrices and consider the u(sl_{2})-module M = u(sl_{2}) ⊗_{u(b)} k, the module obtained from the trivial u(b)-module by coinduction. Then M is free when restricted to the 1-dimensional subalgebra of strictly upper triangular matrices and trivial when restricted to the 1-dimensional subalgebra of strictly lower triangular matrices. □

We conclude this section of explicit examples with a family of modules V_{n}, n > 0 which are modules of constant Jordan type n[2]+1[1] regardless of the prime p. The module V_{n} can be represented by the following diagram:

[figure: •^{⟨v_{1}⟩}; y; x; •^{⟨v_{2}⟩}; y; x; …; •^{⟨v_{n}⟩}; y; x; •; •; •; …; •; •]

The verification of the assertion that V_{n} does indeed have constant Jordan type is a simple computation.

In the following proposition we describe the modules V_{n} explicitly in terms of generators and relations.

Proposition 2.6. Consider a rank 2 elementary abelian p-group E, so that kE = k[x, y]/(x^{p}, y^{p}). Consider the kE-module V_{n} of dimension 2n + 1 generated by v_{1}, …, v_{n} spanned as a k-vector space by {v_{1}, …, v_{n}, x(v_{1}), …, x(v_{n}), y(v_{1})} with x(v_{i}) = y(v_{i+1}), x^{2}(v_{i}) = xy(v_{i}) = y^{2}(v_{i}) = 0. Then V_{n} has constant Jordan type n[2] + 1[1].

3. Specialization, Γ(G)_{M}, and constant Jordan type

In this section, we introduce a strict form of specialization of π-points which was considered briefly in [21, 3.2] for infinitesimal group schemes. We also recall the non-maximal support variety Γ(G)_{M} of a finite dimensional kG-module M. We use both strict specialization and the non-maximal support variety to further explore the class of modules of constant Jordan type.

The condition recalled in Definition 1.2 that the π-point α_{K} specializes to the πpoint β_{L} is equivalent to the algebro-geometric condition that the point [α_{K}] ∈ Π(G) specializes to [β_{L}] ([20]). For some purposes, this notion of specialization is not sufficiently strong. Namely, if Π(G) is reducible, then there exist finite dimensional kG-modules M and π-points α_{K}, β_{L} such that α_{K} specializes to β_{L} but the Jordan type of α_{K}^{∗}(M_{K}) is smaller than the Jordan type of β_{L}^{∗}(M_{L}) (see [20, 4.14]).

The following definition introduces a stricter definition of specialization which is a natural extension of [21, 3.2] from infinitesimal group schemes to arbitrary finite group schemes.

Definition 3.1. Let G be a finite group scheme and α_{K}, β_{L} be π-points of G. We say that α_{K} strictly specializes to β_{L} (and write α_{K} ⇊ β_{L}), if there exists a commutative local domain R over k with field of fractions K and residue field L, together with a map of R-algebras \xymatrix{ν_{R} : R[t]/t^{p} \ar[r] & RG} such that ν_{R} ⊗_{R} K = α_{K}, ν_{R} ⊗_{R} L = β_{L}.

Theorem 3.2. Let R be a commutative local domain with field of fractions K and residue field L. Let A ∈ M_{N}(R) be an N × N matrix which has p-th power 0 and coefficients in R. Then the Jordan type of A ⊗_{R} K ∈ M_{N}(K) is greater or equal to the Jordan type of A ⊗_{R} L ∈ M_{N}(L).

Consequently, if α_{K}, β_{L} are π-points of a finite group scheme G with α_{K} ⇊ β_{L} and if M is a finite dimensional kG-module, then α_{K}^{∗}(M_{K}) has Jordan type greater or equal to that of β_{L}^{∗}(M_{L}). In particular, α_{K} ⇊ β_{L} implies α_{K} ↓ β_{L}.

Proof. Let M[i] denote the cokernel of \xymatrix{A^{i} : R^{N} \ar[r] & R^{N}} for some i < p. Let m_{1}, …, m_{s} ∈ M[i] be chosen so that \overline{m}_{1}, …, \overline{m}_{s} ∈ M[i] ⊗_{R} L is a basis, where \overline{m}_{i} = m_{i} ⊗ 1_{L}. By Nakayama’s Lemma, m_{1}, …, m_{s} generate M[i] as an R-module and thus their images span M[i] ⊗_{R} K. Observe that M[i] ⊗_{R} K is the cokernel of \xymatrix{(A ⊗_{R} K)^{i} : K^{N} \ar[r] & K^{N}} and that M[i] ⊗_{R} L is the cokernel of the homomorphism

\xymatrix{(A ⊗_{R} L)^{i} : L^{N} \ar[r] & L^{N}} . This implies that the rank of (A⊗_{R} K)^{i} is greater or equal to the rank of (A ⊗_{R} L)^{i} for any i < p so that the Jordan type of A ⊗_{R} K ∈ M_{N}(K) is greater or equal to the Jordan type of A ⊗_{R} L ∈ M_{N}(L).

The second statement is a special case of the first. The last statement follows from the observation that if β_{L}^{∗}(M_{L}) is projective then its Jordan type is the maximal possible Jordan type on M and hence α_{K}^{∗}(M_{K}) must also be projective assuming that α_{K} ⇊ β_{L}. □

The next theorem verifies that specialization of points in Π(G) can be represented by strict specialization.

Theorem 3.3. Let G be a finite group scheme and let α_{K}, β_{L} be π-points of G with α_{K} ↓ β_{L}. Then there exist π-points α′_{K′} ∼ α_{K} and β′_{L′} ∼ β_{L} such that α′_{K′} ⇊ β′_{L′}.

Proof. Using [20, 4.13], we can choose an elementary abelian p-group E ⊂ π_{0}(G) such that [α_{K}], [β_{L}] are in the image of the closed map \xymatrix{Π((G^{0})^{E} × E) \ar[r] & Π(G)} . Observe that the group algebra of (G^{0})^{E} × E is isomorphic to the group algebra of some infinitesimal group scheme H. Because the relationship of strict specialization given in Definition 3.1 and the definition of specialization given in Definition 1.2 involve only the group algebra of the given finite group scheme, we may replace G by H. Thus, we assume G is an infinitesimal group scheme.

Let Spec A ⊂ Π(G) be an irreducible affine open subset containing [β_{L}]. Since α_{K} ↓ β_{L}, we have that [β_{L}] is in the closure of [α_{K}] by [20, 4.3]. Hence, any closed irreducible component in the complement of Spec A containing [α_{K}] must contain [β_{L}]. We conclude that [α_{K}] ∈ Spec A.

Replacing A by A_{red}, we may assume that A is a domain. Let R denote the local A-algebra defined as the localization at the prime corresponding to [α_{K}] of the quotient of A by the prime corresponding to [β_{L}]. Set K′ to be the field of fractions of R and L′ to be the residue field of R.

Let r denote the height of the infinitesimal group scheme G and let V_{r}(G) be the scheme of 1-parameter subgroups of G. Recall that the natural morphism \xymatrix{Θ_{G} : V_{r}(G)\\{0\} \ar[r] & Π(G)} , which is determined by sending a 1-parameter subgroup

\xymatrix{μ : 𝔾_{a(r),K} \ar[r] & G_{K}} to the π-point \xymatrix{μ_{∗} ∘ ϵ : K[t]/t^{p} \ar[r] & K𝔾_{a(r)} \ar[r] & KG} , where ϵ

has the property that its composition with the map on group algebras induced by the projection \xymatrix{𝔾_{a(r)} \ar[r] & 𝔾_{a(r)}/𝔾_{a(r−1)}} , is an isomorphism. Because Θ_{G} together with the isogeny H^{•}(G, k) → k[V_{r}(G)] of [26, 5.2] induces the isomorphism \xymatrix{Proj (V_{r}(G)) \ar[r] & Π(G)} of Theorem 1.4, we conclude that the given morphism

\xymatrix{Spec R \ar[r] & Π(G)} lifts to a morphism

\xymatrix{Spec R \ar[r] & (Spec H^{•}(G, k)[1/P])_{0} ⊂ V_{r}(G),}

which corresponds to a morphism of R-group schemes \xymatrix{ν : 𝔾_{a(r),R} \ar[r] & G_{R}} . Here, P is a homogeneous element of positive degree of H^{•}(G, k) which does not vanish on the image of the closed point of Spec R and (A^{•})_{0} denotes the subalgebra of degree 0 elements of the graded algebra A^{•}.

We define ν_{R} to be the map of R-algebras given as the composition

\xymatrix{R[t]/t^{p} \ar[r]^{ϵ} & R(𝔾_{a(r),R}) \ar[r]^{ν_{∗}} & RG.}

By construction, ν_{R} ⊗_{R} K′ is a π-point of G in the equivalence class [α_{K}] (the image of Spec K′) and ν_{R} ⊗_{R} L′ is a π-point of G in the equivalence class [β_{L}]. □

The proof of the theorem in the title of [8] applies with only minor notational change to prove the same assertion, given below, for an arbitrary finite group scheme.

Theorem 3.4. ([8]) Let G be a finite group scheme and M a finite dimensional indecomposable kG-module. Then Π(G)_{M} is connected. In particular, Π(G) = Π(G)_{k} is connected.

Theorems 3.3 and 3.4 enable the following equivalent formulation of the condition that a module has constant Jordan type.

Proposition 3.5. Let G be a finite group scheme and let M be a kG-module. Then M has constant Jordan type if and only if α_{K} and β_{L} have the same Jordan type on M whenever α_{K}, β_{L} are π-points of G satisfying α_{K} ⇊ β_{L}.

Proof. If M has constant Jordan type and α_{K}, β_{L} are π-points of G, then the Jordan types of α_{K} and and β_{L} on M are necessarily equal.

To prove the converse, we recall that if x ∈ Π(G) is a point such that M has maximal Jordan type at some representative of x, then the Jordan type on M is the same for every representative of x (see Remark 1.6). Let x ∈ Π(G) be a point which has maximal Jordan type on M and let y ∈ Π(G) be an arbitrary point. Since Π(G) is connected (by Theorem 3.4) and Noetherian, we may find a chain of irreducible components Z_{0}, …, Z_{r} of Π(G) such that Z_{i} ∩ Z_{i+1} ≠ ∅ and x ∈ Z_{0}, y ∈ Z_{r}. Let η_{i} ∈ Z_{i} be the generic point and choose some z_{i} ∈ Z_{i}∩Z_{i+1}. Then x, η_{0}, z_{1}, η_{1}, …, z_{i}, η_{i}, …, η_{r}, y is a chain of points x = x_{0}, x_{1}, …, x_{n} = y ∈ Π(G) such that there exist morphisms \xymatrix{Spec R_{i} \ar[r] & Π(G)_{red}} sending \{Spec K_{i}, Spec L_{i}\} to the (unordered) pair {x_{i−1}, x_{i}} where R_{i} is a commutative local domain with field of fractions K_{i} and residue field L_{i}.

Our hypothesis in conjunction with Theorem 3.3 implies that there are representing π-points α_{K_{i}}, α_{L_{i}} of x_{i} and x_{i−1} at which M has the same Jordan type. Consequently, M has the same maximal Jordan type at each representative of each x_{i}. Thus, the Jordan type of M at any representative of x_{n} = y equals this same maximal Jordan type; in other words, M has constant Jordan type. □

We give a useful characterization of modules M of constant Jordan type in terms of their non-maximal support varieties Γ(G)_{M} recalled in Remark 1.6.

Proposition 3.6. Let M be a finite dimensional kG-module. Then M has constant Jordan type if and only if Γ(G)_{M} = ∅.

Proof. If M has constant Jordan type, then clearly Γ(G)_{M} = ∅. Conversely, let Γ(G)_{M} = ∅, and let α_{K}, β_{L} be any two π-points of G satisfying α_{K} ⇊ β_{L}. By Theorem 3.2, α_{K}^{∗}(M_{K}) has Jordan type greater or equal to that of β_{L}^{∗}(M_{L}). Since Γ(G)_{M} = ∅, we must have that the Jordan types of α_{K}^{∗}(M_{K}) and β_{L}^{∗}(M_{L}) are the same. Hence, Proposition 3.5 implies that M has constant Jordan type. □

The following “closure property” of modules of constant Jordan type is somewhat striking.

Theorem 3.7. Let M be a kG-module of constant Jordan type. Then any direct summand of M also has constant Jordan type.

Proof. Write M = M′ ⊕ M′′. Let α_{K}, β_{L} be two π-points such that α_{K} ⇊ β_{L}. By Theorem 3.2, we have

(2) JType(α_{K}^{∗}(M′_{K})) ≥ JType(β_{L}^{∗}(M′_{L})) and JType(α_{K}^{∗}(M′′_{K})) ≥ JType(β_{L}^{∗}(M′′_{L})).

Hence,

JType(α_{K}^{∗}(M′_{K})) ⊕ JType(α_{K}^{∗}(M′′_{K})) = JType(α_{K}^{∗}(M)) = JType(β_{L}^{∗}(M_{L})) =

= JType(β_{L}^{∗}(M′_{L})) ⊕ JType(β_{L}^{∗}(M′′_{L})),

where JType(−) is the Jordan type function. Therefore, both inequalities in (2) must be equalities. Since this holds for any pair α_{K} ⇊ β_{L}, the statement now follows immediately from Proposition 3.5. □

4. Behavior with respect to tensor products

We begin with the following “order preserving” property of Jordan types. The proof given below uses an explicit description of the tensor product of indecomposable k[t]/t^{p}-modules which is presented in the Appendix. The coalgebra structure we (implicitly) give k[t]/t^{p} is that for which t is primitive (i.e., ∇(t) = t ⊗ 1 + 1 ⊗ t).

Proposition 4.1. Let M, N and L be k[t]/t^{p}-modules with Jordan types \underline{a} = a_{p}[p]+ ⋯ + a_{1}[1], \underline{b} = b_{p}[p] + ⋯ + b_{1}[1] and \underline{c} = c_{p}[p] + ⋯ + c_{1}[1] respectively such that dim M = dim N. Let \underline{a} ⊗ \underline{c}, \underline{b} ⊗ \underline{c} denote the Jordan types of M ⊗ L, N ⊗ L respectively.

If \underline{a} ≥ \underline{b}, then \underline{a} ⊗ \underline{c} ≥ \underline{b} ⊗ \underline{c}.

If \underline{a} > \underline{b} and if c_{i} ≠ 0 for some i < p, then \underline{a} ⊗ \underline{c} > \underline{b} ⊗ \underline{c}.

Proof. For a k[t]/t^{p}-module M of dimension m, we denote the representation afforded by M by \xymatrix{ρ_{M} : k[t]/t^{p} \ar[r] & End_{k}(A) ≃ M_{m}(k)} . The dominance condition (??) on Jordan types \underline{a} ≥ \underline{b} of M and N both of dimension m can be formulated as the condition that

Rk ρ_{M}(t)^{i} ≥ Rk ρ_{N}(t)^{i}, 1 ≤ i < p

or equivalently that

(3)  dim Ker {ρ_{M}(t)^{i}} ≤ dim Ker {ρ_{N}(t)^{i}}, 1 ≤ i < p.

For \underline{a} > \underline{b}, the additional condition is that Rk ρ_{M}(t)^{i} > Rk ρ_{N}(t)^{i} for some i. If L ≃ L_{1} ⊕ L_{2}, so that \underline{c} = \underline{c}_{1} + \underline{c}_{2}, then we immediately conclude that

\underline{a} ⊗ \underline{c} = \underline{a} ⊗ \underline{c}_{1} + \underline{a} ⊗ \underline{c}_{2}.

Thus, we may assume that L is indecomposable, say L = [ℓ].

Denote by J_{s} ∈ M_{s}(k) the Jordan block of size s. Then J_{s} = ρ_{[s]}(t) once we choose an appropriate basis. Since t acts on M ⊗ [ℓ] as t ⊗ 1 + 1 ⊗ t, we get the formula

ρ_{M⊗[ℓ]}(t) = ρ_{M}(t) ⊗ Id_{[s]} + Id_{M} ⊗ J_{s}

Lemma 1.10 of [21] implies

(4)  Ker {ρ_{M⊗[ℓ]}(t)} ≅ Ker {ρ_{M}(t)^{ℓ}}

Applying the same argument to M ⊗ [ℓ], we obtain

(5)  Ker {ρ_{M⊗[ℓ]}(t)^{s}} = Ker {ρ_{M⊗[ℓ]⊗[s]}(t)}

We write [ℓ] ⊗ [s] = ⨁_{j} C_{ℓs}^{j}[j] where the coefficients C_{ℓs}^{j} are given by the Corollary 10.2 of the Appendix. With this notation,

dim Ker {ρ_{M⊗[ℓ]}(t)^{s}} = dim Ker {ρ_{M⊗(⨁ C_{ℓs}^{j}[j])}(t)} = dim Ker {ρ_{⨁ C_{ℓs}^{j}(M⊗[j])}(t)}

= ∑ C_{ℓs}^{j} dim Ker {ρ_{M⊗[j]}(t)} = ∑ C_{ℓs}^{j} dim Ker {ρ_{M}(t)^{j}}.

Consequently,

dim Ker {ρ_{M⊗[ℓ]}(t)^{s}} = ∑ C_{ℓs}^{j} dim Ker {ρ_{M}(t)^{j}}

and

dim Ker {ρ_{N⊗[ℓ]}(t)^{s}} = ∑ C_{ℓs}^{j} dim Ker {ρ_{N}(t)^{j}}

Hence, (3) implies that

dim Ker {ρ_{M⊗[ℓ]}(t)^{s}} ≤ dim Ker {ρ_{N⊗[ℓ]}(t)^{s}}

for all s. Therefore, \underline{a} ⊗ \underline{c} ≥ \underline{b} ⊗ \underline{c}.

Now assume \underline{a} > \underline{b} and that some c_{i} ≠ 0, i < p. Then, we may assume that L = [ℓ] with ℓ < p. Choose j such that dim Ker {ρ_{M}(t)^{j}} < dim Ker {ρ_{N}(t)^{j}}. The formula of Corollary 4.3 implies that there exists some s such that C_{ℓs}^{j} = 1. That is, if j ≥ ℓ, then take s = j − ℓ + 1, and if j < ℓ, take s = ℓ − j + 1. For such s we get the strict inequality

dim Ker {ρ_{M⊗[ℓ]}(t)^{s}} < dim Ker {ρ_{N⊗[ℓ]}(t)^{s}}

Therefore, \underline{a} ⊗ \underline{c} > \underline{b} ⊗ \underline{c}.

□

Proposition 4.1 enables us to establish various tensor product properties of maximal Jordan type. We should point out that there are some subtleties which must be carefully considered. For example if M and N are kG-modules and α_{K} is a π-point whose image is not a sub-Hopf algebra of KG, then it is not always true that α_{K}^{∗}(M_{K} ⊗ N_{K}) ≃ α_{K}^{∗}(M_{K}) ⊗ α_{K}^{∗}(N_{K}). Hence, Proposition 4.1 can not be applied directly. The somewhat surprising Examples 4.5 and 4.6 indicate the subtle relationship between tensor products and maximal Jordan types. In view of these examples, we take some care in the proofs of the following properties.

Theorem 4.2. Let G be a finite group scheme, and consider two finite dimensional kG-modules M, N and a π-point α_{K} of G.

(1) If α_{K} has maximal Jordan type on both M and N, then

α_{K}^{∗}(M_{K} ⊗ N_{K}) ≃ α_{K}^{∗}(M_{K}) ⊗ α_{K}^{∗}(N_{K}).

(2) If α_{K} has maximal Jordan type on M ⊗ N, then

α_{K}^{∗}(M_{K} ⊗ N_{K}) ≃ α_{K}^{∗}(M_{K}) ⊗ α_{K}^{∗}(N_{K}),

(3) If Π(G) is irreducible and if α_{K} has maximal Jordan type on both M and N, then α_{K} has maximal Jordan type on M ⊗ N and that Jordan type is equal to the Jordan type of α_{K}^{∗}(M_{K}) ⊗ α_{K}^{∗}(N_{K}).

Proof. Let i : C_{K} ⊂ G_{K} be a unipotent abelian subgroup scheme through which α_{K} factors. Since \xymatrix{i : KC_{K} \ar[r] & KG_{K}} is a map of Hopf algebras, i^{∗} commutes with tensor products. Observe that the maximality of α_{K} on M, N or M ⊗ N as kGmodules implies the maximality of α_{K} on M_{K}, N_{K} or M_{K} ⊗_{K} N_{K} as KC_{K}-modules. Because Π(C_{K}) is irreducible, the maximal Jordan types of M_{K}, N_{K}, and M_{K} ⊗_{K} N_{K} are all achieved at a generic π-point of Π(C_{K}). Thus, statement (1) follows from [21, 4.4] applied to C_{K}.

To prove (2), we extend scalars if necessary so that K is perfect. Hence, KC_{K} ≃ K[T_{1}, …, T_{r}]/(T_{1}^{p^{e_{1}}}, …, T_{r}^{p^{er}}) (see [28, 14.4]). Let t_{i} = T_{i}^{p^{e_{i}−1}}.

Let \xymatrix{ρ_{M} : KC_{K} \ar[r] & End_{K}(M_{K})} be the map defined by the representation M_{K} of C_{K}, and similarly define ρ_{N} and ρ_{M⊗N}. Choose a generic π-point η_{Ω} of C_{K}, \xymatrix{η_{Ω} : Ω[t]/t^{p} \ar[r] & ΩC_{K}} , where Ω/K is a field extension. Arguing exactly as in the proof of [21, 4.4], we conclude that the Jordan type of ρ_{M}(η_{Ω}(t)) ⊗ 1 + 1 ⊗ ρ_{N}(η_{Ω}(t)) as an element of End_{Ω}(M_{Ω} ⊗ N_{Ω}) is greater or equal than the Jordan type of any linear combination of elements of the form ρ_{M}(t_{i}) ⊗ 1 and 1 ⊗ ρ_{N}(t_{j}), 1 ≤ i, j ≤ r. Since η_{Ω}^{∗}(M_{Ω} ⊗ N_{Ω}) ≃ η_{Ω}^{∗}(M_{Ω}) ⊗ η_{Ω}^{∗}(N_{Ω}) by [21, 4.4], the same conclusion holds for ρ_{M⊗N}(η_{Ω}(t)). Since α_{K} is maximal on M ⊗ N and η_{Ω} is a generic π-point, the Jordan type of α_{K}^{∗}(M_{K} ⊗ N_{K}) equals the Jordan type of η_{Ω}^{∗}(M_{Ω} ⊗ N_{Ω}). Hence, the Jordan type of ρ_{M⊗N}(α_{K}(t)) is greater or equal than the Jordan type of any linear combination of elements of the form ρ_{M}(t_{i}) ⊗ 1 or 1 ⊗ ρ_{N}(t_{j}), 1 ≤ i, j ≤ r.

Let ∇ : KC_{K} → KC_{K} ⊗KC_{K} be the coproduct on KC_{K}, and let I = (T_{1}, …, T_{r}) be the augmentation ideal of KC_{K}. and let I^{(p)} be the ideal generated by (t_{1}, …, t_{r}). Recall that

α_{K}(t) ⊗ 1 + 1 ⊗ α_{K}(t) − ∇(α_{K}(t)) ∈ I ⊗ I

(see [23, I.2.4]). Since α_{K}(t) has p-th power 0, we can refine this further, concluding that

(6)  α_{K}(t) ⊗ 1 + 1 ⊗ α_{K}(t) − ∇(α_{K}(t)) ∈ I ⊗ I^{(p)} + I^{(p)} ⊗ I.

The action of KC_{K} on the tensor product M_{K} ⊗ N_{K} is given by the formula

ρ_{M⊗N} = (ρ_{M} ⊗ ρ_{N}) ∘ ∇

Hence,

(7)  ρ_{M}(α_{K}(t)) ⊗ 1 + 1 ⊗ ρ_{N}(α_{K}(t)) =
ρ_{M⊗N}(α_{K}(t)) + ⟨multiples of ρ_{M}(t_{i}) ⊗ 1 and 1 ⊗ ρ_{N}(t_{j})⟩

Since the Jordan type of ρ_{M⊗N}(α_{K}(t)) is greater of equal than the Jordan type of any linear combination of elements ρ_{M}(t_{i}) ⊗ 1 and 1 ⊗ ρ_{N}(t_{j}) for 1 ≤ i, j ≤ r, the relation (7) and Theorem 1.12 of [21] imply that ρ_{M⊗N}(α_{K}(t)) and ρ_{M}(α_{K}(t)) ⊗ 1 + 1 ⊗ ρ_{N}(α_{K}(t)) have the same Jordan type. Hence, α_{K}^{∗}(M) ⊗_{K} α_{K}^{∗}(N) ≃ α_{K}^{∗}(M_{K} ⊗_{K} N_{K}).

To prove (3) we assume that Π(G) is irreducible. Let \xymatrix{η_{Ω} : Ω[t]/t^{p} \ar[r] & ΩG} be a generic π-point of G. Applying [21, 4.4] to η_{Ω}, we get

η_{Ω}^{∗}(M_{Ω}) ⊗_{Ω} η_{Ω}^{∗}(N_{Ω}) ≃ η_{Ω}^{∗}(M_{Ω} ⊗_{Ω} N_{Ω})

Since Π(G) is irreducible, the absolute maximal Jordan type of any finite dimensional kG-module is realized at η_{Ω}. Hence, the maximality assumption on α_{K} on M and N implies that α_{K}^{∗}(M_{K}) has the same Jordan type as η_{Ω}^{∗}(M_{Ω}) and α_{K}^{∗}(N_{K}) has the same Jordan type as η_{Ω}^{∗}(N_{Ω}). We conclude that

α_{K}^{∗}(M_{K} ⊗ N_{K}) ≃ α_{K}^{∗}(M_{K}) ⊗ α_{K}^{∗}(N_{K})

has the same Jordan type as

η_{Ω}^{∗}(M_{Ω}) ⊗_{Ω} η_{Ω}^{∗}(N_{Ω}) ≃ η_{Ω}^{∗}(M_{Ω} ⊗_{Ω} N_{Ω})

where the first isomorphism follows from statement (1). Hence, the Jordan type of α_{K} on M ⊗ N is maximal. □

An easy corollary of Theorem 4.2 is the following assertion that the tensor product of modules of constant Jordan type is again of constant Jordan type.

Corollary 4.3. Let G be a finite group scheme and let M, N be finite dimensional kG-modules. If M and N have constant Jordan type, then M ⊗ N also has constant Jordan type.

Proof. We merely observe that α_{K} has maximal Jordan type on both M and N for any π-point \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} whenever M, N are of constant Jordan type. Thus, the corollary follows from the first statement of Theorem 4.2. □

The following consequence of Theorem 4.2 will be used to prove Proposition 4.7 .

Corollary 4.4. Let G be a finite group scheme such that Π(G) is irreducible and let \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} be a π-point of G. Let M be a kG-module such that α_{K}^{∗}(M_{K}) is not projective and let N be another kG-module such that α_{K} does not have maximal Jordan type on N. Then α_{K} does not have maximal Jordan type on M ⊗ N.

Proof. If the Jordan type of α_{K}^{∗}(M_{K} ⊗_{K} N_{K}) were maximal, then the second statement of Theorem 4.2 would imply that this maximal type is the same as that of α_{K}^{∗}(M_{K})⊗_{K} α_{K}^{∗}(N_{K}). However, the hypotheses on α_{K}^{∗}(M_{K}), α_{K}^{∗}(N_{K}) together with Proposition 4.1 would then lead to an immediate contradiction. □

We give two examples to illustrate that naturally formulated improvements of Theorem 4.2 are not valid. Our first example illustrates that the maximality of both M and N at a given π-point is not sufficient to imply the maximality of M ⊗ N at that π-point. Namely, we construct a module W and a π-point β_{L} such that β_{L} has maximal Jordan type on W but not on W ⊗ W (even though we have β_{L}^{∗}(W_{L} ⊗W_{L}) ≃ β_{L}^{∗}(W_{L})⊗_{L} β_{L}^{∗}(W_{L}) by Theorem 4.2). As usual, this anomaly comes from the fact that the ordering on Jordan types is not total.

Example 4.5. Let G be a finite p-group which has two conjugacy classes of maximal elementary abelian subgroups, represented by E and E′ respectively. Furthermore, we require E to be normal. Let e = |E|, f = \frac{|G|}{|E|}. Assume that p > 3.

For example, take G to be the p-Sylow subgroup of the wreath product ℤ/pℤ ≀ S_{p}, so that G is isomorphic to (ℤ/pℤ)^{p} ⋊ℤ/pℤ. Then G has two non-conjugate maximal elementary abelian p-subgroups: E = (ℤ/pℤ)^{p} which is normal and

F = (ℤ/pℤ × ℤ/pℤ × ⋯ × ℤ/pℤ)^{ℤ/pℤ} × ℤ/pℤ ≅ (ℤ/pℤ)^{×2}.

By Quillen stratification, Π(G) = X ∪ Y where X = Π(E)/G, Y = Π(F)/N_{G}(F). Let [α_{K}] ∈ X, [β_{L}] ∈ Y be generic points.

Choose a homogeneous cohomology class ξ ∈ H^{•}(G, k) with the property that ξ vanishes on [β_{L}] but does not vanish on [α_{K}], and let L_{ξ} be Carlson module which has the property that the support of L_{ξ} is the zero locus of ξ. Set M = Ind_{E}^{G}(Ω_{E}^{1}(k)), set N = L_{ξ}^{⊕n} for some positive integer n which is to be determined, and set W = M ⊕ N. It was shown in Example [21, 4.13], that if we pick n to satisfy the inequality

(8)  \frac{f}{p} < n < (p − 1)\frac{f}{p}

then the Jordan types α_{K}^{∗}(W_{K}) and β_{L}^{∗}(W_{L}) are maximal, incomparable generic Jordan types of W.

Let d = dim W. We proceed to deduce a condition on n which would ensure the inequality

JType(α_{K}^{∗}(W_{K}^{⊗2})) > JType(β_{L}^{∗}(W_{L}^{⊗2}))

By the Appendix, [p−1]⊗[p−1] = (p−2)[p]+1[1]. Since α_{K}^{∗}(W_{K}) = m[p]+f[p−1] for some m (see [21, 4.13]), and has dimension d, we get

α_{K}^{∗}(W_{K}^{⊗2}) ≃ (α_{K}^{∗}(W_{K}))^{⊗2} ≃ (\frac{d^{2} − f^{2}}{p})[p] + f^{2}[1].

Similarly, applying the decomposition of β_{L}^{∗}(W_{L}) obtained in [21, 4.13], we get

β_{L}^{∗}(W_{L}^{⊗2}) ≃ (β_{L}^{∗}(W_{L}))^{⊗2} ≃ (\frac{d^{2} − 2n^{2}p}{p})[p] + 2n^{2}[p − 1] + 2n^{2}[1].

In order for the Jordan type of α_{K}^{∗}(W_{K}^{⊗2}) to dominate that of β_{L}^{∗}(W_{L}^{⊗2}) it suffices to choose n such that α_{K}^{∗}(W_{K}^{⊗2}) has more blocks of size p than β_{L}^{∗}(W_{L}^{⊗2}) and fewer blocks altogether. In other words, we need for the following inequalities to hold:

\frac{d^{2} − f^{2}}{p} > \frac{d^{2} − 2n^{2}p}{p}

comparing the number of blocks of size p, and

\frac{d^{2} − f^{2}}{p} + f^{2} < \frac{d^{2} − 2n^{2}p}{p} + 4n^{2}

comparing the overall number of blocks. Simplifying, we get

f^{2} < 2n^{2}p,
(p − 1)f^{2} < 2n^{2}p.

Observe that the second inequality implies the first and simplifies to

\sqrt{\frac{p − 1}{2p}}f < n.

Since p is greater than 3 and divides f, it is possible to choose n to satisfy

\sqrt{\frac{p − 1}{2p}}f < n < \frac{p − 1}{p}f.

Since such n automatically satisfies the inequality (8), we conclude that W has maximal non-comparable types at [α_{K}], [β_{L}] but that the Jordan type of α_{K}^{∗}(W_{K} ⊗_{K} W_{K}) is strictly greater than that of β_{L}^{∗}(W_{L} ⊗_{L} W_{L}). Thus, β_{L} has maximal Jordan type on W but not on W ⊗ W.

Our second example shows that the maximal Jordan type of M ⊗ N can occur at a π-point at which one of M, N does not have maximal Jordan type and neither has projective type. This phenomenon can only occur if Π(G) is reducible.

Example 4.6. As in the previous example, let G be a finite group with exactly two conjugacy classes of maximal elementary abelian p-groups (e.g., the p-Sylow subgroup of the wreath product ℤ/pℤ ≀ S_{p}). Write Π(G) = X ∪ Y with X, Y irreducible closed subsets, and let [α_{K}] ∈ X, [β_{L}] ∈ Y be generic points. Choose cohomology classes ζ, ξ ∈ H^{•}(G, k) such that

α_{K}^{∗}(ζ_{K}) = 0 ≠ β_{L}^{∗}(ζ_{L}), α_{K}^{∗}(ξ_{K}) ≠ 0 = β_{L}^{∗}(ξ_{K}).

Let L_{ζ} (respectively, L_{ξ}) be the corresponding Carlson module so that the support of L_{ζ} (resp., L_{ξ}) is the zero locus of ζ (resp., ξ). Recall that the Jordan type of L_{ζ} on its support is [proj] + 1[p − 1] + 1[1] for some m, and similarly for L_{ξ} [21, 4.8]. Let M = L_{ζ} ⊕ L_{ξ}^{⊕2}, so that M has maximal Jordan type [proj] + 1[p − 1] + 1[1] at α_{K}, and Jordan type [proj] + 2[p − 1] + 2[1] at β_{L}. Similarly, let N = L_{ζ}^{⊕2} ⊕ L_{ξ}, so that N has Jordan type [proj] + 2[p − 1] + 2[1] at α_{K} and maximal Jordan type [proj] + 1[p − 1] + 1[1] at β_{L}. Then M ⊗ N has the same Jordan type at both generic points α_{K} and β_{L}, so that this common generic Jordan type is maximal.

Corollary 4.4 enables us to prove the following property of the non-maximal support variety.

Proposition 4.7. Assume Π(G) is irreducible and let M, N be kG-modules. Then

Γ(G)_{M⊗N} = (Γ(G)_{M} ∪ Γ(G)_{N}) ∩ (Π(G)_{M} ∩ Π(G)_{N}).

Proof. If \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} is a π-point such that either α_{K}^{∗}(M_{K}) or α_{K}^{∗}(N_{K}) is projective, then α_{K}^{∗}(M_{K}) ⊗_{K} α_{K}^{∗}(N_{K}) is projective and thus of maximum type. Consequently, statement (1) of Theorem 4.2 implies that α_{K}^{∗}(M_{K} ⊗_{K} N_{K}) is also projective and thus also maximum. For such α_{K}, [α_{K}] ∉ Γ(G)_{M⊗N}. In other words, Γ(G)_{M⊗N} ⊂ Π(G)_{M} ∪ Π(G)_{N}.

Now suppose that α_{K} has maximal Jordan type on both M and N. The statement (3) of Theorem 4.2 implies that α_{K} has maximal Jordan type on M ⊗ N. Hence, Γ(G)_{M⊗N} ⊂ Γ(G)_{M} ∪ Γ(G)_{N}. We have established the inclusion

Γ(G)_{M⊗N} ⊂ (Γ(G)_{M} ∪ Γ(G)_{N}) ∩ (Π(G)_{M} ∩ Π(G)_{N}).

On the other hand, assume that neither α_{K}^{∗}(M_{K}) nor α_{K}^{∗}(N_{K}) is projective, and that the Jordan type of either α_{K}^{∗}(M_{K}) or α_{K}^{∗}(N_{K}) is not maximal. In other words, we assume that [α_{K}] ∈ (Γ(G)_{M} ∪ Γ(G)_{N}) ∩ (Π(G)_{M} ∩ Π(G)_{N}). Corollary 4.4 implies that α_{K} does not have maximal Jordan type on M⊗N. Hence, [α_{K}] ∈ Γ(G)_{M⊗N}. □

Remark 4.8. The previous examples show the necessity of the hypothesis of irreducibility in Proposition 4.7. Example 4.6 contradicts the inclusion

Γ(G)_{M⊗N} ⊃ (Γ(G)_{M} ∪ Γ(G)_{N}) ∩ (Π(G)_{M} ∩ Π(G)_{N}),

whereas Example 4.5 contradicts the inclusion

Γ(G)_{M⊗N} ⊂ (Γ(G)_{M} ∪ Γ(G)_{N}) ∩ (Π(G)_{M} ∩ Π(G)_{N}).

Next we offer a suggestive characterization of modules of constant Jordan type for those finite group schemes G with Π(G) irreducible.

Proposition 4.9. If Π(G) is irreducible, then a non-projective kG-module M has constant Jordan type if and only if for every finite dimensional kG-module N the tensor product M ⊗ N has the property that Γ(G)_{N} = Γ(G)_{M⊗N}.

Proof. First assume that M does not have constant Jordan type. If N = k (the trivial module), then Γ(G)_{k} is empty whereas Γ(G)_{k⊗M} is not empty.

Conversely, assume that M has constant Jordan type and M is not projective. Let \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} be a π-point such that N has maximal Jordan type at α_{K}. By Theorem 4.2, the Jordan type of α_{K}^{∗}(M_{K} ⊗N_{K}) ≃ α_{K}^{∗}(M_{K})⊗α_{K}^{∗}(N_{K}) is maximal for M ⊗ N. Hence, we get the inclusion Γ(G)_{N} ⊃ Γ(G)_{M⊗N}. To prove the opposite inclusion, assume [α_{K}] ∈ Γ(G)_{N}. Since M is not projective, Corollary 4.4 implies that α_{K} does not have maximal Jordan type on M ⊗ N. Hence, [α_{K}] ∈ Γ(G)_{M⊗N}, and we have established the opposite inclusion. □

5. Endotrivial modules

We recall the definition of an endotrivial module, classically formulated for finite groups but admitting a natural extension to all finite group schemes. Endotrivial modules were introduced by Dade [12], who showed that for an abelian p-group, the only endotrivial kG-modules have the form Ω^{n}(k) ⊕ P where P is a projective module. The endotrivial modules are the building blocks for the endopermutation modules which for many groups are the sources of the simple modules and are also a part of the Picard group of self equivalences of the stable module category. See [9] for references. A classification of the endotrivial modules for finite p-groups was completed in [10].

Definition 5.1. Let G be a finite group scheme over k. A kG-module M is an endotrivial module provided End_{k}(M) is stably isomorphic as a kG-module to the trivial module. In other words, M is endotrivial provided that there exists a kGprojective module P and a kG-isomorphism

Hom_{k}(M, M) ≅ k ⊕ P.

As can readily be verified (for example, using formula (10.2) of the Appendix) an indecomposable k[t]/t^{p}-module is endotrivial if and only if it is stably isomorphic to either 1[1] or 1[p − 1], the trivial module k or the Heller shift Ω^{1}(k) of k. More generally, Theorem 5.6 below implies that for any finite group scheme the Heller shifts Ω^{i}(k) of the trivial module are endotrivial modules. As mentioned earlier, for elementary abelian p-groups, these are the only indecomposable endotrivial modules. On the other hand, there do exist sporadic examples of finite groups admitting other endotrivial modules. For example, if G is a dihedral group of order 8, then rad(kG)/ rad^{4}(kG) is the direct sum of two modules of dimension three that are endotrivial (see [9]).

As we show in Theorem 5.6 below, every endotrivial module is a module of constant Jordan type. As seen in section 2 and as well in the next section, there exist many examples of modules of constant Jordan type which are not direct sums of endotrivial modules.

If M, N are kG-modules, then we may identify Hom_{k}(M, N) as a kG-module with M^{#} ⊗ N, where M^{#} = Hom_{k}(M, k). For our purposes, it suffices to analyze M^{#} and then apply Section 4 in order to investigate Hom_{k}(M, N).

Let G be a finite group scheme and (as usual) let kG denote the group algebra of G. Denote by S the antipode of the Hopf algebra kG. If \xymatrix{ρ_{M} : kG \ar[r] & End_{k}(M)} ≃ M^{#} ⊗ M is a finite dimensional representation of G determining the kG-module M, then

(9)  ρ_{M^{#}} = ϕ ∘ ρ_{M} ∘ S : \xymatrix{kG \ar[r] & kG \ar[r] & M^{#} ⊗ M \ar[r] & M ⊗ M^{#}} ,

where ϕ exchanges factors (and thus is the transpose from the point of view of matrices).

Observe that the dual [i]^{#} of the indecomposable k[t]/t^{p}-module [i] is indecomposable, and thus isomorphic to [i] as can be seen by comparing dimensions. The following proposition enables us to work with the generic and maximal Jordan types of Hom_{k}(M, N) for finite dimensional kG-modules M, N.

Proposition 5.2. Let G be a finite group scheme, M be a finite dimensional kGmodule, and \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} be a π-point of G. Assume that α_{K} has maximal Jordan type on M. Then

α_{K}^{∗}(M_{K}^{#}) ≃ α_{K}^{∗}(M_{K}) ≃ (α_{K}^{∗}(M_{K}))^{#}.

Moreover, α_{K} has maximal Jordan type on M if and only if it has maximal Jordan type on M^{#}.

Proof. Let C_{K} ⊂ G_{K} be a unipotent abelian group scheme through which α_{K} factors, so that α_{K} has maximal Jordan type for M_{K} as a KC_{K}-module. The restriction of the KG-module M_{K}^{#} to KC_{K} is the dual of the restriction of M_{K}, since \xymatrix{KC_{K} \ar[r] & KG_{K}} is a map of Hopf algebras. Thus, to prove the first asserted isomorphism, it suffices to assume G is a unipotent abelian finite group scheme.

Extending scalars if necessary so that k is perfect, we get

kG ≃ k[T_{1}, …, T_{r}]/(T_{i}^{p^{e_{i}}}) (see [28, 14.4]). Let t_{i} = T_{i}^{p^{e_{i}−1}}. Let I = (T_{1}, …, T_{r}) be the augmentation ideal of kG, and let I^{(p)} be the ideal generated by (t_{1}, …, t_{r}). Let ∇ : kG → kG ⊗ kG be the coproduct in kG. Recall that ∇(α_{K}(t)) − 1 ⊗ α_{K}(t) − α_{K}(t) ⊗ 1 ∈ I ⊗ I ([23, I.2.4]). Since α_{K}(t) has p-th power 0, we can refine this further, concluding that

(10)  ∇(α_{K}(t)) − 1 ⊗ α_{K}(t) − α_{K}(t) ⊗ 1 ∈ I ⊗ I^{(p)} + I^{(p)} ⊗ I.

Let μ : kG ⊗ kG → kG be the multiplication map in kG. We have (μ ∘ (S ⊗ Id))(∇(α_{K}(t)) = ϵ(α_{K}(t)) = 0 by one of the Hopf algebra axioms (see [23, I.2.3]) where ϵ : kG → k is the counit map, and S is the antipode. Hence, applying μ ∘ (S ⊗ Id) to (10), we get

α_{K}(t) + S(α_{K}(t)) ⊂ I · I^{(p)} + I^{(p)} · I.

Moreover, we obtain the same inclusion if we change the base field from K to any field extension L over K. Therefore, we can apply [21, 1.12] to conclude that ρ_{M}(α_{K}(t)) and ρ_{M}(S(α_{K}(t))) have the same (maximal) Jordan type. Hence, ρ_{M}(α_{K}(t)) and ρ_{M^{#}}(α_{K}(t)) = ϕ(ρ_{M}(S(α_{K}(t)))) have the same (maximal) Jordan type. We conclude that α_{K}^{∗}(M_{K}^{#}) ≃ α_{K}^{∗}(M_{K}).

The second isomorphism follows immediately from the observation that [i]^{#} = [i].

□

Since sending M to M^{#} is idempotent, we get the following corollary.

Corollary 5.3. Let G be a finite group scheme, and let M be a finite dimensional kG-module. Then

Γ(G)_{M} = Γ(G)_{M^{#}}.

The next corollary follows immediately from Corollary 4.3, Proposition 5.2 and the isomorphism Hom_{k}(M, N) ≃ M^{#} ⊗ N.

Corollary 5.4. Let G be a finite group scheme, and let M, N be finite dimensional kG-modules of constant Jordan types \underline{a} = a_{p}[p] + ⋯ + a_{1}[1], \underline{b} = b_{p}[p] + ⋯ + b_{1}[1], respectively. Then Hom_{k}(M, N) has constant Jordan type \underline{a} ⊗ \underline{b} (given explicitly by the formula in the Appendix).

Corollary 5.5. Let G be a finite group scheme, and consider two finite dimensional kG-modules M, N, and a π-point α_{K} of G. If α_{K} has maximal Jordan type on Hom(M, N), then

α_{K}^{∗}(Hom_{K}(M_{K}, N_{K})) ≃ Hom_{K}(α_{K}^{∗}(M_{K}), α_{K}^{∗}(N_{K}))

Proof. Let i : C_{K} ↪ G_{K} be a unipotent abelian group scheme through which α_{K} factors, so that α_{K} is maximal on Hom_{K}(M_{K}, N_{K}) as a KC_{K}-module. Since i : C_{K} ↪ G_{K} is a map of Hopf algebras, it commutes with Hom. Hence, we may assume that G is a unipotent abelian group scheme. In particular, Π(G) is irreducible.

Since α_{K} has maximal Jordan type on Hom(M, N) ≃ M^{#} ⊗ N, Theorem 4.2(2) implies that

α_{K}^{∗}(M_{K}^{#} ⊗ N_{K}) ≃ α_{K}^{∗}(M_{K}^{#}) ⊗ α_{K}^{∗}(N_{K}).

If α_{K}^{∗}(M_{K}^{#}) ⊗ α_{K}^{∗}(N_{K}) is projective, then either α_{K}^{∗}(N_{K}) or α_{K}^{∗}(M_{K}^{#}) is projective. Since projectivity of α_{K}^{∗}(M_{K}^{#}) implies projectivity of α_{K}^{∗}(M_{K}^{#}), we conclude that in this case Hom_{K}(α_{K}^{∗}(M_{K}), α_{K}^{∗}(N_{K})) is projective.

Assume that α_{K}^{∗}(M_{K}^{#}) ⊗ α_{K}^{∗}(N_{K}) is not projective. In this case neither α_{K}^{∗}(N_{K}) nor α_{K}^{∗}(M_{K}^{#}) is projective. Since Π(G) is irreducible, Corollary 4.4 implies that α_{K} has maximal Jordan types on both N and M^{#}. Hence, α_{K}^{∗}(M_{K}^{#}) ≃ (α_{K}^{∗}(M_{K}))^{#}. Therefore, Hom_{K}(α_{K}^{∗}(M_{K}), α_{K}^{∗}(N_{K})) ≃ (α_{K}^{∗}(M_{K}))^{#} ⊗ α_{K}^{∗}(N_{K}) ≃ α_{K}^{∗}(M_{K}^{#}) ⊗ α_{K}^{∗}(N_{K}) ≃ α_{K}^{∗}(Hom_{K}(M_{K}, N_{K})).

□

We now conclude that endotrivial modules are modules of constant Jordan type. The second statement of the theorem provides a “local” criterion of endotriviality, similar to the projectivity criterion given by the Dade’s lemma (see [12]).

Theorem 5.6. Let G be a finite group scheme, and let M be a finite dimensional kG-module.

(1) If M is endotrivial, then M has constant Jordan type of the form either m[p]+1[1] or m[p]+1[p−1] for some m ≥ 0, and thus α_{K}^{∗}(M_{K}) is endotrivial for every π-point α_{K} of G.

(2) Conversely, if α_{K}^{∗}(M_{K}) is endotrivial for each π-point α_{K} of G (and hence of the form either m[p] + 1[1] or m[p] + 1[p − 1]), then M is endotrivial.

Proof. Observe that any endotrivial module must have dimension whose square is congruent to 1 modulo p and thus must have dimension congruent to either 1 or p − 1 modulo p. To prove the first assertion, we assume that M is endotrivial, so that End_{k}(M) = k ⊕ (proj). Thus, α_{K}^{∗}(End_{K}(M_{K})) has Jordan type m[p] + 1[1] at each π-point α_{K}. In particular, every π-point α_{K} has maximal Jordan type on End_{k}(M). By Corollary 5.5, α_{K}^{∗}(End_{K}(M_{K})) ≃ End_{K}(α_{K}^{∗}(M_{K})). Hence, α_{K}^{∗}(M_{K}) is an endotrivial K[t]/t^{p}-module. The statement now follows from the fact that the only such modules are of the form m[p] + 1[1] or m[p] + 1[p − 1] for some m ≥ 0.

To prove the converse, we assume that α_{K}^{∗}(M_{K}) is an endotrivial K[t]/t^{p}-module for each π-point \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} . Thus, for each α_{K}, α_{K}^{∗}(M_{K}) has Jordan type either m[p] + 1[1] or m[p] + 1[p − 1] for some m ≥ 0. Since the dimension of M can not be congruent to both 1 and p − 1 modulo p, we conclude that M has constant Jordan type. Consider the short exact sequence

(11)  \xymatrix{0 \ar[r] & X \ar[r] & End_{k}(M) \ar[r]^{Tr} & k \ar[r] & 0,}

where Tr is the trace map. By Corollary 5.5, α_{K}^{∗}(End_{K}(M_{K})) ≃ End_{K}(α_{K}^{∗}(M_{K})). Moreover, because the dimension of End_{k}(M) is relatively prime to p, the trace map of (11) splits. Pulling back the split short exact sequence (11) via α_{K}, we conclude that

α_{K}^{∗}(X) ≃ ker\{End_{K}( \xymatrix{α_{K}^{∗}(M_{K})) \ar[r] & K} \} ≃ m[p]

is projective for all π-points α_{K} and thus X is projective ([20, 5.4]). Hence, M is endotrivial. □

6. Constructing modules of constant Jordan type

In this section, we consider two different methods of constructing modules of constant Jordan type. Proposition 6.1 presents the observation that an extension of modules of constant Jordan type has “total module” also of constant Jordan type if the extension splits when pulled back along any π-point. This observation fits well with the Auslander-Reiten theory of almost split exact sequences as we discuss in §8. Proposition 6.6 presents a method of constructing extensions of constant Jordan type whose pull-backs along π-points are not split.

We shall frequently utilize the Tate cohomology ring Ĥ^{∗}(G, k) for a finite group scheme G, and The Tate Ext groups Êxt^{∗}(M, N) for G-modules M, N. Tate cohomology ring Ĥ^{∗}(G, k) is a graded commutative k-algebra which coincides with regular cohomology in positive degrees. One advantage of Tate cohomology that we will exploit throughout this section is that it allows for arbitrary degree shifts. Namely, we have the following formulas

Êxt^{n}(M, N) ≃ Êxt^{0}(Ω^{n}M, N) = Hom_{stmod(kG)}(Ω^{n}M, N) ≃
Hom_{stmod(kG)}(M, Ω^{−n}N) = Êxt^{0}(M, Ω^{−n}N)

for any n ∈ ℤ. We refer the reader to [3] for further details.

Proposition 6.1. Suppose that G is a finite group scheme over k. Let M and N be kG-modules of constant Jordan type, and suppose that

(12)  \xymatrix{0 \ar[r] & M \ar[r] & B \ar[r] & N \ar[r] & 0}

is an exact sequence. Let ζ ∈ Ext_{kG}^{1}(N, M) be the class of (12). If for every π-point \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} the restriction α_{K}^{∗}(ζ) is zero, then B has constant Jordan type. Moreover, if the Jordan types of M and N are ∑_{i=1}^{p} m_{i}[i] and ∑_{i=1}^{p} n_{i}[i], then the Jordan type of B is ∑_{i=1}^{p}(m_{i} + n_{i})[i].

Proof. Because the cohomology class vanishes under restriction to a π-point α_{K}, we have that the restriction of (12) along α_{K} splits and α_{K}^{∗}(B_{K}) ≃ α_{K}^{∗}(M_{K})⊕α_{K}^{∗}(N_{K}). The result is now obvious. □

Proposition 6.1 does not always produce “new” examples of modules of constant Jordan type as we observe in the special case G = ℤ/2ℤ × ℤ/2ℤ.

Example 6.2. Let G be ℤ/2ℤ×ℤ/2ℤ. As presented in [2], [22], there is a complete classification of the kG-modules. Using this classification, we observe that the only indecomposable kG-modules of constant Jordan type are of the form Ω^{n}(k) for some n.

Namely, it is shown in [22] that all of the indecomposable kG-modules of odd dimension are of the form Ω^{n}(k) for some n. On the other hand, the non-projective indecomposable modules of even dimension are all isomorphic to L_{ζ} for ζ ∈ H^{n}(G, k).

The support varieties of these even dimensional modules are proper non-trivial subvarieties of Π(G), so that none of these have constant Jordan type.

It is instructive to look more closely to see why Proposition 6.1 does not determine other modules of constant Jordan type in this example. Observe that H^{∗}(G, k) ≅ Ext_{kG}^{∗}(k, k) is a polynomial ring in two variables having no non-zero element whose restriction to every π-point vanishes. Hence the only possible application of Proposition 6.1 would be in a situation where N ≅ Ω^{n}(k), M ≅ Ω^{m}(k) and n < m. Then (12) represents an element of negative Tate cohomology, ζ of Ĥ^{n−m+1}(G, k). By Proposition 6.3 which follows, ζ restricts to zero at every π-point, but the middle term of this non-split short exact sequence splits as Ω^{n+a}(k) ⊕ Ω^{n+b} ⊕ (proj) where a and b are nonnegative integers such that a + b = m − n (see the proof of Theorem 6.13).

The preceding example is special since ℤ/2ℤ×ℤ/2ℤ has tame representation type. For more general groups, the observation of Proposition 6.1 in conjunction with the following proposition does give new examples.

Proposition 6.3. Let G be a finite group scheme with the property that every πpoint factors by way of a flat map through a unipotent abelian group scheme whose cohomology has Krull dimension at least 2. Let ζ ∈ Ĥ^{n}(G, k) for n < 0 be an element in negative Tate cohomology of G corresponding to a short exact sequence of the form

(13)  \xymatrix{0 \ar[r] & k \ar[r] & E \ar[r] & Ω^{n−1}(k) \ar[r] & 0.}

Then for any π-point \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} , the pull-back of (13) along α_{K} is split (i.e., α_{K}^{∗}(ζ_{K}) = 0).

Proof. Recall that every element ζ of Ĥ^{n}(G, k) is represented by a homomorphism \xymatrix{ζ′ : k \ar[r] & Ω^{−n}(k)} . In order to prove the lemma, it is sufficient to show that for any

π-point \xymatrix{α_{K} : K[t]/(t^{p}) \ar[r] & KG} , the restriction of ζ′ factors through a projective K[t]/(t^{p})-module. We proceed to establish such a factorization using our knowledge of the modules Ω^{−n}(K) for n < 0. By Theorem 1.4, equivalent π-points induce the same map in cohomology. Thus, it suffices to prove the statement for some representative α_{K} of each [α_{K}] ∈ Π(G).

Our hypothesis immediately allows us to replace G by some abelian unipotent group scheme (defined over K) whose cohomology has Krull dimension at least 2. After possibly passing to some finite extension of K, [28, 14.1] enables us to conclude that KG ≃ K[T_{1}, …, T_{r}]/(T_{1}^{p^{e_{1}}}, …, T_{r}^{p^{er}}), where r ≥ 2. Because neither our hypothesis nor our conclusion depends upon the coalgebra structure on KG, we may assume that G is an abelian p-group. Let t_{i} = T_{i}^{p^{e_{i}−1}} and let E ⊂ G be the (unique) elementary abelian p-group with KE = k[t_{1}, …, t_{r}]/(t_{1}^{p}, …, t_{r}^{p}) ⊂ KG. By [18, 4.1], any K-rational π-point of KG has a representative factoring through KE. Thus, we may assume that G = E is an elementary abelian p-group of rank at least 2. Changing the generators of KE, we may further assume t_{1} = α_{K}(t); moreover, it suffices to assume that E has rank 2, for if the proposition is valid for an elementary abelian subgroup of G of rank 2, then it is valid for G itself. Thus, we are reduced to the case that KG is isomorphic to K[u, v]/(u^{p}, v^{p}) with u = α_{K}(t).

The structure of a minimal KG-projective resolution P_{•} → K is well known [11], with P_{m−1} = KG^{×m}. A set of generators a_{1}, …, a_{m} for P_{m−1} can be chosen so that Ω^{m}(K) is the submodule generated by the elements

u^{p−1}a_{1}, va_{1} − ua_{2}, v^{p−1}a_{2} − u^{p−1}a_{3}, va_{3} − ua_{4}, …, va_{m−1} − ua_{m}, v^{p−1}a_{m}

for m even and

ua_{1}, va_{1} − u^{p−1}a_{2}, v^{p−1}a_{2} − ua_{3}, va_{3} − u^{p−1}a_{4}, …, v^{p−1}a_{m−1} − ua_{m}, va_{m}

for m odd. Every KG-fixed point of P_{m−1}, and thus also of Ω^{m}(K) is a linear combination of the elements u^{p−1}v^{p−1}a_{i}. Moreover, for i odd and m even,

u^{p−1}v^{p−1}a_{i} = u^{p−1}v^{p−2}(va_{i} − ua_{i+1})

and for i even and m even,

u^{p−1}v^{p−1}a_{i} = u^{p−1}(v^{p−1}a_{i} − u^{p−1}a_{i+1}).

Similar formulas hold for m odd. Consequently, every KG-fixed point of Ω^{m}(K) has the form u^{p−1}z for some z ∈ Ω^{m}(K). Thus, as a map of K[t]/t^{p}-modules (via \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} ), ζ′_{K} factors as the composition of \xymatrix{K \ar[r] & K[t]/t^{p}} sending 1

to t^{p−1} and \xymatrix{K[t]/t^{p} \ar[r] & Ω^{m}(K)} sending 1 to z. Hence, α_{K}^{∗}(ζ′) factors through the rank 1 projective module K[t]/t^{p} as required. □

The following property is an immediate corollary of Proposition 6.3.

Corollary 6.4. Let E be an elementary abelian p-group of rank at least 2, and let

\xymatrix{θ : Ω^{m}(k) \ar[r] & Ω^{n}(k)}

be a homomorphism. If the restriction of θ along some π-point α_{K} : K[t]/t^{p} → KE does not factor through a projective K[t]/t^{p}-module, then m ≥ n.

The following proposition shows that there are limitations on the Jordan types which can be realized as extensions.

Proposition 6.5. Let G be a finite group scheme over a field k of characteristic p > 2 with the property that every π-point factors by way of a flat map through a unipotent abelian group scheme whose cohomology has Krull dimension at least 2. Then there does not exist a kG-module M of constant Jordan type n[p] + 1[2] which is an extension of a kG-module of constant Jordan type m[p] + 1[1] and one of constant Jordan type (n − m)[p] + 1[1].

Proof. As argued above in the proof of Proposition 6.3, we may assume that G is an elementary abelian p-group of rank 2. Consider a short exact sequence of the form

(14)  \xymatrix{E :  0 \ar[r] & L \ar[r] & M \ar[r] & N \ar[r] & 0}

in which both L and N have stable constant Jordan type 1[1]. By Theorem 5.6, L and N are endotrivial modules, so that L ≅ Ω^{a}(k), N ≅ Ω^{b}(k) for some integers a and b. Since p > 2, both a and b must be even integers.

Thus the class of (14) is a cohomology class ζ in

Ext_{kG}^{1}(Ω^{b}(k), Ω^{a}(k)) ≅ Ĥ^{b−a+1}(G, k).

If M has constant Jordan type n[p] + 1[2], then the restriction of (14) along any π-point α_{K} does not split; equivalently, the restricted class α_{K}^{∗}(ζ) does not vanish. However, the only such cohomology classes ζ are scalar multiples of the identity in degree b − a + 1 = 0. Since both a and b are even integers, such classes ζ can not occur. □

We now proceed to describe a second method of constructing modules of stable constant Jordan type n[1] that cannot be created by the methods of Proposition 6.1. The construction can be summarized as follows: if a certain map represented by a matrix with coefficients in Ĥ^{∗}(G, k) has the maximal possible rank when restricted to any π-point of G, then it has a kernel of constant Jordan type.

Theorem 6.6. Let G be a finite group scheme, and choose integers m > n, m_{i} and n_{j} such that all m_{i} and n_{j} are even if p > 2. Choose cohomology classes ζ_{i,j} ∈ Ĥ^{m_{j}−n_{i}}(G, k) and let ζ̂_{i,j} : Ω^{m_{j}}(k) → Ω^{n_{i}}(k) represent ζ_{i,j}. We consider an exact sequence

\xymatrix{0 \ar[r] & L \ar[r] & M \ar[r]^{φ=(ζ̂_{i,j})} & N \ar[r] & 0}

of kG-modules where

M ≅ ∑_{j=1}^{m} Ω^{m_{j}}(k) ⊕ (proj) and N ≅ ∑_{i=1}^{n} Ω^{n_{i}}(k).

Assume that for every π-point \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} , the restriction of the matrix

of cohomology elements (α_{K}^{∗}(ζ_{i,j})) ∈ M_{n,m}(Ĥ^{∗}(K[t]/t^{p}, K)) has rank n. Then the module L has stable constant Jordan type (m − n)[1].

Proof. Let \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} be a π-point. For any s, which is even if p > 2, the restriction of the module Ω^{s}(k) along α_{K} has the form α_{K}^{∗}(Ω^{s}(K)) ≅ K ⊕(proj). Moreover, if we have a map \xymatrix{ζ : Ω^{s}(k) \ar[r] & Ω^{t}(k)} whose cohomology class when restricted along α_{K} is not zero, then the composition

\xymatrix{K \ar[r]^{ι} & α_{K}^{∗}(Ω^{s}(K)) \ar[r]^{α_{K}^{∗}(ζ_{K})} & α_{K}(Ω^{t}(K)) \ar[r]^{ρ} & K,}

where ι and ρ are the split inclusion and projection maps, is an isomorphism. Thus our hypothesis asserts that the composition

\xymatrix{K^{m} \ar[r]^{ι} & α_{K}^{∗}(M_{K}) \ar[r]^{α^{∗}(φ_{K})} & α_{K}^{∗}(N_{K}) \ar[r]^{ρ} & K^{n}}

has rank n. Since α_{K}^{∗}(M_{K}) = (proj) +m[1] and α_{K}^{∗}(N_{K}) = (proj) +n[1], it follows that the kernel of φ, α_{K}^{∗}(L), has stable Jordan type (m − n)[1]. As this happens for any α_{K}, we are done. □

Theorem 6.6 is stated for even dimensional cohomology classes (for p > 2), yet as we observe in the following proposition a very similar argument can also be applied to odd dimensional classes to yield additional modules of constant Jordan type.

Proposition 6.7. Let G be a finite group scheme, and choose positive integers m > n, m_{1}, … m_{m} all odd, n_{1}, …, n_{n} all even. Assume p > 2. Choose cohomology classes ζ_{i,j} ∈ Ĥ^{m_{j}−n_{i}}(G, k) and let ζ̂_{i,j} : Ω^{m_{j}}(k) → Ω^{n_{i}}(k) represent ζ_{i,j}. We consider an exact sequence

\xymatrix{0 \ar[r] & L \ar[r] & M \ar[r]^{φ=(ζ̂_{i,j})} & N \ar[r] & 0}

of kG-modules where

M ≅ ∑_{j=1}^{m} Ω^{m_{j}}(k) ⊕ (proj) and N ≅ ∑_{i=1}^{n} Ω^{n_{i}}(k).

Assume that for every π-point \xymatrix{α_{K} : K[t]/(t^{p}) \ar[r] & KG} , the restriction of the ma-

trix of cohomology elements α_{K}^{∗}(ζ_{i,j}) ∈ M_{n,m}(Ĥ^{∗}(K[t]/t^{p}, K)) has rank n. Then the module L has stable constant Jordan type (m − n)[p − 1] + n[p − 2].

Proof. Our hypothesis implies that the restriction of (ζ̂_{i,j}) via any α_{K} is a map of K[t]/t^{p}-modules which remains surjective after free summands are dropped. We write such a map of K[t]/t^{p}-modules symbolically as a map of their (stable) Jordan types m[p − 1] ↠ n[1]. Such a surjective map of K[t]/t^{p}-modules with indicated stable Jordan type necessarily has kernel with Jordan type (m−n)[p−1]+n[p−2]. □

In general, it is very easy to construct examples for which Theorem 6.6 and Proposition 6.7 are relevant. These modules are multi-parametrized versions of important modules introduced and studied by the first author.

Example 6.8. Let ξ_{1}…. , ξ_{r} be homogeneous elements in H^{•}(G, k) such that the radical of the ideal generated by the ξ_{i}’s is the augmentation ideal of H^{•}(G, k). Alternatively, for p odd, let ξ_{1}…. , ξ_{r} be homogeneous elements in H^{∗}(G, k) of odd degree whose Bocksteins generate an ideal of H^{•}(G, k) whose radical is the augmentation ideal. For each i, let \xymatrix{ξ̂_{i} : Ω^{n_{i}}(k) \ar[r] & k} be a cocycle representing ξ_{i}, where n_{i} is the degree of ξ_{i}. Define

φ : ⨁_{i=1}^{r} Ω^{n_{i}}(k) → k

by the formula φ(a_{1}, …, a_{r}) = ξ̂_{1}(a_{1}) + ⋯ + ξ̂_{r}(a_{r}). The condition that either the ξ_{i}’s or the Bockstein’s of the ξ_{i}’s generate an ideal of H^{•}(G, k) whose radical is the augmentation ideal implies that for each π-point α_{K}, there is some i such that α_{K}^{∗}(ξ̂) ≠ 0. (Namely, no point [α_{K}] ∈ Π(G) ≅ Proj H^{•}(G, k) lies in the union of the zero loci of either the ξ_{i}’s or the Bockstein’s of the ξ_{i}’s.) Thus,

(15)  L_{ξ_{1},...,ξ_{r}} ≡ Ker φ

has constant Jordan type.

In Theorem 6.13 below, we give an explicit example of such an L_{ξ_{1},...,ξ_{r}} which can not be constructed using the technique of Proposition 6.1. The detailed verification of this example will occupy the remainder of this section, and involves the following four lemmas.

Recall that if E ≃ (ℤ/pℤ)^{×r} is an elementary abelian p-group of rank r, then

(16)  H^{∗}(E, k) ≅ \begin{cases}k[ζ_{1}, …, ζ_{r}] ⊗ Λ(η_{1}, …, η_{r}) & if p > 2 \\ k[η_{1}, …, η_{r}] & if p = 2. \end{cases}

where each η_{j} has degree one and each ζ_{i} has degree 2. Here, Λ(η_{1}, …, η_{r}) is the exterior algebra on the generators η_{1}, …, η_{r}.

Lemma 6.9. Let E be an elementary abelian p-group of rank r, for r > 1. For n > 0,

(1) dim H^{n}(E, k) = \binom{n+r−1}{r−1}.

(2) dim P_{n} = \binom{n+r−1}{r−1}·p^{r}, where P_{n} is the n-th term of a minimal kE-projective resolution of k.

(3) dim Ω^{n}(k) = p^{r} · a_{r,n} + (−1)^{n}, n > 0 where

a_{r,n} = \binom{n + r − 2}{r − 1} − \binom{n + r − 3}{r − 1} + ⋯ + (−1)^{n−1}\binom{r − 1}{r − 1}.

(4) If r = 2, then dim(Ω^{2n}(k)) = p^{2}n + 1.

(5) If r = 3, then dim(Ω^{2n}(k)) = p^{3}n(n + 1) + 1 Finally, if n < 0, then dim(Ω^{n}(k)) = dim(Ω^{−n}(k)).

Proof. The computation of dim H^{n}(E, k) is a straightforward and familiar computation. In a minimal projective kE-resolution of k,

\xymatrix{… \ar[r] & P_{2} \ar[r]^{∂_{2}} & P_{1} \ar[r]^{∂_{1}} & P_{0} \ar[r]^{ε} & k \ar[r] & 0,}

each P_{n} is a direct sum of copies of kE, the number of copies equal to the dimension of H^{n}(E, k). Using the vanishing of the Euler characteristic of an exact sequence, we conclude the asserted formula for the dimension of Ω^{n}(k). The assertions for r = 2, 3 are special cases. □

Lemma 6.10. Let E be an elementary abelian p-group of rank 3, n be a negative integer, and \xymatrix{Q \ar[r] & Ω^{2n}(k)} be the projective cover of the 2n^{th} Heller shift of k. Then dim Q = p^{3}(2n^{2} − n).

Proof. Taking duals we have an injection \xymatrix{Ω^{−2n}(k) \ar[r] & Q^{∗}} , where Q^{∗} is the injective hull of Ω^{−2n}(k). Hence, Q^{∗} ≃ P_{−2n−1}, the (−2n − 1)^{st} term of the minimal projective resolution of k. As seen in Lemma 6.9(2), the dimension of P_{−2n−1} is p^{3}\binom{−2n−1+2}{2} = p^{3}(2n^{2} − n). □

Lemma 6.11. Suppose that E is an elementary abelian p-group of rank at least 2. Let

\xymatrix{θ : Ω^{m}(k) \ar[r] & Ω^{n}(k)}

be a homomorphism for some nonnegative integers m and n, which are both even in case p > 2. Assume that for some π-point \xymatrix{α_{K} : K[t]/(t^{p}) \ar[r] & KE} the restriction along α of θ does not factor through a projective K[t]/(t^{p})-module. Then θ is surjective.

Proof. By Corollary 6.4, we have m − n ≥ 0. Let θ̂ denote the cohomology class of θ in

Ext_{kE}^{m−n}(k, k) ≅ H^{m−n}(E, k).

The condition on the restriction of θ along α together with the fact that m−n is even if p > 2 implies that θ̂ is a non-nilpotent element of H^{m−n}(E, k). Hence, multiplication by θ̂induces an injective map \xymatrix{H^{n}(E, k) \ar[r] & H^{m}(E, k)} . Since Hom_{kE}(Ω^{ℓ}(k), k) ≅ H^{ℓ}(E, k) for ℓ ≥ 0, it follows that θ induces an injective map

\xymatrix{θ′ : Hom_{kE}(Ω^{n}(k), k) \ar[r] & Hom_{kE}(Ω^{m}(k), k)} .

So θ must be surjective. □

Lemma 6.12. Suppose that E is an elementary abelian p-group of rank 2. Let ξ_{1} ∈ H^{2m}(E, k), ξ_{2} ∈ H^{2n}(E, k), and assume that the radical of the ideal generated by ξ_{1}, ξ_{2} is the augmentation ideal of H^{•}(E, k). Consider the exact sequence

\xymatrix{0 \ar[r] & L_{ξ_{1},ξ_{2}} \ar[r] & Ω^{2m}(k) ⊕ Ω^{2n}(k) \ar[r]^{^{(\begin{matrix}ξ_{1} \\ ξ_{2} \end{matrix})}} & k \ar[r] & 0.}

Then L_{ξ_{1},ξ_{2}} ≃ Ω^{2m+2n}(k).

Proof. The condition on ξ_{1}, ξ_{2} is equivalent to the condition that the matrix (ξ_{1}, ξ_{2}) has rank 1 when restricted to any π-point of G. Hence, by Theorem 6.6, L_{ξ_{1},ξ_{2}} has stable constant Jordan type 1[1], and it is an endotrivial module. By Lemma 6.9, the dimension of L_{ξ_{1},ξ_{2}} is p^{2}(m + n) + 1, and hence L_{ξ_{1},ξ_{2}} ≃ Ω^{±(2m+2n)}(k).

Now suppose that L_{ξ_{1},ξ_{2}} ≃ Ω^{−(2m+2n)}(k). Then the sequence represents a nonzero element γ ∈ H^{2m+2n+1}(E, k) which has the property that α_{K}^{∗}(γ_{K}) is zero for any π-point \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KE} . However, because 2m + 2n + 1 is both positive and odd, there is no such element. Therefore, we must have that L_{ξ_{1},ξ_{2}} ≃ Ω^{2m+2n}(k) as desired. □

As we show in the following proposition, there exist examples of modules of constant Jordan types constructed as in Proposition 6.6 which are not middle terms of extensions of endotrivial modules as in Proposition 6.1.

Theorem 6.13. Let E be an elementary abelian p-group of rank 3 and consider the kE-module L = L_{ζ_{1},ζ_{2},ζ_{3}} as in (15), where {ζ_{i}}_{i=1,2,3} ⊂ H^{2}(E, k) form a system of generators of H^{•}(E, k)_{red}. Then there does not exist a projective kE-modules P such that M = L ⊕ P fits in a short exact sequence of the form

(17)  \xymatrix{0 \ar[r] & L′ \ar[r] & M \ar[r] & L′′ \ar[r] & 0}

with both L′, L′′ endotrivial modules.

Proof. Observe that M has stable Jordan type 2[1], so that in any short exact sequence of the form (17) both L′, L′′ must have stable Jordan type 1[1]. We assume that such a short exact sequence exists and proceed to obtain a contradiction. First, by eliminating projective summands we may reduce to the case when both L′ and L′′ are projective-free. Since E is a p-group, we have P ≅ (kE)^{t} for some t ≥ 0. Since L′, L′′ have stable constant Jordan type 1[1], they are endotrivial by Theorem 5.6. If p > 2, then we immediately conclude L′ ≅ Ω^{2n_{1}}(k), L′′ ≅ Ω^{2n_{2}}(k) for some integers n_{1}, n_{2} since L′, L′′ have stable Jordan type 1[1]. If p = 2, then dim L′ + dim L′′ = dim L + dim P ≡ 2 (mod dim kE). Since dim Ω^{n}(k) ≡ (−1)^{n} (mod dim kE), we get that L′, L′′ must be even syzygies in the case p = 2 as well. Thus, the sequence becomes

(18)  \xymatrix{0 \ar[r] & Ω^{2n_{1}}(k) \ar[r] & L ⊕ (kE)^{t} \ar[r] & Ω^{2n_{2}}(k) \ar[r] & 0}

Let {g_{1}, g_{2}, g_{3}} be group generators of E, and consider the subgroup F = ⟨g_{1}, g_{2}⟩ ⊆ E. The restriction of ζ_{3} to F vanishes, so that

L_{|F} ≅ Ω^{2}(k) ⊕ L_{|F,ζ_{1},ζ_{2}} ⊕ (proj),

where L_{|F,ζ_{1},ζ_{2}} is constructed as in (15) with respect to the group F. By Lemma 6.12, L_{|F,ζ_{1},ζ_{2}} ≅ Ω^{4}(k) as a kF -module. Consequently,

L_{|F} ≅ Ω^{2}(k) ⊕ Ω^{4}(k) ⊕ (proj) .

Restricting (18) to F and eliminating the projective summands at the ends, we obtain an exact sequence of kF -modules

(19)  \xymatrix{0 \ar[r] & Ω^{2n_{1}}(k) \ar[r] & Ω^{2}(k) ⊕ Ω^{4}(k) ⊕ (proj) \ar[r] & Ω^{2n_{2}}(k) \ar[r] & 0}

By performing a shift and eliminating excess projectives, we get the sequence of kF -modules

(20) \xymatrix{0 \ar[r] & Ω^{2n_{1}−2n_{2}}(k) \ar[r] & Ω^{4−2n_{2}}(k) ⊕ Ω^{2−2n_{2}}(k) ⊕ (proj) \ar[r]^{θ} & k \ar[r] & 0}

where θ restricts to θ_{1} on Ω^{4−2n_{2}}(k) and to θ_{2} on Ω^{2−2n_{2}}(k). Since the kernel of θ has stable Jordan type 1[1], at least one of θ_{1}, θ_{2} does not factor through a projective module when restricted to any π-point of F. Hence, Proposition 6.3 implies that n_{2} ≤ 2. By the same argument applied to the other end of Sequence (20), we have that 2n_{1} − 2n_{2} cannot be less than 2 − 2n_{2} (i.e., n_{1} ≥ 1). We further observe that if there were a non-trivial projective summand in the middle term of (20), then θ restricted to the projective summand would be surjective, and, hence, the kernel would have stable Jordan type different from 1[1]. It follows that there is no projective summand in the middle term of (20). Hence, we get p^{3}(|4 − 2n_{2}| + |2 − 2n_{2}|) + 2 = dim(Ω^{4−2n_{2}}(k) ⊕ Ω^{2−2n_{2}}(k)) = dim Ω^{2n_{1}−2n_{2}}(k) + 1 = p^{3}(|2n_{1} − 2n_{2}|) + 2 by Lemma 6.9. Hence,

(21)  |2 − n_{2}| + |1 − n_{2}| = |n_{1} − n_{2}|

We conclude that n_{1} +n_{2} = 3 when n_{2} ≤ 1. Moreover, n_{1} and n_{2} must have different parity.

We consider two cases:

(I) n_{2} ≥ 0, and (II) n_{2} < 0.

Case I: n_{2} ≥ 0. We first show that the middle term of the sequence (18) does not have a projective summand, that is t = 0.

The sequence (18) represents a cohomology class in Ĥ^{2n_{2}−2n_{1}+1}(E, k) which we denote by η. Restriction of (18) to any π-point of E has the form

\xymatrix{0 \ar[r] & n[p] + 1[1] \ar[r] & (n + m)[p] + 2[1] \ar[r] & m[p] + 1[1] \ar[r] & 0}

This sequence of ℤ/pℤ-modules is necessarily split. Hence, η vanishes upon restriction to any π-point of E.

Applying Hom_{stmod(kE)}(−, k) to the short exact sequence (18), viewed as a distinguished triangle in stmod(kE), we obtain a long exact sequence

\xymatrix{… \ar[r] & Êxt_{kE}^{−1}(Ω^{2n_{1}}(k), k) \ar[r]^{δ} & Êxt_{kE}^{0}(Ω^{2n_{2}}(k), k) \ar[r] & Êxt_{kE}^{0}(L, k) \ar[r] & ….}

which is equivalent to

(22)  \xymatrix{… \ar[r] & H^{2n_{1}−1}(E, k) \ar[r]^{·η} & H^{2n_{2}}(E, k) \ar[r] & Êxt_{kE}^{0}(L, k) \ar[r] & ….}

The rank of the free summand (kE)^{t} in the middle term of the sequence (18) equals

the dimension of the kernel of the map \xymatrix{Êxt_{kE}^{0}(Ω^{2n_{2}}(k), k) \ar[r] & Êxt_{kE}^{0}(L, k)} . Hence, it also equals the dimension of the image of the connecting homomorphism δ, which is multiplication by η, a cohomology class of degree deg η = 2n_{2}−2n_{1}+1. Therefore, to show that t = 0 we need to show that multiplication by η on H^{2n_{1}−1}(E, k) is trivial.

If 2n_{2} − 2n_{1} + 1 > 0 then for p > 2 the fact that η vanishes upon restriction to any π-point implies that η is divisible by the product η_{1}η_{2}η_{3} where η_{1}, η_{2}, η_{3} are the nilpotent generators in degree 1 of H^{∗}(E, k). Thus, the multiplication by η of any odd-dimensional class of positive degree is zero. Hence, the image of ·η on H^{2n_{1}−1}(E, k) is trivial since 2n_{1} − 1 ≥ 1. If p = 2 then the only class which vanishes upon restriction to any π-point is the zero class. We conclude that t = 0 in this case.

Now assume that 2n_{2} − 2n_{1} + 1 < 0. Since cohomology of E is not periodic, multiplication by η on H^{− deg η}(E, k) is trivial ([4, 2.2]). Since any monomial in H^{2n_{1}−1}(E, k) factors as a product of a class of degree − deg η = 2n_{1} − 2n_{2} − 1 and a class of degree 2n_{2}, we further conclude that multiplication by η on H^{2n_{1}−1}(E, k) is trivial. Hence, t = 0 in this case as well.

We now compare the dimensions of the terms of the sequence (18) in which we take t = 0. Note that n_{1} and n_{2} are both non-negative in the case that we are considering. We get

6p^{3} + 2 = dim L = dim Ω^{2n_{1}}(k) + dim Ω^{2n_{2}}(k) = n_{1}(n_{1} + 1)p^{3} + 1 + n_{2}(n_{2} + 1)p^{3} + 1

using Lemma 6.9. Hence, we get the following equation on n_{1}, n_{2}:

n_{1}^{2} + n_{2}^{2} + n_{1} + n_{2} = 6

The only non-negative integer solutions are (2, 0) and (0, 2). This is impossible since (n_{1}, n_{2}) must have the opposite parity by (21). Hence, we obtain a contradiction as desired, and the case n_{2} ≥ 0 is finished.

Case II. We now consider the case of negative n_{2}. The projective summand (kE)^{t} in the middle term of the sequence (18) can not be any bigger than a projective cover of Ω^{2n_{2}}(k) as otherwise, kernel of the map \xymatrix{(kE)^{t} \ar[r] & Ω^{2n_{2}}(k)} has a projective submodule which is then a direct summand of the right hand term of (18). By Lemma 6.10, t ≤ 2n_{2}^{2} − n_{2}. Hence,

dim L ≥ dim Ω^{2n_{2}}(k) + dim Ω^{2n_{1}}(k) − p^{3}(2n_{2}^{2} − n_{2})
= p^{3}(n_{2}^{2} − n_{2}) + p^{3}(n_{1}^{2} + n_{1}) + 2 − p^{3}(2n_{2}^{2} − n_{2})

Simplifying, and using the fact that n_{2} + n_{1} = 3, we get

dim L ≥ p^{3}(n_{1}^{2} − n_{2}^{2} + 3 − n_{2}) + 2
= p^{3}(3(n_{1} − n_{2}) + 3 − n_{2}) + 2 = p^{3}(3n_{1} + 3 − 4n_{2}) + 2

Since n_{1} ≥ 1, and n_{2} < 0, we conclude that dim L ≥ 10p^{3} + 2. On the other hand, dim L = 6p^{3} + 2 by Lemma 6.9 and the definition of L, and we have a contradiction. □

7. Constraint on ranks

A consequence of our constructive techniques is a new proof of a special case of Macaulay’s Generalized Principle Ideal Theorem. The point is that if the coefficients are in a field of finite characteristic, then we can represent homogeneous elements of a multivariable polynomial ring as elements in the cohomology ring of an elementary abelian p-group. We can represent a matrix of such elements as a map of modules over the group algebra. Specifically, we have the following

Theorem 7.1. (See Exercise 10.9 of [13]) Suppose that k is an algebraically closed field. Fix integers n, r and d_{1}, …, d_{r+1}, with n ≥ 3, r ≥ 2, and d_{j} > 0 for all j. Let P = k[x_{1}, …, x_{n}] be a polynomial ring in n variables. Let A = A(x_{1}, …, x_{n}) be a (r + 1) × r matrix with the property that every entry in column i of A is a homogeneous polynomial in P of degree d_{j} for all j = 1, …, r + 1. Then there is some point α ∈ k^{n}\{0} such that A(α) has rank less than r. Equivalently, the determinants of the r × r minors of A (which are elements of P) have a common non-trivial zero.

Proof. Assume first that the characteristic of k is p > 0, as in the rest of the paper. Let E denote an elementary abelian p-group of rank n. As recalled in (16), H^{∗}(E, k) contains a polynomial subring Q ≅ k[ζ_{1}, …, ζ_{n}], where the elements ζ_{i} are in degree 2 if p > 2 and in degree 1 if p = 2. For the purposes of the argument, we assume that p > 2. The proof in the even characteristic case is very similar.

Let A = (a_{i,j}) where for each i and j, a_{i,j} = a_{i,j}(x_{1}, …, x_{n}) is a homogeneous polynomial of degree d_{j}. Then, a_{i,j}(ζ_{1}, …, ζ_{n}) is an element in H^{2d_{j}}(E, k). Moreover, such an element is uniquely represented by a cocycle

\xymatrix{a′_{i,j}(ζ_{1}, …, ζ_{n}) : Ω^{2d_{j}}(k) \ar[r] & k.}

Now we let A′ be the map

\xymatrix{A′ : ⨁_{t=1}^{r+1} Ω^{2d_{t}}(k) \ar[r] & k^{r}}

whose matrix is A′ = (a′_{i,j}(ζ_{1}, …, ζ_{n})).

We proceed to prove the theorem by contradiction, observing that if A(α) had rank r for all α ∈ k^{n}\{0}, then, because k is algebraically closed (so that the krational points of Π(E) are dense), the kernel L of A′ would be a module of constant Jordan type (with stable Jordan type 1[1]) as in Theorem 6.6. By Theorem 5.6, L

is an endotrivial module. Hence, L ≅ Ω^{2m}(k) + (proj) for some m. Since ⨁_{t=1}^{r+1} Ω^{2d_{t}}(k)

does not have projective summands, we conclude that L ≅ Ω^{2m}(k).

As in the proof of Theorem 6.13, we can use a dimension argument to ascertain the value of m. Let H ⊂ E be an elementary abelian subgroup of rank 2. Restricting to H and eliminating the projective summand in L and in the domain of A′, we get that the dimension of the projective-free part of L ↓_{H} is precisely p^{2} ∑_{i=1}^{r+1} d_{i} + 1 by Lemma 6.9(4). Consequently, by the same lemma m = ∑_{i=1}^{r+1} d_{i}.

Now let H′ ⊂ E be an elementary abelian p-subgroup of rank 3, and let L′ be the projective-free part of the restriction of L to H′. Since L′ ≃ Ω^{2m}(k) as H′-modules, Lemma 6.9(5) implies that

(23)  dim L′ = p^{3}m(m + 1) + 1 = p^{3}(∑_{i=1}^{r+1} d_{i})(∑_{i=1}^{r+1} d_{i} + 1) + 1

On the other hand, L′ is the projective-free part of the kernel of the map A′ restricted to H′. Applying Lemma 6.9(4) to compute the dimension of the H′-module

⨁_{t=1}^{r+1} Ω^{2d_{i}}(k) we get

(24)  dim L′ = p^{3} ∑_{t=1}^{r+1} d_{i}(d_{i} + 1) + 1.

As all d_{i} > 0, the formula (23) clearly yields a greater value than (24). Thus, we get a contradiction.

Now, we consider a field of characteristic 0, still denoted k. Let R ⊂ k be the ring finitely generated over ℤ by the coefficients of the (homogeneous polynomial) entries of A. The r + 1 determinants of the r × r minors of A define a closed subscheme Z of the projective scheme ℙ_{R}^{n−1}. By the preceding argument for fields of positive characteristic, Z intersects each geometric fiber of \xymatrix{ℙ_{R}^{n−1} \ar[r] & Spec R} above a point of Spec R with positive residue characteristic.

Observe that the geometric points of Spec R whose residue characteristics are positive are dense in Spec R. Hence, the image of Z ⊂ ℙ_{R}^{n−1} in Spec R is both closed and dense and thus all of Spec R. We conclude that Z intersects every geometric fiber of \xymatrix{ℙ_{R}^{n−1} \ar[r] & Spec R} , including that given by \xymatrix{R \ar[r] & k} . □

8. Auslander Reiten Components

Our objective in this section is to show how the Auslander-Reiten theory of almost split sequences can be used to construct modules having certain specific constant Jordan types. One of our main results is that if one module in a component of the Auslander-Reiten quiver of kG has constant Jordan type and if k is perfect, then every module in the component has constant Jordan type.

We begin this section by briefly recalling a few basic facts concerning the Auslander-Reiten theory of almost split sequences, sometimes called Auslander-Reiten sequences. We refer the reader to [1] or [3, I.4] for more detailed accounts.

Suppose that A is a finite dimensional algebra over the field k. A sequence of A-modules

\xymatrix{0 \ar[r] & L \ar[r]^{ζ} & M \ar[r]^{γ} & N \ar[r] & 0}

is an almost split sequence if M and N are indecomposable, the sequence is not split, and one of the following two equivalent conditions holds:

(1) Any A-module map σ : X ⟶ N which is not a splittable epimorphism, admits a factorization σ = γθ for some homomorphism θ : X ⟶ M.

(2) Any A-module map σ : L ⟶ Y which is not a splittable monomorphism,

admits a factorization σ = θζ for some homomorphism θ : M ⟶ Y. (see [1, V.1]). If M is an indecomposable non-projective A-module, then the almost split sequence of A-modules ending in M is unique up to isomorphism and has the form

\xymatrix{0 \ar[r] & τM \ar[r] & B \ar[r] & M \ar[r] & 0}

where τ = D ∘ Tr is the translation operator on the set of isomorphism classes of indecomposable non-projective A-modules defined as the composition of the transpose and duality functors (see [1, VII.1]). Similarly, there is a unique up to isomorphism almost split sequence starting with an indecomposable non-injective module L. If A is a symmetric algebra (in particular, if A = kG for a finite group G), then τ = Ω^{2}. We give description of τ in the case of any finite group scheme G in the proof of Lemma 8.6.

Let L, M be indecomposable A-modules. A map \xymatrix{γ : L \ar[r] & M} is an irreducible map if the following two conditions hold:

(a) γ is neither a splittable monomorphism nor a splittable epimorphism

(b) for any factorization γ = μν either μ is a splittable monomorphism (has a

right inverse) or ν is a splittable epimorphism (has a left inverse) (see [1, V.5]). Irreducible maps are closely related to almost split sequences as the following lemma indicates.

Lemma 8.1. ([1, V.5.3]) Let 0⟶L ⟶^{ζ} M ⟶^{γ} N ⟶ 0 be an almost split se-

quence, and let M = ⨁_{i=1}^{r} M_{i} where each M_{i} is an indecomposable module. Then the maps \xymatrix{ζ_{i} : L \ar[r] & M_{i}} and \xymatrix{γ_{i} : M_{i} \ar[r] & N} that make up ζ and γ are irreducible maps. Moreover, every irreducible map occurs in such a way in some almost split sequence.

The next result shows that irreducible maps originating in projective modules, and consequently, almost split sequences with projective summands in the middle term have a very special form. We now take the algebra A to be the group algebra kG for a finite group scheme G. Since kG is a Frobenius algebra, projective modules are injective and vice versa.

Lemma 8.2. ([1, V.5.5]) Let G be a finite group scheme, and P be an indecomposable projective kG-module. Then, up to scalar multiple, the only irreducible map originating at P is the quotient map \xymatrix{P \ar[r] & P/ soc(P)} . Consequently, the only almost split sequence in which P occurs is the almost split sequence ending in P/ soc(P). This sequence has the form

(25)  \xymatrix{0 \ar[r] & rad(P) \ar[r] & P ⊕ rad(P)/ soc(P) \ar[r] & rad(P)/ soc(P) \ar[r] & 0}

For a finite group scheme G, the stable Auslander-Reiten quiver of G, denoted Γ_{G}^{s}, is a valued quiver whose vertices are isomorphism classes of indecomposable nonprojective kG-modules, and whose arrows are induced by irreducible morphisms (see [1, VII.1,4]). Namely, if there is an irreducible map M → N for non-projective indecomposable modules M,N, then there is an arrow \xymatrix{[M] \ar[r] & [N]} in the Auslander-

Reiten quiver of G. By Lemma 8.1, there is an arrow \xymatrix{[M] \ar[r] & [N]} if and only if

there exists an almost split sequence 0 ⟶ τN ⟶ M̃ ⟶ N ⟶ 0 such that M is an indecomposable direct summand of M̃. A (stable) Auslander-Reiten component of an indecomposable non-projective module M is a connected component of the Auslander-Reiten quiver Γ_{G}^{s} containing the vertex [M]. In addition, the Auslander-Reiten quiver is equipped with the translation automorphism \xymatrix{τ : Γ_{G}^{s} \ar[r] & Γ_{G}^{s}} induced by the translation operator τ.

An Auslander-Reiten component Θ of Γ_{G}^{s} has the form

(26)  Θ ≃ ℤ[Δ⃗_{Θ}]/H

where Δ⃗_{Θ} is a directed tree, and H ⊂ Aut(ℤ[Δ⃗_{Θ}]) is a group acting “admissibly” on Δ⃗_{Θ} (see [1, VII.4] for definitions and [3, I.4.15.6] for the structure isomorphism). The underlying non-directed tree Δ_{Θ} is uniquely determined by the component Θ and Θ is said to have tree class or type of Δ_{Θ}.

Later in the section we use the following result, which is due to Erdmann [14] in the case of finite groups. For arbitrary finite group schemes, there are similar results of Farnsteiner [16], [17].

Theorem 8.3. [14] Suppose that G is a finite group and that M is a kG-module which lies in a block of wild representation type. Then the Auslander-Reiten component of M has tree class A_{∞}.

The tree class A_{∞} tells us a lot about the structure of the almost split sequences. The vertices of a stable Auslander-Reiten component of such a tree class are obtained by iterated applications of the translation operator τ to a distinguished collections of vertices which correspond to isomorphism classes of non-projective indecomposable modules X_{0}, X_{1}, X_{2}, … such that X_{0} is the beginning vertex and for each i ≥ 0 there is an irreducible map X_{i} → X_{i+1}. That is, we have a tree

\xymatrix{Δ : X_{0} \ar[r] & X_{1} \ar[r] & … \ar[r] & X_{i} \ar[r] & X_{i+1} \ar[r] & …}

such that the component is isomorphic to ℤ[Δ⃗]/H as in (26). An almost split sequence involving modules in the component has the form

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

for some n as long as i ≥ 1. Here, τ^{0} is the identity operator, τ^{−1} = T r ∘ D, etc. On the boundary of the component (i = 0), the almost split sequence has the form

\xymatrix{0 \ar[r] & τ^{n+1}X_{0} \ar[r] & τ^{n+1}X_{1} ⊕ P \ar[r] & τ^{n}X_{0} \ar[r] & 0}

where P might be zero or might be an indecomposable projective module (see [1, VII.4]). However, as stated in Lemma 8.2, each indecomposable projective module occurs in a unique almost split sequence (which is not necessarily in the component under consideration).

Before showing how almost split sequences can be used to generate indecomposable modules of constant Jordan type, we need the following technical lemma. We fix some notation which differs from standard notation in the case of finite groups: In what follows, let A ⊂ B be rings, and M be a left A-module. Then

Coind_{A}^{B}M = B ⊗_{A} M

Let G be a finite group scheme, and \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} be a π-point. We denote by K⟨α_{K}(t)⟩ the subalgebra of KG = KG_{K} generated by α_{K}(t).

Lemma 8.4. Let G be a finite group scheme and let α_{K} : K[t]/t^{p} → KG be a π-point of G. Let N be a finite dimensional K[t]/t^{p}-module which is not projective and set M = Coind_{K⟨α_{K}(t)⟩}^{KG}N. Then

Π(G_{K})_{M} = {[α_{K}]} ⊂ Π(G_{K}).

Proof. If Π(G_{K})_{M} were empty, then M would be projective, and, hence, injective. On the other hand, the Eckmann-Shapiro Lemma ([3, I.2.8.4]) would enable us to then conclude that Ext_{G_{K}}^{∗>0}(M, K) = Ext_{K⟨α_{K}(t)⟩}^{∗>0}(N, K) = 0 which would contradict our assumption that N is not projective. Thus, Π(G_{K})_{M} is non-empty.

Let U_{K} ⊂ G_{K} be a unipotent abelian subgroup scheme through which α_{K} factors. The proof of [21, 4.12] which is stated for induction rather than coinduction implies that

Π(G_{K})_{M} ⊂ im{Π(U_{K}) → Π(G_{K})}

so that we may assume that G_{K} = U_{K} is a unipotent abelian finite group scheme over K.

After possibly replacing K by some purely inseparable extension which does not change the space Π(G_{K})_{M}, we may by [28, 14.4] assume that KG_{K} is isomorphic (as an algebra) to K[T_{1}, …, T_{n}]/(T_{1}^{p^{e_{1}}}, …, T_{n}^{p^{en}}) for suitable choice of n, e_{1}, …, e_{n}. Let t_{i} = T_{i}^{p^{e_{i}−1}}, and recall that any π-point β_{L} : L[t]/t^{p} → LG_{K} must send t to a sum of monomials in T_{1}, …, T_{n} at least one of which is a non-linear scalar multiple of some t_{i} and each of which are divisible by some (possibly varying) t_{i}. By a change of generators, we may arrange that α_{K}(t) = t_{1} + p(T) where each monomial of the polynomial p(T) is a non-scalar multiple of some t_{i}.

In order to verify that [β_{L}] ∉ Π(G_{K})_{M} for [β_{L}] ≠ [α_{K}], we may choose a representative β_{L} of [β_{L}] which is linear in the t_{i}’s. Assuming [β_{L}] ≠ [α_{K}], we may change generators once again so that α_{K}(t) retains the form t_{1} + p(T) as above and β_{L}(t) = t_{2}. The condition that [β_{L}] ∉ Π(G_{K})_{M} is equivalent to the condition that β_{L}^{∗}(M_{L}) is free. Clearly, it suffices to assume that N is indecomposable of the form K[t]/t^{i}, i < p. Then M_{L} ≅ L[T_{1}, …, T_{n}]/(t_{1}^{p}, …, t_{n}^{p}, (t_{1} + p(T))^{i}) which is free over L⟨t_{2}⟩ with a monomial basis {T_{1}^{j_{1}}, T_{2}^{j_{2}}, …, T_{n}^{j_{n}}}, where 0 ≤ j_{1} < ip^{e_{1}−1}, 0 ≤ j_{2} < p^{e_{2}−1}, 0 ≤ j_{3} < p^{e_{3}} … 0 ≤ j_{n} < p^{e_{n}}. □

We now prove a local restriction result for almost split exact sequences of kGmodules.

Proposition 8.5. Let G be a finite group scheme such that the dimension of Π(G) is at least 1, and let M be an indecomposable non-projective kG-module of constant Jordan type. Assume that one of the following conditions hold: either M is absolutely indecomposable or k is perfect. Consider the almost split sequence of kG-modules

\xymatrix{E : 0 \ar[r] & N \ar[r] & B \ar[r] & M \ar[r] & 0.}

Then for any π-point \xymatrix{α_{K} : K[t]/t^{p} \ar[r] & KG} , α_{K}^{∗}(E_{K}) is a split short exact sequence of K[t]/t^{p}-modules.

Proof. If M_{K} is decomposable, write M_{K} ≅ ⊕M_{K}^{i} as a direct sum of indecomposable KG_{K}-modules. Since in this case k is perfect, Theorem [24, 3.8] implies that the almost split sequence E_{K} is a direct sum of almost split sequences

\xymatrix{E_{K}^{i} : 0 \ar[r] & N_{K}^{i} \ar[r] & B_{K}^{i} \ar[r] & M_{K}^{i} \ar[r] & 0.}

Thus, it suffices to prove that α_{K}^{∗}(E_{K}^{i}) is split for each i. Hence, we may assume that M_{K} is neither projective nor decomposable, and that

\xymatrix{E_{K} : 0 \ar[r] & N_{K} \ar[r] & B_{K} \ar[r] & M_{K} \ar[r] & 0}

is an almost split sequence of KG_{K}-modules.

Let M̃_{K} = Coind_{K⟨α_{K}(t)⟩}^{KG}(α_{K}^{∗}(M_{K})). We have a commutative diagram

(27) \xymatrix{Hom_{KG_{K}}(M̃_{K}, B_{K}) \ar[d]^{≅} \ar[r] & Hom_{KG_{K}}(M̃_{K}, M_{K}) \ar[d]^{≅} \\ Hom_{K⟨α_{K}(t)⟩}(α_{K}^{∗}(M_{K}), α_{K}^{∗}(B_{K})) \ar[r] & Hom_{K⟨α_{K}(t)⟩}(α_{K}^{∗}(M_{K}), α_{K}^{∗}(M_{K}))}

where the vertical arrows are isomorphisms by the Eckmann-Shapiro Lemma. If α_{K}^{∗}(E_{K}) were not split, then the lower horizontal arrow of (27) and thus also the upper horizontal arrow of (27) would not be surjective. On the other hand, the defining property of almost split sequences would then imply that M̃_{K} must have M_{K} as a direct summand. If so, then Π(G_{K})_{M_{K}} ⊂ Π(G_{K})M̃_{K}. Since M_{K} is a non-projective module of constant Jordan type, we have Π(G_{K})_{M_{K}} = Π(G_{K}), and, hence, the support of M_{K} has dimension at least 1. Since Π(G_{K})M̃_{K} consists of only 1 point by Lemma 8.4, we obtain a contradiction. □

Lemma 8.6. Let G be a finite group scheme, and M be an indecomposable nonprojective finite-dimensional kG-module. Then M is a module of constant Jordan type if and only if τM is a module of constant Jordan type. Moreover, if M is a module of constant Jordan type then the stable Jordan types of M and τM are the same.

Proof. Since kG is self-injective, the translation operator τ is isomorphic to N ∘ Ω^{2} where N is the Nakayama functor defined as

N = Hom_{kG}(−, kG)^{#}

(see [1, IV.3]).

The Heller shift Ω^{2} preserves the property of being of constant Jordan type, and, moreover, preserves the stable Jordan type of a module of constant Jordan type (1.8). Hence, to show that τ preserves modules of constant Jordan type, we need to demonstrate this for the Nakayama functor.

By [23, I.8.12] we can exhibit a character δ of G with the property that kG^{#} ⊗ k_{δ} ≃ kG is a two-sided kG-isomorphism, where k_{δ} is the one-dimensional kGrepresentation with kG acting trivially on the right and by the character δ on the left. Hence, we get the following isomorphisms of kG-modules

Hom_{kG}(M, kG) ≃ Hom_{kG}(M, kG^{#} ⊗ k_{δ}) ≃ Hom_{kG}(M ⊗ k_{δ}^{#}, kG^{#}) ≃
Hom_{kG}(M ⊗ k_{δ}^{#}, Hom_{k}(kG, k)) ≃ Hom_{k}(M ⊗ k_{δ}^{#}, k) = M^{#} ⊗ k_{δ}

Hence,

N (M) = Hom_{kG}(M, kG)^{#} ≃ (M^{#} ⊗ k_{δ})^{#} ≃ M ⊗ k_{δ}^{#}

as kG-modules. Let M be a module of constant Jordan type. Since the module k_{δ} is 1-dimensional, it is of trivial constant Jordan type. Hence, the module N (M) ≃ M ⊗k_{δ}^{#} is also of constant Jordan type by Corollary 4.3. Moreover, the Jordan type of M ⊗ k_{δ}^{#} is the same as that of M.

To show that τM being of constant Jordan type implies that M is of constant Jordan type, we merely observe that the operator τ has an inverse given by Ω^{−2} ∘ N^{−1}. Hence, if τM is of constant Jordan type then so is M. □

The following theorem asserts that whether or not an indecomposable kG-module M has constant Jordan type is a function of the connected components of the stable Auslander-Reiten quiver of kG.

Theorem 8.7. Let G be a finite group scheme, and let M be an indecomposable non-projective module of constant Jordan type. Let Θ be a component of the stable Auslander-Reiten quiver of kG containing the module M. Assume that one of the following conditions hold: either all vertices of Θ are absolutely indecomposable, or k is perfect. Then for any [N] ∈ Θ, the module N has constant Jordan type.

Proof. We first consider the case when dim Π(G) = 0. By Theorem 3.4, Π(G) consists of one point. Hence, any module is tautologically a module of constant Jordan type.

We may therefore assume that dim Π(G) ≥ 1. Let [N] be any predecessor of [M] in the stable quiver component Θ. Then there exists an almost split sequence

\xymatrix{E : 0 \ar[r] & τM \ar[r] & B \ar[r] & M \ar[r] & 0}

such that N is a direct summand of B. By Lemma 8.6, τM is a module of constant Jordan type. Let α_{K} : K[t]/t^{p} → KG_{K} be a π-point. By Proposition 8.5, α_{K}^{∗}(E_{K}) splits. Thus, α_{K}^{∗}(B_{K}) = α^{∗}(τM_{K}) ⊕ α^{∗}(M_{K}). We conclude that B is a module of constant Jordan type, so that Theorem 3.7 implies that N has constant Jordan type.

Now let [N] be any successor of [M], i.e., there is an arrow \xymatrix{[M] \ar[r] & [N]} . By [1,

V.1.12] and [1, V.5.3], there is an arrow \xymatrix{[τN] \ar[r] & [M]} . Applying the argument above to τN and M, we conclude that τN has constant Jordan type. By Lemma 8.6, N has constant Jordan type.

Since Θ is connected, the argument is finished by induction. □

To prove the following “realization of constant types” result, we appeal to the work of K. Erdmann [14] in the case of finite groups and that of R. Farnsteiner [16],

[17] for arbitrary finite group schemes (see 8.3). Namely, a result of [15] (see also [17]) following earlier work of Webb [29] asserts that if kG has wild representation type then the Auslander-Reiten component of the trivial module has a very restricted form. Results of Erdmann and Farnsteiner assert that under hypotheses specified in the theorem below, the Auslander-Reiten component of the trivial module must have type A_{∞}.

Theorem 8.8. Let G be a finite group scheme (over k algebraically closed) satisfying one of the following conditions: either G is a finite group which has p-rank at least 2 and whose Sylow p-subgroup is not dihedral or semi-dihedral or Π(G) has dimension at least 2. Then for any n there exists an indecomposable module of stable constant Jordan type n[1].

Proof. By [14, 2] in the case of finite groups and [17, 3.3] for arbitrary finite group schemes, our assumptions imply that Auslander-Reiten quiver of the component containing the trivial module must have type A_{∞}.

Let V_{n} be an indecomposable module representing the n^{th} vertex of the tree underlying the component containing the trivial module. The bottom vertex has label 0. By Proposition 8.7, V_{n} has constant Jordan type for every n. Let \underline{a}_{n} be the stable Jordan type of V_{n}. Proposition 8.5 and Lemma 8.6 imply that the middle term B_{n} of the almost split sequence

\xymatrix{0 \ar[r] & τV_{n} \ar[r] & B_{n} \ar[r] & V_{n} \ar[r] & 0}

has stable constant Jordan type 2\underline{a}_{n}.

Since the tree class is A_{∞}, V_{1} must be the only non-projective indecomposable summand of the middle term of the almost split sequence for V_{0}:

\xymatrix{0 \ar[r] & τV_{0} \ar[r] & B_{0} \ar[r] & V_{0} \ar[r] & 0}

Hence, \underline{a}_{1} = 2\underline{a}_{0}. The middle term of the almost split sequence for V_{1}

\xymatrix{0 \ar[r] & τV_{1} \ar[r] & B_{1} \ar[r] & V_{1} \ar[r] & 0}

has two indecomposable non-projective summands, one of which is isomorphic to τV_{0}. Hence, the other summand has stable constant Jordan type 2\underline{a}_{1} − \underline{a}_{0} = 3\underline{a}_{0}. Proceeding by induction, we see that the module which represents the n^{th} node in this A_{∞} tree has stable constant Jordan type \underline{a}_{n} = n\underline{a}_{0}.

We immediately conclude that k must be at the bottom node since the stable Jordan type of k is 1[1]. Hence, \underline{a}_{0} = 1[1]. Therefore, V_{n} is an indecomposable module of stable constant Jordan type n[1]. □

9. Questions and Conjectures

We offer a few broad questions as well as specific conjectures which provide challenges for further investigation.

Question 9.1. For a given finite group scheme G, what Jordan types are realized as the Jordan type of finite dimensional kG-modules with constant Jordan type?

Certainly, there are constraints as the following examples illustrate.

Example 9.2. Let G be a quasi-elementary abelian group scheme, G = 𝔾_{a(s)} × E with E and elementary abelian p-group of rank r and p > 2. Then any kG-module M of constant Jordan type of stable type 1[1] is endotrivial, and hence of the form Ω^{i}(k) [12]. The Jordan type of such a module has the form 1[1] + mp^{r+s−1}[p] for some m ≥ 0 (see Lemma 6.9).

Example 9.3. We verify that there does not exist a finite dimensional kE-module of constant Jordan type [2] + [p] for p > 3 for E an elementary abelian p-group of rank 2, which the reader can see as very limited evidence for Conjecture 9.5. Suppose V is such a kE-module and write kE = k[x, y]/(x^{p}, y^{p}). Consider the k-vector space basis u, xu, v, xv, x^{2}v, …, x^{p−1}v for V.

We will show that some linear combination y − bx satisfies (y − bx)^{p−1}V = 0, so that the Jordan form associated to y − bx has no block of size p. Observe that y(v) written in our given basis has coefficient 0 for v because y is nilpotent and only v in our basis satisfies x^{p−1}v ≠ 0. Second, suppose that y(v) has coefficient b for xv and consider y − bx. Then once again (y − bx)(v) has coefficient 0 for v and by construction coefficient 0 for xv. Let us replace y by y − bx, so that y(v) ∈ span{u, xu, x^{2}v, …, x^{p−1}v}.

If we apply y to the given basis, the only basis element with the property that y applied to it can have non-zero coefficient for u is u itself, since any of the other basis elements would have to be annihilated by x^{2} and thus in the image of x. Since y is nilpotent, we conclude that y(u) ∈ span{xu, x^{2}v, …, x^{p−1}v} so that y^{2}(u) ∈ span{x^{4}v, …, x^{p−1}v}.

Thus, y^{3}u and y^{3}v are both contained in x^{4}V, and we conclude that y^{p−1}(x^{i}v) = 0 = y^{p−1}(x^{i}u).

The challenge of Question 9.1 seems more interesting if we work stably, so that we identify two Jordan types \underline{a} = a_{p}[p] + ⋯ + a_{1}[1], \underline{b} = b_{p}[p] + ⋯ + b_{1}[1] provided that a_{i} = b_{i}, i ≠ p.

Question 9.4. For which finite group schemes G is every stable Jordan type the Jordan type of a finite dimensional kG-module of constant Jordan type?

The following is a specific conjecture would be a step towards answering the previous question.

Conjecture 9.5. Let E be an elementary abelian p-group of rank ≥ 2, with p > 3. Then there does not exist a finite dimensional kE-module of stable constant Jordan type [2].

Andrei Suslin has formulated the following intriguing question whose affirmative answer would in particular verify the preceding conjecture.

Question 9.6. Let E be an elementary abelian p-group of rank 2, with p > 3. Let M be a kE-module with constant Jordan type ∑_{i} a_{i}[i] and let i be an integer, 1 < i < p. Is it the case that if a_{i} ≠ 0, then either a_{i+1} ≠ 0 or a_{i−1} ≠ 0?

Of course, if Conjecture 9.5 is valid, then it follows that there is no module of stable constant Jordan type [2] for any finite group scheme G containing a quasielementary subgroup scheme H = 𝔾_{a(r)} × E such that the rank s of E plus r is greater or equal to 2.

We can make many other “non-existence conjectures” such as the following. We recall (see Example 2.2) that for E an elementary abelian p-group of rank n ≥ 2 there exists a kE-module whose stable type is constant of type 1[2] + (n − 1)[1]

Conjecture 9.7. Let E be an elementary abelian p-group of rank n ≥ 2, with p > 3. There does not exist a kE-module whose stable type is constant of type 1[2] + j[1] with j ≤ n − 2.

We next formulate questions of a more qualitative nature.

Definition 9.8. Let G be a finite group scheme over a field k (of characteristic p > 0). We denote by L = ℕ^{p} the (additive) lattice of Jordan types over k. We denote by

R(G) ⊂ L

the sublattice of those Jordan types which can be realized as the Jordan types of kG-modules of constant Jordan type.

Question 9.9. For which finite group schemes G is L/R(G) finite? Among such finite group schemes, how does the invariant L/R(G) behave?

For those finite group schemes G for which L/R(G) is infinite, can we give some interpretation of the rank of this quotient in more familiar terms?

In view of our discussion involving Auslander-Reiten almost split sequences, it seems of considerable interest to consider I(G) as defined below.

Definition 9.10. Let G be a finite group scheme over a field k (of characteristic p > 0) and let \overline{R}(G) ⊂ \overline{L} = ℕ^{p−1} denote the subset of those stable Jordan types realizable as the stable Jordan type of a finite dimensional kG-module of constant Jordan type. Further, let us denote by I(G) ⊂ \overline{R}(G) ⊂ \overline{L} = ℕ^{p−1} the subset of stable Jordan types which are the Jordan types of indecomposable kG-modules of constant Jordan type.

Question 9.11. For which G is I(G) closed under addition?

Remark 9.12. If G is the Klein four group, G ≃ ℤ/2ℤ × ℤ/2ℤ, then the only non-projective indecomposable modules of constant Jordan type are Heller shifts of the trivial module k. Thus, for this choice of G, I(G) is not closed under addition.

One is tempted to ask many questions concerning how the realizability of modules of constant type behaves with respect to change of finite group scheme. We ask one such question.

Question 9.13. For which H ⊂ G does restriction induce a bijection from the set of Jordan types realized as Jordan types of kG-modules of constant Jordan type to the set of Jordan types realized as Jordan types of kH-modules of constant Jordan type?

10. APPENDIX: Decomposition of tensor products of k[t]/t^{p}-modules

The purpose of this appendix is to derive a closed form for tensor products of k[t]/t^{p}-modules, presumably implicit in [27]. Here, we view k[t]/t^{p} as a self-dual Hopf algebra; in other words, as the restricted enveloping algebra of the 1-dimensional p-restricted Lie algebra (with trivial p-restriction operator). Thus, the coproduct is given by the formula t ↦ 1 ⊗ t + t ⊗ 1.

Let V (λ) denote the simple restricted sl_{2}-module of highest weight λ where 0 ≤ λ ≤ p − 1. Let e, h, f be the standard generators for sl_{2}. Let ⟨e⟩ ⊂ sl_{2} be the 1-dimensional p-restricted Lie algebra generated by the element e. Hence, u(⟨e⟩) ≃ k[t]/t^{p} is a Hopf subalgebra of u(sl_{2}).

The sl_{2}-representation theory implies that

(28)  V (λ) ↓_{u(⟨e⟩)}≃ [λ + 1]

as k[t]/t^{p}-modules. Hence, the tensor product formulas for indecomposable k[t]/t^{p} modules follow from the tensor product formulas for simple sl_{2}-modules. Such formulas were obtained in [25] (see also [6]).

Proposition 10.1. (See [25, Lemma 2.3, 2.4] Let {V (λ)}_{0≤λ≤p−1} be the collection of simple restricted sl_{2}-modules.

• If 0 ≤ μ ≤ λ ≤ p − 2 and λ + μ ≤ p − 1 then

V (λ) ⊗ V (μ) ≃ V (λ − μ) ⊕ V (λ − μ + 2) ⊕ … ⊕ V (λ + μ)

• If 0 ≤ μ ≤ λ ≤ p − 2 and λ + μ ≥ p then

V (λ) ⊗ V (μ) ≃ V (λ − μ) ⊕ V (λ − μ + 2) ⊕ … ⊕ V (2(p − 2) − λ − μ) + (proj)

where the projective summand has dimension λ + μ − (p − 2).

Restricting the tensor decompositions for the simple restricted sl_{2}-modules V (λ) to the indecomposable k[t]/t^{p}-modules [i] via the formula (28) we obtain the following formulas.

Corollary 10.2. Let [i] be an indecomposable k[t]/t^{p}-module of dimension i for 1 ≤ i ≤ p. Then

[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}

We conclude this appendix with the analogous formula for the tensor product of indecomposable kC_{p}-modules where C_{p} is a cyclic group of order p. A subtlety here is that even though the module categories for kC_{p} and the algebra k[t]/t^{p} with the coproduct t ↦ 1 ⊗ t + t ⊗ 1 are equivalent, the tensor product structure comes from two different coproducts. Nonetheless, the tensor multiplicities turn out to be the same.

Corollary 10.3. Let [i], 1 ≤ i ≤ p, be indecomposable kC_{p}-modules. Then the decomposition of the tensor product [i] ⊗ [j] into indecomposable kC_{p}-modules is given by the formulas as in Corollary 10.2.

Proof. By [21, 4.5] the tensor product of any two kC_{p}-modules M, N is isomorphic as k[t]/t^{p}-module to the tensor product M ⊗N using the coproduct t ↦ 1⊗t+t⊗1. The statement now follows from Corollary 10.2. □

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Department of Mathematics, University of Georgia, Athens, GA

  E-mail address: jfc@math.uga.edu

Department of Mathematics, Northwestern University, Evanston, IL

  E-mail address: eric@math.northwestern.edu

Department of Mathematics, University of Washington, Seattle, WA

  E-mail address: julia@math.washington.edu