GENERIC AND MAXIMAL JORDAN TYPES

ERIC M. FRIEDLANDER^{∗}, JULIA PEVTSOVA^{∗}, AND ANDREI SUSLIN^{∗}

Abstract. For a finite group scheme G over a field k of characteristic p > 0, we associate new invariants to a finite dimensional kG-module M. Namely, for each generic point of the projectivized cohomological variety Proj H^{•}(G, k) we exhibit a “generic Jordan type” of M. In the very special case in which G = E is an elementary abelian p-group, our construction specializes to the non-trivial observation that the Jordan type obtained by restricting M via a generic cyclic shifted subgroup does not depend upon a choice of generators for E. Furthermore, we construct the non-maximal support variety Γ(G)_{M}, a closed subset of Proj H^{•}(G, k) which is proper even when the dimension of M is not divisible by p.

0. Introduction

Elementary abelian p-subgroups of a finite group G capture significant aspects of the cohomology and representation theory of G. For example, if k is a field of characteristic p > 0, then a theorem of D. Quillen [18] asserts that the Krull dimension of the cohomology algebra H^{•}(G, k) is equal to the maximum of the ranks of elementary abelian p-subgroups of G and a theorem of L. Chouinard [6] asserts that a kG-module is projective if and only if its restrictions to all elementary abelian p-subgroups of G is projective. The cohomology algebra H^{•}(E, k) of an elementary abelian p-group E is easily computed, whereas kE is of wild representation type provided that the rank of E is at least 2 (at least 3, for p = 2). Nonetheless, Chouinard’s theorem and Quillen’s geometric description of Spec H^{•}(G, k) [18] provide the basis for interesting invariants of kG-modules, most notably the cohomological support variety |G|_{M} of a kG-module M.

The investigation of the cohomology and representation theory of finite groups is naturally extended to other algebraic structures. Friedlander and B. Parshall developed a theory of support varieties for finite dimensional p-restricted Lie algebras g over a field k of characteristic p > 0 (e.g., [9]). For restricted Lie algebras, the role of the group algebra kG of the finite group G is played by the restricted enveloping algebra u(g). Indeed, restricted Lie algebras lead one to more interesting geometrical structures than do finite groups, and seemingly lead to stronger results. For example, the theorem of G. Avrunin and L. Scott [1] identifying the cohomological support variety |E|_{M} of a finite dimensional kE-module M for an elementary abelian p-group E with the rank variety of J. Carlson [3] admits a formulation in the case of a restricted Lie algebras g in terms of closed subvarieties of the p-nilpotent cone of g ([9], [14], [20]).

2 ERIC M. FRIEDLANDER^{∗}, JULIA PEVTSOVA^{∗}, AND ANDREI SUSLIN^{∗}

Finite groups and finite dimensional p-restricted Lie algebras are examples of finite group schemes. The representation theory of a finite group scheme is the study of kG-modules, where kG is the group algebra of G defined to be the k-linear dual of the representing Hopf algebra k[G]. A fundamental theorem of Friedlander and Suslin [13] is that the cohomology algebra H^{∗}(G, k) is finitely generated for any such finite group scheme.

∗ partially supported by the NSF.

If G is an affine algebraic group over k, then the r^{th} Frobenius kernel G = G_{(r)} ≡ ker{F^{r} : G → G^{(r)}} is an infinitesimal group scheme (i.e. a connected finite group scheme). Such Frobenius kernels play an important role in the study of (rational) representations of G: the family {G_{(r)}; r > 0} faithfully captures the representation theory of G. In [19], [20], Suslin, Friedlander, and C. Bendel extend to all infinitesimal group schemes G earlier work on the cohomology and support varieties of finite dimensional p-restricted Lie algebras (whose restricted enveloping algebras are the group algebras of infinitesimal group schemes of height 1).

In [10], [12], Friedlander and Pevtsova present a uniform approach to the study of the cohomology and related representation theory of all finite group schemes. This approach involves the use of π-points of G, which are finite flat maps of Kalgebras K[t]/t^{p} → KG for field extensions K/k; these play the role of “cyclic shifted subgroups” in the case that G is an elementary abelian p-group and the role of 1-parameter subgroups in the case that G is an infinitesimal group scheme. In [12], the space Π(G) of equivalence classes [α_{K}] of π-points α_{K} : K[t]/t^{p} → KG of G is given a scheme structure without reference to cohomology such that Π(G) is isomorphic as a scheme to Proj H^{•}(G, k). In particular, there is a natural bijection between such equivalence classes of π-points and homogeneous prime ideals of H^{•}(G, k). The perspective of π-points leads to the considerations of this present paper.

As was first suggested in [8], this representation-theoretic interpretation of support varieties can be refined to determine invariants of kG-modules which provide more detailed information than that given by support varieties. Namely, for any πpoint α_{K} : K[t]/t^{p} → KG, one can consider the Jordan type of α_{K}(t) as a nilpotent operator on M_{K}. As one quickly discovers, the answer may vary considerably depending on the representative α_{K} of an equivalence class of π-points in Π(G). Even in the case of G = ℤ/pℤ × ℤ/pℤ the answer depends on the choice of generators of E which in turn determines the choice of representative of a π-point (Example 2.3).

The following theorem (Theorem 4.2) provides a new intrinsic invariant of modular representations of finite group schemes. In particular, in the case of an elementary abelian group, this invariant is independent of the choice of generators.

Theorem 0.1. Let G be a finite group scheme, let M be a finite dimensional Gmodule and let α_{K} be a π-point of G which represents a generic point [α_{K}] ∈ Π(G). Then the Jordan type of α_{K}(t) viewed as a nilpotent operator on M_{K} depends only upon [α_{K}] and not the choice of α_{K} representing [α_{K}].

For a given generic point [α_{K}] ∈ Π(G) and a given finite dimensional kG-module M, we call the Jordan type of α_{K}(t) on M_{K} the generic Jordan type of M (at [α_{K}]). As we verify in Proposition 4.7, sending a module M to its generic Jordan type [α_{K}]^{∗}(M_{K}) for generic [α_{K}] ∈ Π(G) determines a well defined tensor triangulated functor on stable module categories [α_{K}]^{∗} : stmod(kG) → stmod(K[t]/t^{p}). In other words, generic Jordan type is well-behaved with respect to short exact sequences, direct sums, tensor products, and Heller shifts.

We establish Theorem 0.1 by successively considering more general finite group schemes, beginning with elementary abelian p-groups and then abelian finite group schemes in §2 and then infinitesimal group schemes in §3 before treating the general case in §4. The following theorem (Theorem 1.12) concerning commuting nilpotent matrices provides the key to our proof of Theorem 0.1.

Theorem 0.2. Let k be an infinite field and let α, α_{1}, …, α_{n} be a set of commuting nilpotent n × n matrices such that the Jordan type of α is greater or equal to the Jordan type of any α + c_{1}α_{1} + ⋯ + c_{n}α_{n} for all c_{i} ∈ k. Then the Jordan type of α is greater or equal to the Jordan type of any polynomial in α, α_{1}, …, α_{n} without constant term.

For a given kG-module M, we define a notion of a maximal Jordan type (Definition 2.4). Some but not necessarily all of the generic Jordan types are maximal, but maximal Jordan types are also realized at non-generic points. The locus of non-maximal Jordan types for a given finite dimensional kG-module M provides us with a geometric invariant Γ(G)_{M} which agrees with the support variety of M if and only if the maximal type of M is projective. (Theorems 4.10, 5.2).

Theorem 0.3. Let G be a finite group scheme over k and M a finite dimensional kG-module. Then whether or not the Jordan type of α_{K}^{∗}(M_{K}) is maximal for M depends only upon the equivalence class of α_{K}, [α_{K}] ∈ Π(G). The set of those equivalence classes of π-points α_{K} : K[t]/t^{p} → KG such that the Jordan type of α_{K}^{∗}(M) is not maximal for M is a closed subvariety

Γ(G)_{M} ⊂ Π(G)

of Π(G) ≃ Proj H^{•}(G, k).

The prototype of our non-maximal support variety Γ(G)_{M} was introduced by W. Wheeler ([22]) for elementary abelian p-groups. Our construction significantly strengthens Wheeler’s result even in the case of an elementary abelian p-group E, since we do not rely upon a fixed set of generators for E.

The reader will find various examples throughout the paper intended to illustrate some of these new invariants and their behavior. More sophisticated examples which reflect deeper properties of the representation theory of G are considered in [4].

In §1, k will denote an infinite field and in subsequent sections k will denote an arbitrary field of characteristic p > 0. If M is a k-vector space and K/k is a field extension, we use the notation M_{K} to denote M ⊗_{k} K. If M is a finite dimensional k vector space equipped with the structure of a k[t]/t^{p}-module, then the data of the block sizes of M as a k[t]/t^{p}-module (or, equivalently, of the Jordan form of t viewed as an endomorphism of M) will be called the Jordan type of M. In particular, the Jordan type of the k[t]/t^{p}-module M is the same as that of the K[t]/t^{p}-module M_{K} for any field extension K/k. We employ the notation H^{•}(G, k) for the commutative k-algebra given as the full cohomology algebra H^{∗}(G, k) of G if char (k) = 2 and for the subalgebra of H^{∗}(G, k) generated by classes homogeneous of even degree if p ≠ 2. For a map f : H → G of finite group schemes, we denote by the same letter f the induced map on group algebras, f : kH ⟶ kG, and by f^{∗} : G-mod ⟶ H-mod the pull-back via f on the categories of modules.

We are pleased to acknowledge useful conversations with Steve Smith. Friedlander and Suslin gratefully acknowledge the support of the Institute for Advanced Study, and Friedlander and Pevtsova gratefully acknowledge the hospitality of the Max-Planck Institut in Bonn.

1. Maximality for commuting nilpotent matrices

In this section, we consider a finite set of pair-wise commuting nilpotent matrices α_{1}, …, α_{n} with coefficients in an infinite field k. Theorem 1.12 establishes that maximality of the Jordan type among all polynomials in the α_{i}’ s is realized by some linear combination of the α_{i}’ s. This theorem is the essential step in our investigation in later sections of maximality of Jordan types associated to finite dimensional representations of finite group schemes.

Given two partitions \underline{n} = n_{1} ≥ n_{2} ≥ … ≥ n_{N}, and \underline{m} = m_{1} ≥ m_{2} ≥ … ≥ m_{N} of N we say that \underline{n} dominates \underline{m}, or \underline{n} ≥ \underline{m}, if

∑_{1≤j≤k} n_{j} ≥ ∑_{1≤j≤k} m_{j} for 1 ≤ k ≤ N

(see [7, 6.2.1]).

Definition 1.1. Let α ∈ M_{N}(k) be an N × N nilpotent matrix with coefficients in k. The Jordan type of α is the partition of N, \underline{n} = n_{1} ≥ n_{2} ≥ ⋯ ≥ n_{s} with ∑_{j} n_{j} = N, such that the canonical Jordan form of α is a direct sum of indecomposable blocks of size n_{j}.

If α, β ∈ M_{N}(k) are two nilpotent matrices, we write α ≥ β (and say that the Jordan type of α is greater or equal to the Jordan type of β) if the partition associated to α is greater or equal to the partition associated to β. The condition that α ≥ β is equivalent to the condition that

(1.1.1)  rank(α^{s}) ≥ rank(β^{s}), ∀s > 0,

(see [7, 6.2.2]) which in turn is equivalent to the condition that

dim_{k} Ker(α^{s}) ≤ dim_{k} Ker(β^{s}), ∀s > 0.

We denote by ∼ the equivalence relation associated to this partial ordering:

α ∼ β iff α ≤ β and β ≤ α.

Observe that α ∼ β if and only if α and β have the same Jordan type if and only if they are conjugate (under the adjoint action of GL_{N}(k)) elements of M_{N}(k).

The data of the Jordan type of a nilpotent matrix α ∈ M_{N}(k) is equivalent to each of the following:

• the isomorphism class of k^{N} as a k[t]-module, with t acting as α;

• the data of the ranks of α^{s}, ∀s > 0;

• the data of the dimensions of the kernels of α^{s}, ∀s > 0. If M is an N-dimensional k[t]-module with t acting nilpotently, then we shall frequently refer to the “Jordan type of M” rather than the Jordan type of the matrix in M_{N}(k) given by the action of t.

Lemma 1.2. Let α, α_{1}, ⋯, α_{n} ∈ M_{N}(k) be a family of commuting nilpotent matrices with coefficients in an infinite field k. The following conditions are equivalent:

(1) For all λ_{1}, ⋯, λ_{n} ∈ k,

rank(α) ≥ rank(α + λ_{1}α_{1} + ⋯ + λ_{n}α_{n}),
where α, α + λ_{1}α_{1} + ⋯ + λ_{n}α_{n} are viewed as matrices in M_{N}(k).

(2) For every field extension K/k, all λ_{1}, ⋯, λ_{n} ∈ K,

rank(α) ≥ rank(α + λ_{1}α_{1} + ⋯ + λ_{n}α_{n}),

where α, α + λ_{1}α_{1} + ⋯ + λ_{n}α_{n} are viewed as matrices in M_{N}(K).

(3) For t_{1}, …, t_{n} independent variables in the field k(t_{1}, …, t_{n}) of rational functions in n variables over k,

rank(α) = rank(α + t_{1}α_{1} + ⋯ + t_{n}α_{n}).

Proof. For any A ∈ M_{N}(k) and any pair of subsets I, J ⊂ {1, …, N} of cardinality s ≤ N, denote by A_{I,J} the associated minor, an s × s submatrix of A. The condition rank A ≥ s is equivalent to the condition that there exist I and J such that det(A_{I,J}) ≠ 0. For any I and J as above, consider

d_{I,J}(t_{1}, …, t_{n}) = det((α + t_{1}α_{1} + ⋯ + t_{n}α_{n})_{I,J})

where t_{1}, …, t_{n} are independent variables in the field k(t_{1}, …, t_{n}). Let K/k be an arbitrary field extension. Then d_{I,J}(t_{1}, …, t_{n}) = 0 if and only if d_{I,J}(λ_{1}, …, λ_{n}) = 0 for all (λ_{1}, …, λ_{n}) in an open dense subset in 𝔸_{K}^{n}. Thus,

rank(α + t_{1}α_{1} + ⋯ + t_{n}α_{n}) ≥ rank(α + λ_{1}α_{1} + ⋯ + λ_{n}α_{n}) for all λ_{1}, ..., λ_{n} ∈ K,

rank(α + t_{1}α_{1} + ⋯ + t_{n}α_{n}) = rank(α + λ_{1}α_{1} + ⋯ + λ_{n}α_{n}) for all (λ_{1}, …, λ_{n}) in an open dense subset in 𝔸_{K}^{n}. □

Since the argument of the proof of Lemma 1.2 applies equally to all powers of α, we immediately conclude the following proposition.

Proposition 1.3. Let α, α_{1}, …, α_{n} ∈ M_{N}(k) be a family of commuting nilpotent matrices with coefficients in an infinite field k. Then the following conditions are equivalent:

(1) For all λ_{1}, …, λ_{n} ∈ k,

α ≥ α + λ_{1}α_{1} + ⋯ + λ_{n}α_{n}.
(2) For every field extensions K/k, all λ_{1}, …, λ_{n} ∈ K,
α ≥ α + λ_{1}α_{1} + ⋯ + λ_{n}α_{n}.

(3) For t_{1}, …, t_{n} independent variables in the field k(t_{1}, …, t_{n}) of rational functions in n variables over k,

α ∼ α + t_{1}α_{1} + ⋯ + t_{n}α_{n}

in M_{N}(k(t_{1}, …, t_{n})).

Definition 1.4. We say that α is rank maximal with respect to α_{1}, …, α_{n} provided that the equivalent conditions of Lemma 1.2 are satisfied for α, α_{1}, …, α_{n} ∈ M_{N}(k).

Similarly, we say that α is maximal with respect to α_{1}, …, α_{n} if the equivalent conditions of Proposition 1.3 are satisfied for α, α_{1}, …, α_{n} ∈ M_{N}(k).

Remark 1.5. Let α_{1}, …, α_{n} ∈ M_{N}(k) be a family of nilpotent commuting matrices with coefficients in an infinite field k. There exists α ∈ M_{N}(k) satisfying the conditions of Proposition 1.3. Indeed, let s_{1}, …, s_{n} be a set of indeterminants. Let

α\underline{s} = s_{1}α_{1} + ⋯ + s_{n}α_{n} ∈ M_{N}(k(s_{1}, …, s_{n})).

Observe that α\underline{s} = s_{1}α_{1}+⋯+s_{n}α_{n} ∼ (s_{1}+t_{1})α_{1}+⋯+(s_{n}+t_{n})α_{n} = α\underline{s}+t_{1}α_{1}+⋯+ t_{n}α_{n} as elements of M_{N}(k(s_{1}, …, s_{n}, t_{1}, …, t_{n})). Indeed, (s_{1} + t_{1})α_{1} + ⋯ + (s_{n} + t_{n})α_{n} can be specialized to s_{1}α_{1}+⋯+s_{n}α_{n} by setting t_{i} = 0. On the other hand, if one makes a change of variables s′_{i} = s_{i} +t_{i}, then α_{s} = (s′_{1} −t_{1})α_{1} +⋯+(s′_{n} −t_{n})α_{n} can be specialized to α\underline{s} + t_{1}α_{1} + ⋯ + t_{n}α_{n} = s′_{1}α_{1} + ⋯ + s′_{n}α_{n} by taking t_{i} = 0. Thus, α\underline{s} has the same Jordan type as α\underline{s} + t_{1}α_{1} + ⋯ + t_{n}α_{n}. Proposition 1.3 implies that α\underline{s} is maximal with respect to α_{1}, …, α_{n}.

By upper semi-continuity of the Jordan type, there exists an open dense subset in 𝔸_{k}^{n} such that for any \underline{λ} = (λ_{1}, …, λ_{n}) from that subset we have α\underline{λ} = λ_{1}α_{1} + ⋯ + λ_{n}α_{n} ∼ α\underline{s}. For any such \underline{λ}, α = α\underline{λ} satisfies the conditions of Proposition 1.3.

We proceed to investigate the rank maximality condition in the special case of two commuting matrices. For any commuting nilpotent matrices α, β ∈ M_{N}(k) set

W_{α,β} = ∑_{μ∈k^{∗}} Ker(α + μβ) ⊂ k^{m},

the subspace spanned by the union of the subspaces Ker(α + μβ). Since α, β commute, each Ker(α + μβ) is invariant under both α and β.

Lemma 1.6. Let α, β ∈ M_{N}(k) be a pair of commuting matrices and set W = W_{α,β}. Then β(W) = α(W) ⊂ W and hence

(α + λβ)(W) ⊂ α(W) for all λ ∈ k.

Proof. For any μ ∈ k^{∗}, we have β|_{Ker(α+μβ)} = −μ^{−1}α|_{Ker(α+μβ)}. Hence β(Ker(α + μβ)) = α(Ker(α+μβ)). Thus β(W) = α(W). The second claim is now obvious. □

Using Lemma 1.6, we obtain a useful criterion for when α is rank maximal with respect to α, β.

Lemma 1.7. Let k be an infinite field, let α, β ∈ M_{N}(k) be two commuting nilpotent matrices, and set W = W_{α,β}. Then the following conditions are equivalent

• α is rank maximal with respect to β.

• Ker(α) ⊂ W.

Proof. Set r = max_{λ∈k} rank(α + λβ). As seen in the proof of Lemma 1.2, for all but finitely many λ ∈ k we have an equality rank(α + λβ) = r. Hence, α is rank maximal with respect to β if and only if rank(α) = r, that is dim(Ker(α)) = N − r. By Lemma 1.6, rank(α|_{W}) ≥ rank((α + λβ)|_{W}) for all λ ∈ k with equality holding for all but finitely many λ’s. Thus for a “generic” λ ∈ k, we have the following chain of equalities and inequalities:

dim(Ker α) ≥ dim((Ker α) ∩ W) = dim(Ker(α|_{W}))
= dim(Ker((α + λβ)|_{W}) = dim(Ker(α + λβ)) = N − r

where the last but one equality holds since Ker(α + λβ) ⊂ W for all λ ∈ k^{∗} by definition of W. This chain shows that dim(Ker α) = N − r if and only if Ker α ⊂ W. □

Lemma 1.7 will now enable us to prove the following key property of rank maximality. In some sense, the property proved in Proposition 1.8 is the essential case of the general maximality result of Theorem 1.12.

Proposition 1.8. Let k be an infinite field and let α, β ∈ M_{N}(k) be two commuting nilpotent matrices with α rank maximal with respect to β. Then for any nilpotent matrix γ ∈ M_{N}(k) commuting with both α and β,

rank(α) = rank(α + βγ).

Proof. Set W = W_{α,β}. Since γ commutes with α and β, W is γ-invariant.

We first verify that α(W) = (α + βγ)(W). To prove this formula it suffices to show that α(V_{μ}) = (α+βγ)(V_{μ}), where μ ∈ k^{∗} and V_{μ} = Ker(α+μβ). This follows immediately from the following relations:

α|_{V_{μ}} = −μ · β|_{V_{μ}} , (α + βγ)|_{V_{μ}} = −μ · β|_{V_{μ}} ∘ (1 − μ^{−1} · γ|_{V_{μ}})

since the matrix 1 − μ^{−1} · γ|_{V_{μ}} is invertible.

Next we show that Ker(α+βγ) ⊂ W. Let v ∈ Ker(α+βγ) be an arbitrary vector in k^{N}. We are going to show that γ^{i}(v) ∈ W for all i ≥ 0 using descending induction on i. Since γ is nilpotent, this statement is trivial for i large enough. Assume now that γ^{i+1}(v) ∈ W. Then α(γ^{i}(v)) = −βγ^{i+1}(v) = β(−γ^{i+1}(v)) ∈ β(W) = α(W). Thus there exists w ∈ W such that α(γ^{i}(v)−w) = 0. Hence γ^{i}(v) ∈ W +Ker α = W, since Ker α ⊂ W according to Lemma 1.7.

Using α(W) = (α + βγ)(W) and Ker(α + βγ) ⊂ W, we conclude that

rank(α + βγ) = N − dim(Ker(α + βγ)) = N − dim(Ker((α + βγ)|_{W})) =

= N − dim W + dim (α + βγ)(W) = N − dim W + dim α(W) =
= N − dim Ker α|_{W} = N − dim Ker α = rank α.
□

The following theorem establishes that rank maximality with respect to linear combinations of α_{1}, …, α_{n} implies rank maximality with respect to all polynomials in α_{1}, …, α_{n}. Our ultimate theorem, Theorem 1.12, refines this by replacing rank maximality by maximality.

Theorem 1.9. Let α, α_{1}, ⋯, α_{n}, β_{1}, …, β_{n} ∈ M_{N}(k) be a family of commuting nilpotent matrices with coefficients in an infinite field k. Assume that α is rank maximal with respect to the family α_{1}, ..., α_{n}. Then α is also rank maximal with respect to the family α_{1}, ..., α_{n}, α_{1}β_{1}, …, α_{n}β_{n}.

Moreover, rank α = rank(α + α_{1}β_{1} + ⋯ + α_{n}β_{n}).

Proof. Our procedure is to expand the family of matrices with respect to which α is rank maximal.

Our first (and key) step is to show that we can add α_{1}β_{1}. We proceed to verify that

rank α = rank(α + t_{1}α_{1} + ⋯ + t_{n}α_{n} + tα_{1}β_{1})

for independent variables t_{1}, ..., t_{n}, t. Observe that α + t_{1}α_{1} + ... + t_{n}α_{n} is obviously rank maximal with respect to α_{1}, so that Proposition 1.8 thus implies that

rank(α + t_{1}α_{1} + ⋯ + t_{n}α_{n} + tα_{1}β_{1}) = rank(α + t_{1}α_{1} + ⋯ + t_{n}α_{n}).

On the other hand,

rank(α + t_{1}α_{1} + ⋯ + t_{n}α_{n}) = rank α

in view of rank maximality of α with respect to α_{1}, …, α_{n}.

Iterating this procedure, we see that we can add all terms of the form α_{i}β_{i} to our family, thereby establishing the rank maximality of α with respect to the family α_{1}, …, α_{n}, α_{1}β_{1}, …, α_{n}β_{n}.

To prove the rank maximality of α+α_{1}β_{1} +⋯+α_{n}β_{n} and the equality rank α = rank(α+α_{1}β_{1} +⋯+α_{n}β_{n}), we first verify these assertions with α_{1}β_{1}. The equality of ranks follows immediately from Proposition 1.8. There remains to show that for independent variables t_{1}, …, t_{n} we have the equality

rank(α + α_{1}β_{1}) = rank(α + α_{1}β_{1} + t_{1}α_{1} + ⋯ + t_{n}α_{n}).

Since α + t_{1}α_{1} + ⋯ + t_{n}α_{n} is rank maximal with respect to α_{1}, two applications of Proposition 1.8 imply the equalities

rank(α + t_{1}α_{1} + ⋯ + t_{n}α_{n} + α_{1}β_{1}) = rank(α + t_{1}α_{1} + ⋯ + t_{n}α_{n}) = rank α =

= rank(α + α_{1}β_{1}).

Finally, we get the general case by iterating this procedure. □

To extend Theorem 1.9 to establish maximality of Jordan types, we employ the following auxilliary construction which reduces maximality to rank maximality of associated matrices. Denote by J_{s} ∈ M_{s}(k) the Jordan block of size s, i.e. J_{s} is a nilpotent matrix which acts on the standard basis of k^{s} according to the rule J_{s}(e_{1}) = 0, J_{s}(e_{i}) = e_{i−1} (2 ≤ i ≤ s).

Lemma 1.10. For any finite dimensional k-vector space V and any α ∈ End_{k}(V), there is a natural isomorphism

Ker(α^{s}) ≃ Ker(α ⊗ 1_{s} + 1_{V} ⊗ J_{s}).

where 1_{s}, 1_{V} are identity operators on k^{s}, V respectively, and α ⊗ 1_{s} + 1_{V} ⊗ J_{s} ∈ End_{k}(V ⊗ k^{s}).

Proof. Any v ∈ V ⊗ k^{s} can be written uniquely in the form v_{1} ⊗ e_{1} + ⋯ + v_{s} ⊗ e_{s} with appropriate v_{1}, …, v_{s} ∈ V. Moreover

(α ⊗ 1_{s} + 1_{V} ⊗ J_{s})(v_{1} ⊗ e_{1} + ⋯ + v_{s} ⊗ e_{s}) = ∑_{i=1}^{s−1}(α(v_{i}) + v_{i+1}) ⊗ e_{i} + α(v_{s}) ⊗ e_{s}.

Thus, equations defining Ker(α ⊗ 1_{s} + 1_{V} ⊗ J_{s}) look as follows:

α(v_{1}) = −v_{2}, α(v_{2}) = −v_{3}, …, α(v_{s−1}) = −v_{s}, α(v_{s}) = 0.

We conclude that v_{1} determines all other components via the formulae

v_{i} = (−1)^{i−1}α^{i−1}(v_{1}),

and the only equation for v_{1} is α^{s}(v_{1}) = 0, i.e. v_{1} ∈ Ker α^{s}. □

Corollary 1.11. Let α, α_{1}, …, α_{n} ∈ M_{N}(k) be a family of commuting nilpotent matrices with coefficients in an infinite field k. Then the following conditions are equivalent.

• rank(α^{s}) ≥ rank(α + λ_{1}α_{1} + ⋯ + λ_{n}α_{n})^{s} ∀λ_{1}, …, λ_{n} ∈ k

• α ⊗ 1_{s} + 1_{N} ⊗ J_{s} is rank maximal with respect to α_{1} ⊗ 1_{s}, …, α_{n} ⊗ 1_{s}

We now formulate the main theorem of this section, a result which will play a key role in later sections in our consideration of “local maximality” for finite dimensional modules for an arbitrary finite group scheme.

Theorem 1.12. Let α, α_{1}, …, α_{n}, β_{1}, … β_{n} ∈ M_{N}(k) be a finite set of commuting nilpotent matrices with coefficients in an infinite field k. Assume that α is maximal with respect to the family α_{1}, …, α_{n}. Then α is also maximal with respect to the family α_{1}, ..., α_{n}, α_{1}β_{1}, …, α_{n}β_{n}.

Moreover,

α ∼ α + α_{1}β_{1} + ⋯ + α_{n}β_{n}.

Proof. This follows immediately by combining Theorem 1.9 and Corollary 1.11. □

By taking β_{i} to be polynomials in α_{j} without constant term, we obtain the following special case of Theorem 1.12.

Corollary 1.13. Assume that α is maximal with respect to a family α_{1}, …, α_{n} ∈ M_{N}(k) of commuting nilpotent matrices. Then

α ≥ α + q(α_{1}, …, α_{n})

for any polynomial q ∈ k[X_{1}, …, X_{n}] with no non-zero constant term. Moreover if q ∈ k[X_{1}, …, X_{n}] with no non-zero constant or linear term, then

α ∼ α + q(α_{1}, …, α_{n}).

Remark 1.14. A special case of Corollary 1.13 follows from the work of George McNinch [17] via the theory of nilpotent orbits. Namely, one needs to make an assumption that the Jordan type of all the matrices involved does not have blocks of size p. We thank George McNinch for his interest in our problem.

Remark 1.15. In a few degenerate cases considered below, we shall (implicitly) have to include the possibility that α, α_{1}, …, α_{n} ∈ M_{N}(k) with k finite. Then we shall say α is maximal with respect to α_{1}, …, α_{n} provided that this is the case when viewing these matrices in M_{N}(K) for some infinite field extension K/k.

2. Maximal Jordan types for abelian finite group schemes

If E = (ℤ/pℤ)^{×r} is an elementary abelian p-group of rank r, then a choice of generators g_{1}, …, g_{r} ∈ E determines an isomorphism kE ≃ k[t_{1}, …, t_{r}]/(t_{1}^{p}, …, t_{r}^{p}) given by sending g_{i} to t_{i}+1. A cyclic shifted subgroup of E with respect to the set of generators \{g_{1}, …, g_{r}\} is a map of k-algebras specified by some \underline{λ} = (λ_{1}, …, λ_{r}) ∈ k^{r}

α\underline{λ} : k[t]/t^{p} → k[t_{1}, …, t_{r}]/(t_{1}^{p}, …, t_{r}^{p}), t ↦ ∑_{i=1}^{r} λ_{i}t_{i}.

A theorem of W. Wheeler [22] asserts that for any finite dimensional kE-module M, the subset of those \underline{λ} ∈ k^{r} for which α∗ ; \underline{λ}(M) is not of maximal Jordan type is Zariski closed in 𝔸^{r}.

In this section, we use Theorem 1.12 to verify that this maximal Jordan type is an invariant of the kE-module M independent of the choice of generators of kE. In particular, the intrinsic, basis-free nature of this maximal Jordan type tells us that this Jordan type is also maximal if we view the action of kE on M as the action of the restricted enveloping algebra u(E) of a rank r abelian Lie algebra E with trivial p-restriction determined by an arbitrary choice of k-algebra isomorphism of kE ≃ k𝔾_{a(1)}^{×r} = u(E).

In Theorem 2.9, this intrinsic maximal Jordan type is shown to extend to the data of an arbitrary finite abelian group scheme over k acting upon a finite dimensional k-vector space.

In [10] (as corrected in [11]) and [12], the first two authors consider “π-points”, an alternative to shifted subgroups of E that generalizes to any finite group scheme. We recall this alternative in the following definition.

Definition 2.1. Let G be a finite group scheme over k with group algebra kG. A π-point of G is a (left) flat map of K-algebras

α_{K} : K[t]/t^{p} → KG

which factors through the group algebra KC_{K} ⊂ KG of some unipotent abelian subgroup scheme C_{K} ⊂ G_{K} defined over some field extension K/k.

A π-point α_{K} is said to specialize to the π-point β_{L} if for every finite dimensional kG-module M the condition that β_{L}^{∗}(M_{L}) is free as an L[t]/t^{p}-module implies that α_{K}^{∗}(M_{K}) is free as a K[t]/t^{p}-module.

If α_{K} specializes to β_{L} and β_{L} specializes to α_{K}, then we say that α_{K} is equivalent to β_{L} (denoted by α_{K} ∼ β_{L}). The equivalence class of the π-point α_{K} is denoted [α_{K}].

One of the fundamental properties of π-points is that there is a natural homeomorphism

(2.1.1)  \xymatrix{Ψ_{G} : Π(G) \ar[r]^{∼} & Proj H^{•}(G, k)}

from the space Π(G) of equivalence classes of π-points of G (with topology intrinsically determined by kG-modules) to Proj H^{•}(G, k) with its Zariski topology, where H^{•}(G, k) is the commutative algebra of even dimensional cohomology of G for p odd and the commutative algebra of all cohomology of G for p = 2.

The following proposition makes explicit the equivalence relation α_{K} ∼ β_{L} on π-points of an elementary abelian p-group E.

Proposition 2.2. Let E = (ℤ/pℤ)^{×r} be an elementary abelian p-group viewed as a finite group scheme over k. Choose some k–algebra isomorphism

kE ≃ k[t_{1}, …, t_{r}]/(t_{1}^{p}, …, t_{r}^{p}).

Let α_{K} : K[t]/t^{p} → KE, β_{L} : L[t]/t^{p} → LE be two π-points of kE. Then α_{K} ∼ β_{L} if and only if there exist embeddings K ⊂ Ω, L ⊂ Ω over k of K, L into some field extension Ω over k and some 0 ≠ ω ∈ Ω such that

(α_{K} ⊗_{K} Ω)(t) − ω(β_{L} ⊗_{L} Ω)(t) ∈ Ω[t_{1}, …, t_{r}]/(t_{1}^{p}, …, t_{r}^{p})

is a polynomial in the t_{i}’s with no non-zero constant or linear term.

Proof. The projectivized rank variety of E can be identified with the projective space Proj I/I^{2} (see [2, II.4]) over an algebraically closed field Ω of characteristic p where I denotes the augmentation ideal of ΩE. Since Ω-rational points on the projectivized rank variety of E are precisely equivalence classes of π-points defined over Ω ([10, 2.6,2.9]), one concludes that two Ω-rational π-points α, β : Ω[t]/t^{p} → ΩE, are equivalent if and only if there exists is 0 ≠ ω ∈ Ω such that α(t) − ωβ(t) ∈ I^{2}.

To prove the “if” direction, we observe that non-equivalent π-points remain non-equivalent after base change. Thus, the argument of the preceding paragraph implies that the asserted condition is sufficient to imply α_{K} ∼ β_{L}.

Conversely, assume that α_{K}, β_{L} are equivalent as π-points of kE. Let Ω/k be an algebraically closed field containing both K and L. Let σ : Ω → Ω be a field automorphism of Ω over k, and consider the action on π-points

σ : α_{Ω} ↦ α_{Ω}^{σ}

given by sending α_{Ω} to the Ω-algebra map which sends t to (α_{Ω}(t))^{σ}. If ρ : kG → End_{k}(M) specifies a kG-module M, then ρ(α_{Ω}^{σ}(t)) when viewed as a matrix is simply the result of applying σ to the matrix entries of ρ(α_{Ω}(t)).

The assertion that α_{Ω}, β_{Ω} are equivalent as π-points of kE implies the existence of some automorphism σ of Ω/k with the property that α_{Ω}, β_{Ω}^{σ} are equivalent as π-points of ΩE. This is easily verified (as in [12, 4.6]) by viewing equivalence classes of α_{Ω}, β_{Ω} as Ω-rational points of Proj H^{•}(G, Ω) which map to the same point in Proj H^{•}(G, k). Thus, another application of the argument in the first paragraph implies that we may find some ω ∈ Ω such that α_{Ω}(t) − ω · β_{Ω}^{σ}(t) ∈ Ω[t_{1}, …, t_{r}]/(t_{1}^{p}, …, t_{r}^{p}) is a polynomial in the t_{i}’s with no non-zero constant or linear term. Now twisting the embedding of L ⊂ Ω by the automorphism σ we obtain the desired relationship between α_{K} and β_{L}. □

The following example reflects the “typical” situation in which equivalent πpoints α, β of an elementary abelian group E lead to non-isomorphic k[t]/t^{p}-modules α^{∗}(M), β^{∗}(M).

Example 2.3. Assume p > 2. Let E = ℤ/pℤ × ℤ/pℤ be an elementary abelian group of rank 2, choose an identification of kE with k[x, y]/(x^{p}, y^{p}), and let M be the kE-module (with respect to this identification of kE) defined as the quotient k[x, y]/(x^{2} − y, x^{p}) of kE. Let α : k[t]/t^{p} → k[x, y]/(x^{p}, y^{p}) be defined by sending t to x^{2} − y and let β : k[t]/t^{p} → k[x, y]/(x^{p}, y^{p}) be defined by sending t to y. One readily checks that the k[t]/t^{p}-module α^{∗}(M) is trivial whereas β^{∗}(M) consists of two Jordan blocks of sizes \frac{p+1}{2} and \frac{p−1}{2}. Then α ∼ β by Proposition 2.2, but α^{∗}(M) is not isomorphic to β^{∗}(M).

The observation that the Jordan type can change when one replaces a cyclic shifted subgroup by another within the same equivalence class was made earlier by S. Kaptanoglu, who also observed that for E = ℤ/2ℤ × ℤ/2ℤ this phenomenon does not occur ([16]).

A major conclusion of our work is that if α_{K} has “maximal Jordan type” on a finite dimensional kG-module M as defined below, then β_{L}^{∗}(M_{K}) has the same Jordan type as α_{K}^{∗}(M_{K}) whenever α_{K} ∼ β_{L}. We prove this for an arbitrary finite group scheme G, proceeding in incremental steps of increasing generality, beginning with the case of an elementary abelian p-group.

Let M_{1} be a K[t]/t^{p}-module, and M_{2} be L[t]/t^{p}-module where L, K are field estensions of k. Assume that dim_{K} M_{1} = dim_{L} M_{2} = N. Let the action of t on M_{1} and M_{2} be given by nilpotent matrices of Jordan types \underline{n}_{1} and \underline{n}_{2} respectively, where \underline{n}_{1}, \underline{n}_{2} are partitions of N. We say that the Jordan type of M_{1} is greater or equal than the Jordan type of M_{2}, also written as M_{1} ≥ M_{2}, if \underline{n}_{1} ≥ \underline{n}_{2}.

We shall frequently apply this convention to the situation where we compare the Jordan types of α_{K}^{∗}(M_{K}) and β_{L}^{∗}(M_{L}) for two π-points α_{K}, β_{L} and a finite dimensional kG-module M.

Definition 2.4. Let G be a finite group scheme over k and let M be a kG-module of dimension N. We say that a partition \underline{n} of N is a maximal Jordan type for M if there exists some π-point α_{K} : K[t]/t^{p} → KG such that the Jordan type of α_{K}^{∗}(M_{K}) equals \underline{n} and if there does not exist a π-point β_{L} : L[t]/t^{p} → LG such that the Jordan type of β_{L}^{∗}(M_{L}) is strictly greater than \underline{n}. For any π-point α_{K} such that the Jordan type of α_{K}^{∗}(M_{K}) equals \underline{n}, we say that α_{K}^{∗}(M_{K}) has maximal Jordan type for M, that the Jordan type of α_{K}^{∗}(M_{K}) is maximal for M, and that α_{K} has maximal Jordan type on M.

Furthermore, we say that a partition \underline{n} of N is the absolute maximal Jordan type for M if there exists some π-point α_{K} : K[t]/t^{p} → KG such that the Jordan type of α_{K}^{∗}(M_{K}) equals \underline{n} and if \underline{n} is greater or equal to the Jordan type of β_{L}^{∗}(M_{L}) for every π-point β_{L} of G.

Remark 2.5. If Π(G) is irreducible and if α_{K} is a generic π-point of G, then α_{K}^{∗}(M_{K}) has absolute maximal Jordan type for M by Corollary 4.11. On the other hand, we shall see in Example 4.13 that a kG-module can have more than one maximal Jordan type if Π(G) is reducible.

Remark 2.6. Let M be a kG-module and α_{K} : K[t]/t^{p} → KG be a π-point of G such that α_{K}^{∗}(M_{K}) has maximal Jordan type for M. Let i : C_{K} ⊂ G_{K} be a subgroup scheme such that α_{K} factors as i ∘ α′_{K} : K[t]/t^{p} → KC_{K} → KG. Then α_{K}^{∗}(M_{K}) = α_{K}^{′∗}(i^{∗}(M_{K})) has maximal Jordan type for the restriction i^{∗}(M_{K}) of M_{K} to C_{K}.

The following theorem is a re-interpretation of Theorem 1.12 in terms of π-points of an elementary abelian p-group E = (ℤ/pℤ)^{×r} viewed as a finite group scheme over k. We say that a π-point of a finite group scheme G is generic if its equivalence class is a generic point of Π(G). One example of a generic π-point of E = (ℤ/pℤ)^{×r} is the map

(2.6.1) ν_{k(x_{1},...,x_{r})} : k(x_{1}, …, x_{r})[t]/t^{p} → k(x_{1}, …, x_{r})E, t ↦ ∑_{i=1}^{r} x_{i} · (g_{i} − 1)

where {g_{1}, …, g_{r}} is a set of generators for E.

Theorem 2.7. Let E be an elementary abelian p-group of rank r, and let α_{K} : K[t]/t^{p} → KE be any generic π-point of E. Then for any finite dimensional kEmodule M, the Jordan type of α_{K}^{∗}(M_{K}) is the absolute maximal Jordan type for M.

Moreover, let M be a finite dimensional kE-module and α_{K} be a π-point of E such that the Jordan type of α_{K}^{∗}(M_{K}) is the absolute maximal Jordan type for M. Then for any β_{L} ∼ α_{K}, the Jordan type of β_{L}^{∗}(M_{L}) equals that of α_{K}^{∗}(M_{K}).

Proof. Let {g_{1}, …, g_{r}} be a set of generators for E, and let {t_{i} = g_{i} − 1}_{1≤i≤r} be the corresponding polynomial generators of kE. Let ρ : kG → End_{k}(M) specify the kG-module structure of M. According to Remark 1.5, the π-point ν_{k(x_{1},...,x_{r})} of (2.6.1) has the property that ρ(ν_{k(x_{1},...,x_{r})}(t)) is maximal with respect to the set {ρ(t_{1}), …, ρ(t_{r})}. Theorem 1.12 thus implies that the Jordan type of ρ(ν_{k(x_{1},...,x_{r})}(t)) is greater or equal to the Jordan type of ρ(β_{K}(t)) for any π-point β_{K} : K[t]/t^{p} → KG. Thus, ν_{k(x_{1},...,x_{r})}^{∗}(M_{k(x_{1},...,x_{r})}) has absolute maximal Jordan type for M.

As mentioned above, ν_{k(x_{1},...,x_{r})} is a generic π-point of E. Thus, to complete the proof of the theorem, it suffices to prove the second assertion. Let M be a given finite dimensional kE-module M with kE-structure specified by ρ : kE → End_{k}(M). Assume that α_{K}^{∗}(M_{K}) has maximal Jordan type for M so that ρ(α_{K}(t)) is maximal with respect to the set {ρ(t_{1}), …, ρ(t_{r})}. Since multiplying β_{Ω} by a scalar does not change the Jordan type of β_{Ω}^{∗}(M_{Ω}), Proposition 2.2 enables us to assume that α_{Ω}(t)−β_{Ω}(t) ∈ Ω[t_{1}, …, t_{r}]/(t_{1}^{p}, …, t_{r}^{p}) is a polynomial in the t_{i}’s with no non-zero constant or linear term for some field extension Ω/k. Since extension of scalars preserves the “maximality” property by Proposition 1.3, we still have that ρ(α_{Ω}(t)) is maximal with respect to the set {ρ(t_{1}), …, ρ(t_{r})}. Theorem 1.12 implies that ρ(α_{Ω}(t)) has the same Jordan type as ρ(β_{Ω}(t)), thereby verifying that α_{Ω}^{∗}(M_{Ω}) and β_{Ω}^{∗}(M_{Ω}) have the same Jordan type. Since extension of scalars preserves Jordan type, we conclude that α_{K}^{∗}(M_{K}) and β_{L}^{∗}(M_{L}) have the same Jordan type. □

The following lemma enables us to frequently replace k by a conveniently chosen finite field extension F/k.

Lemma 2.8. Let G be a finite group scheme defined over k, let α_{K}, β_{L} be π-points of G, and let F/k be a finite normal extension. Then α_{K} ∼ β_{L} as π-points of G if and only if there exist embeddings of the composites K̃ = F K, L̃ = F L in some field extension Ω/F such that αK̃ ⊗K̃ Ω, βL̃ ⊗L̃ Ω are equivalent π-points of G_{F}.

Moreover, for any finite field extension F/k, we have that α_{K} is a generic π-point of G if and only if αK̃ is a generic π-point of G_{F}, where K̃ is again a composite of F and K.

Proof. Observe that α_{K}, β_{L} are equivalent π-points of G if and only if their base changes αK̃ : K̃[t]/t^{p} → K̃G, βL̃ : L̃[t]/t^{p} → L̃G remain equivalent π-points of G. Thus, it suffices to assume that K̃ = K, L̃ = L, so that F ⊂ K ∩ L.

We first consider the case in which F/k is separable and thus Galois. In this case, H^{•}(G, k) → H^{•}(G, F) = H^{•}(G, k) ⊗_{k} F is a Galois map, so that

Π(G_{F}) ≃ Proj H^{•}(G_{F} , F) → Proj H^{•}(G, k) ≃ Π(G)

is a finite Galois covering. In particular, the pre-image of any generic point of Π(G) is a generic point of Π(G_{F}) and any two points in the pre-image of a given point of Π(G) (such as [α_{K}] = [β_{L}]) are conjugate by an element of Gal(F/k). Since the isomorphism Π(G_{F}) ≃ Proj H^{•}(G_{F} , F) is compatible with the Galois action (see [12, 4.5]), we get α_{K} ∼ β_{L} as π-points of G if and only if α_{Ω} ∼ β_{Ω}^{σ} as π-points of G_{F}. Twisting the embedding of L into Ω by σ, we obtain the desired result.

More generally, F/k factors as F/F^{s}/k where F^{s}/k is separable and F/F^{s} is purely inseparable. Now, the map H^{•}(G, F^{s}) → H^{•}(G, F) is a purely inseparable isogeny, so that

Π(G_{F}) ≃ Proj H^{•}(G_{F} , F) → Proj H^{•}(G_{F^{s}}, F^{s}) ≃ Π(G_{F^{s}})

is a bijection. Thus, the general case follows. □

We now generalize Theorem 2.7 to an arbitrary finite abelian group scheme.

Theorem 2.9. Let C be an abelian finite group scheme over k and let α_{K} : K[t]/t^{p} → KC be any generic π-point of C. Then for any finite dimensional kC-module M, α_{K}^{∗}(M_{K}) has absolute maximal Jordan type for M.

Moreover, let M be a finite dimensional kC-module and α_{K} be a π-point of C such that α_{K}^{∗}(M_{K}) has maximal Jordan type for M. Then for any β_{L} ∼ α_{K}, the Jordan type of β_{L}^{∗}(M_{L}) equals that of α_{K}^{∗}(M_{K}).

Proof. Let C^{D} be the Cartier dual of C. By [12, 4.2], we may find some finite extension F/k such that C_{F}^{D} = (C_{F}^{D})^{0} × π_{0}(C_{F}^{D}), where (C_{F}^{D})^{0} is the connected component of C_{F}^{D}. Dualizing again, we obtain

C_{F} = (C_{F}^{D})^{D} = ((C_{F}^{D})^{0})^{D} × (π_{0}(C_{F}^{D}))^{D}.

The first factor is coconnected which is equivalent to unipotent, and the second factor is dual to an etale group scheme, and, hence, is semi-simple. Lemma 2.8 enables us to assume that C itself is a product of a unipotent abelian finite group scheme and a semi-simple abelian finite group scheme.

Since any π-point of C necessarily factors (by definition) through a unipotent abelian subgroup scheme of C, it must factor through ((C^{D})^{0})^{D}. Hence, we may assume that C is itself unipotent. Let k̄ be the perfect closure of k. By [21, 14.4], the group algebra k̄C is of the form k̄[t_{1}, …, t_{r}]/(t_{1}^{p^{e_{1}}} , …, t_{r}^{p^{er}}). Since every generator t_{i} is defined over some finite subextension k′/k, we can find a finite purely inseparable L/k such that LC = L[t_{1}, …, t_{r}]/(t_{1}^{p^{e_{1}}} , …, t_{r}^{p^{er}}). Hence, applying Lemma 2.8 again, we may assume

kC ≃ k[t_{1}, …, t_{r}]/(t_{1}^{p^{e_{1}}} , …, t_{r}^{p^{er}}).

Let T_{i} = t_{i}^{p^{e_{i}−1}}. Any π-point α_{K} : K[t]/t^{p} → KC must send t to a polynomial in the t_{i}’s, each monomial of which is divisible by some T_{j} = t_{j}^{p^{e_{j} −1}} since t^{p} = 0. As seen in Remark 1.5, the π-point

α_{k(x_{1},...,x_{r})} : k(x_{1}, …, x_{r})[t]/t^{p} → k(x_{1}, …, x_{r})[t_{1}, …, t_{r}]/(t_{1}^{p^{e_{1}}} , …, t_{r}^{p^{er}})

sending t to ∑_{i} x_{i}T_{i} is a generic π-point of C; moreover, ρ(α_{k(x_{1},...,x_{r})}(t)) is maximal with respect to {ρ(T_{1}), …, ρ(T_{r})} for any finite dimensional kC-module with structure specified by ρ by Theorem 2.7. Thus, α_{k(x_{1},...,x_{r})}^{∗}(M_{k(x_{1},...,x_{r})}) has absolute maximal Jordan type for any finite dimensional kC-module M by Theorem 1.12. Moreover, if M is a specified finite dimensional kC-module with structure specified by ρ and if α_{K}^{∗}(M_{K}) has absolute maximal Jordan type for M and if β_{L} ∼ α_{K}, then as in the proof of Theorem 2.7 we conclude that Theorem 1.12 implies that α_{K}^{∗}(M_{K}) and β_{L}^{∗}(M_{L}) have the same Jordan type. □

3. Generic and maximal Jordan types for infinitesimal group schemes

Recall that a finite group scheme G over k is said to be infinitesimal if its coordinate algebra k[G] is a (Artinian) local k-algebra. The height of such an infinitesimal group scheme is the least integer r such that f^{p^{r}} = 0 for all f in the maximal ideal of k[G]. A 1-parameter subgroup of an infinitesimal group scheme G of height ≤ r over some field extension K/k is a morphism 𝔾_{a(r),K} → G_{K} of finite group schemes over K. If G is an infinitesimal group scheme of height ≤ r, then the functor

(f.g. commutative k-algebras) → (sets)

A ↦ Hom_{grpsch/A}(𝔾_{a(r),A}, G_{A})

is representable by an affine scheme denoted V_{r}(G) = Spec k[V_{r}(G)] ([19, 1.5]). Here, G_{A} is the group scheme over A given by base change, G_{A} = G ×_{Spec k} Spec A.

In particular, there is a universal 1-parameter subgroup

(3.0.1)  \xymatrix{u : 𝔾_{a(r),k[V_{r}(G)]} \ar[r] & G_{k[V_{r}(G)]}}

associated to the identity map of k[V_{r}(G)] for any infinitesimal group scheme G. In some ways, the existence of this universal 1-parameter subgroup makes the study of representations of infinitesimal group schemes more tractable than the representations of finite groups.

In order to relate 1-parameter subgroups to π-points, we use the map of group algebras

\xymatrix{k𝔾_{a(1)} = k[t]/t^{p} \ar[r]^{t↦u_{r−1}} & k[u_{0}, …, u_{r−1}]/(u_{0}^{p}, …, u_{r−1}^{p}) = k𝔾_{a(r)}}

where u_{i} is the basis element dual to t^{p^{i}} (so that u_{i} = \frac{1}{p^{i}!} \frac{d^{p^{i}}}{dt^{p^{i}}}). This is a map of Hopf algebras if and only if r = 1 (in which case, we view this map as the identity). We employ the notation

\xymatrix{ϵ : k𝔾_{a(1)} \ar[r] & k𝔾_{a(r)}}

for this map of group algebras, and stay alert to the fact that ϵ is not necessarily a map of Hopf algebras.

There is a natural map

\xymatrix{Θ̃_{G} : V_{r}(G)\\{0\} \ar[r] & Π(G)}

which induces a homeomorphism (see [20, 5.2] and [12, 3.6]) of Zariski topological spaces

(3.0.2)  \xymatrix{Θ_{G} : Proj(V_{r}(G)) \ar[r] & Π(G).}

This construction sends \xymatrix{x : Spec k(x) \ar[r] & V_{r}(G)} to

\xymatrix{μ_{x} ∘ ϵ : k(x)[t]/t^{p} \ar[r] & k(x)𝔾_{a(r),k(x)} \ar[r] & k(x)G_{k(x)} = k(x)G}

where \xymatrix{μ_{x} : k(x)𝔾_{a(r),k(x)} \ar[r] & k(x)G_{k(x)}} is the map on group algebras induced by the 1-parameter subgroup μ_{x} associated to the point x ∈ V_{r}(G) with residue field k(x). Indeed, the composition Ψ_{G} ∘ Θ_{G} of the homeomorphisms (2.1.1) and (3.0.2) is induced by the isogeny exhibited in [20, 5.2].

1-parameter subgroups give us “natural” representatives for equivalence classes of π-points. As we now verify, for each finite dimensional kG-module M and each [α_{K}] ∈ Π(G), the Jordan type of α_{K}^{∗}(M_{K}) is independent of the 1-parameter subgroup representing [α_{K}].

Proposition 3.1. Let G be an infinitesimal group scheme. For any π-point α_{K}, let x ∈ V_{r}(G)\{0} project onto [α_{K}] ∈ Π(G). Then

μ_{x} ∘ ϵ : k(x)[t]/t^{p} → k(x)G ∼ α_{K} : K[t]/t^{p} → KG.

Moreover, if μ : 𝔾_{a(r),K} → G_{K}, ν : 𝔾_{a(r),L} → G_{L} are non-zero 1-parameter subgroups of G which map to the same equivalence class of π-points in Π(G), then the Jordan type of (μ ∘ ϵ)^{∗}(M_{K}) is the same as the Jordan type of (ν ∘ ϵ)^{∗}(M_{L}).

Proof. The first assertion is merely a recollection of the definition of Θ_{G} , granted that Θ_{G}(x) = [α_{K}].

If μ : 𝔾_{a(r),K} → G_{K} is a 1-parameter subgroup for some field extension K/k and if k(x) is the residue field at the point x ∈ V_{r}(G) determined by the map Spec K → V_{r}(G) corresponding to μ, then μ determines an extension K/k(x) of fields over k and

(3.1.1)  μ = μ_{x} ×_{Spec k(x)} Spec K.

In particular, (μ ∘ ϵ)^{∗}(M_{K}) has the same Jordan type as (μ_{x} ∘ ϵ)^{∗}(M_{k(x)}).

Assume now that μ : 𝔾_{a(r),K} → G_{K}, ν : 𝔾_{a(r),L} → G_{L} are 1-parameter subgroups determining the same point of Π(G). By the preceding paragraph, we may assume that K = L. Then μ, ν are related by the action V_{r}(G) × 𝔸^{1} → V_{r}(G) reflected in the grading of the coordinate algebra k[V_{r}(G)]. As can be seen using [20, 1.12], the action of some a ∈ 𝔸^{1}(K) sends μ ∘ ϵ : K[t]/t^{p} → KG to a^{p^{r−1}}(μ ∘ ϵ) : K[t]/t^{p} → KG. Thus, (μ ∘ ϵ)^{∗}(M_{K}) has the same Jordan type as (ν ∘ ϵ)^{∗}(M_{K}). □

We say that a point η ∈ V_{r}(G) specializes to a point ζ ∈ V_{r}(G) if there exists a map ψ : Spec R → V_{r}(G) where R is a local, integral k-algebra whose field of fractions has image η and whose residue field has image ζ. More generally, if μ : 𝔾_{a(r),K} → G, ν : 𝔾_{a(r),L} → G are 1-parameter subgroups obtained by base change from η, ζ and if η specializes to ζ, then we say μ specializes to ν.

Proposition 3.2. Let G be an infinitesimal group scheme. Assume that the 1parameter subgroup μ : 𝔾_{a(r),K} → G specializes to the 1-parameter subgroup ν : 𝔾_{a(r),L} → G. Then for any finite dimensional kG-module M, the Jordan type of (μ ∘ ϵ)^{∗}(M_{K}) is greater or equal to the Jordan type of (ν ∘ ϵ)^{∗}(M_{L}).

Proof. Let η, ζ ∈ V_{r}(G) be such that μ : 𝔾_{a(r),K} → KG, ν : 𝔾_{a(r),L} → LG are obtained by base change from μ_{η}, ν_{ζ}. Let ψ : Spec R → V_{r}(G) be a morphism with R a local, integral k-algebra whose field of fractions Ω has image η and whose residue field F has image ζ. This corresponds to an R-group scheme homomorphism ψ : 𝔾_{a(r),R} → G_{R}. By Proposition 3.1, it suffices to consider the R[t]/t^{p}-module obtained as the pull-back of M_{R} via ψ ∘ ϵ and show that the Jordan form of the endomorphism t on M_{R} ⊗_{R} Ω is greater or equal to the Jordan form of the endomorphism t on M_{R} ⊗_{R} F. This in turn is equivalent to proving for all i, 1 ≤ i < p, that the rank of t^{i} on M_{R} ⊗_{R} Ω is greater or equal to the rank of t^{i} on M_{R} ⊗_{R} F. This “upper-semicontinuity” property follows easily from Nakayama’s Lemma. □

A simple geometric argument shows that if η ∈ V_{r}(G) projects onto [α_{K}], ζ ∈ V_{r}(G) projects onto [β_{L}], and [β_{L}] is a specialization of [α_{K}] as points of Π(G), then η specializes to ζ as points of V_{r}(G) as formulated prior to Proposition 3.2. The following property of Jordan types is now an immediate corollary of Propositions 3.1 and 3.2.

Corollary 3.3. Let G be an infinitesimal group scheme, and consider a π-point α_{K} : K[t]/t^{p} → KG specializing to the π-point β_{L} : L[t]/t^{p} → LG. Let μ : 𝔾_{a(r),Ω} → G_{Ω}, ν : 𝔾_{a(r),Σ} → G_{Σ} be non-zero 1-parameter subgroups of G which project to [α_{k}], [β_{L}] ∈ Π(G). Then, for any finite dimensional kG-module M, the Jordan type of (μ ∘ ϵ)^{∗}(M_{Ω}) is greater or equal to the Jordan type of (ν ∘ ϵ)^{∗}(M_{Σ})

Using Theorem 2.9, we now establish the well-definedness of the generic Jordan type (i.e., independence of the choice of representing generic π-point) of a finite dimensional kG module for any infinitesimal group scheme G.

Theorem 3.4. Let G be an infinitesimal group scheme over k and let

α_{K} : K[t]/t^{p} → KG, β_{L} : L[t]/t^{p} → LG

be equivalent generic π-points of G. Then for any finite dimensional kG-module M, α_{K}^{∗}(M_{K}) has the same Jordan type as β_{L}^{∗}(M_{L}).

Proof. Write \xymatrix{α_{K} = i_{C} ∘ α′_{K} : K[t]/t^{p} \ar[r]^{α′_{K}} & KC_{K} \ar[r]^{i_{C}} & KG} where C_{K} ⊂ G_{K} is

an abelian subgroup scheme (which is necessarily unipotent). Choose a 1-parameter subgroup μ : 𝔾_{a(r),K} → C_{K} such that [μ ∘ ϵ] = [α′_{K}] ∈ Π(C_{K}) and let η : 𝔾_{a(r),Ω} → C_{Ω} be another 1-parameter subgroup such that η ∘ ϵ is a generic π-point of C_{K}. Let M be a kG-module. By Theorem 2.9, (η ∘ ϵ)^{∗}(i_{C}^{∗}M_{Ω}) has maximal Jordan type among all γ_{F}^{∗} (i_{C}^{∗}M_{F}) as F/K ranges over all field extensions over K, and γ_{F} ranges over all π-points of C_{K}. Since i_{C} ∘ η ∘ ϵ ∼ i_{C} ∘ μ ∘ ϵ as π-points of G, Proposition 3.1 implies that (μ ∘ ϵ)^{∗}(i_{C}^{∗}M_{K}) has the same Jordan type as (η ∘ ϵ)^{∗}(i_{C}^{∗}M_{Ω}), and thus is maximal among all γ_{F}^{∗} (i_{C}^{∗}M_{F}) as γ_{F} ranges over all π-points of C_{K}. Since μ∘ϵ ∼ α′_{K} as π-points of C_{K}, Theorem 2.9 implies that the Jordan type of α_{K}^{∗}(M_{K}) = α′_{K}(i_{C}^{∗}M_{K}) is also the same as the Jordan type of (μ ∘ ϵ)^{∗}(i_{C}^{∗}M_{K}).

Consider \xymatrix{β_{L} = i_{D} ∘ β′_{L} : L[t]/t^{p} \ar[r]^{β′_{L}} & LD_{L} \ar[r]^{i_{D}} & LG} equivalent to α_{K} (where D_{L} ⊂ G_{L} is a unipotent abelian subgroup scheme) and let ν : 𝔾_{a(r),L} → G_{L} be a 1-parameter subgroup scheme such that [ν ∘ ϵ] = [β′_{L}] ∈ Π(D_{L}). The preceding argument tells us that the Jordan type of β_{L}^{∗}(M_{L}) is the same as the Jordan type of (ν ∘ϵ)^{∗}(i_{D}^{∗}M_{L}). On the other hand, Proposition 3.1 implies that the Jordan type of (μ ∘ ϵ)^{∗}(i_{C}^{∗}M_{K}) equals that of (ν ∘ ϵ)^{∗}(i_{D}^{∗}(M_{L}) since μ ∘ ϵ ∼ α_{K} ∼ β_{L} ∼ ν ∘ ϵ. □

As seen implicitly in the proof of Theorem 2.9, Π(C) is irreducible if C is an abelian finite group scheme. However, this is not true for a general finite group scheme, even in the case of a height 1 infinitesimal group scheme (i.e., a restricted Lie algebra). For example, the nullcone of a Borel subalgebra of a reductive Lie algebra is not necessarily irreducible for small primes (see [5]). If Π(G) is not irreducible, then the Jordan type at a generic point need not be maximal, but Corollary 4.11 asserts that maximal Jordan types for a given finite dimensional module M are realized at generic points.

Of course, maximal Jordan types can be realized at π-points which are not generic. We see in the following theorem that if the Jordan type for a given kGmodule is maximal at some π-point then this Jordan type is independent of the representative of that π-point.

Theorem 3.5. Let G be an infinitesimal group scheme over k and M a finite dimensional kG-module. Let β_{L} : L[t]/t^{p} → LG be a π-point of G with the property that the Jordan type of β_{L}^{∗}(M_{L}) is maximal for M. Then for any π-point α_{K} : K[t]/t^{p} → KG which specializes to β_{L}, the Jordan type of α_{K}^{∗}(M_{K}) equals the Jordan type of β_{L}^{∗}(M_{L}).

Proof. As in the proof of 3.4 we write \xymatrix{α_{K} = i_{C} ∘ α′_{K} : K[t]/t^{p} \ar[r]^{α′_{K}} & KC_{K} \ar[r]^{i_{C}} & KG}

and \xymatrix{β_{L} = i_{D} ∘ β′_{L} : L[t]/t^{p} \ar[r]^{β′_{L}} & LD_{L} \ar[r]^{i_{D}} & LG} with C_{K} ⊂ G_{K}, D_{L} ⊂ G_{L} abelian

unipotent subgroup schemes. Let μ : 𝔾_{a(r),K} → C_{K}, ν : 𝔾_{a(r),L} → D_{L} be 1parameter subgroups such that [β′_{L}] = [ν ∘ ϵ] ∈ Π(D_{L}) and [α′_{K}] = [μ ∘ ϵ] ∈ Π(C_{K}). The maximality of β_{L}^{∗}(M_{L}) = (β′_{L})^{∗}(i_{D}^{∗}(M_{L})) for M and the fact that β′_{L} ∼ ν ∘ ϵ imply by Theorem 2.9 that the Jordan type of β_{L}^{∗}(M_{L}) equals that of (ν ∘ ϵ)^{∗}(i_{D}^{∗}(M_{L})) = ((i_{D} ∘ ν) ∘ ϵ)^{∗}(M_{L}). Because α_{K} ∼ (i_{C} ∘ μ) ∘ ϵ specializes to β_{L} ∼ (i_{D} ∘ ν) ∘ ϵ, Proposition 3.2 implies that ((i_{C} ∘ μ) ∘ ϵ)^{∗}(M_{K}) is likewise maximal for M. Thus, applying Theorem 2.9 to α_{K} ∼ (i_{C} ∘ μ) ∘ ϵ, we conclude that α_{K}^{∗}(M_{K}) is also maximal for M.

□

4. Generic and maximal Jordan types for arbitrary finite group schemes

Let G be a finite group scheme of the form G^{0} ⋊ τ, where the connected component G^{0} ⊂ G is geometrically connected and where τ = π_{0}(G) is the (constant) group of connected components of G. For each elementary abelian p-subgroup E ⊂ τ, define Π_{0}((G^{0})^{E} × E) ⊂ Π((G^{0})^{E} × E) to be the subset of those π-points which do not admit a representative factoring through (G^{0})^{E} ×E′ with E′ a proper subgroup of E.

Recall the “Quillen decomposition” for Π(G) by locally closed subspaces ([12, 4.13]):

(4.0.1)  ∐ Π_{0}((G^{0})^{E} × E)/N_{τ} (E) ≃ Π(G)

where the disjoint union is indexed by conjugacy classes of elementary abelian psubgroups of τ, and where N_{τ} (E) denotes the normalizer of E in τ.

In order to consider the Jordan types α_{K}^{∗}(M_{K}) for specific π-points representing [α_{K}] ∈ Π(G), we sharpen somewhat this decomposition by considering π-points up to the finer equivalence defining Π(H) for subgroups H of the form (G^{0})^{E} × E.

Proposition 4.1. Let G be a finite group scheme of the form G^{0} ⋊ τ, with τ = π_{0}(G) constant and G^{0} geometrically connected. Assume given [α_{K}] ∈ Π(G) and a representative

\xymatrix{α_{K} = i ∘ α′_{K} : K[t]/t^{p} \ar[r]^{α′_{K}} & K((G^{0})^{E} × E) \ar[r]^{i} & KG}

of [α_{K}].

Then there exists a π-point α′′_{K} : K[t]/t^{p} → K((G^{0})^{E} × E′) equivalent to α′_{K} as a π-point of (G^{0})^{E} × E with E′ ⊂ E such that [α_{K}] ∈ Π_{0}((G^{0})^{E′} × E′).

Proof. Let α′_{K} : K[t]/t^{p} → K((G^{0})^{E} × E) be given by

t ↦ ∑_{i=1}^{m} a_{i}f_{i} + ∑_{j=1}^{r} b_{j}s_{j} + p(\underline{f}, \underline{s}) ∈ K(G^{0})^{E} ⊗_{K} KE,

where {f_{i}; 1 ≤ i ≤ n} is a set of commuting nilpotent elements of K((G^{0})^{E}), where s_{j} = g_{j} − 1 with {g_{1}, …, g_{r}} ⊂ E a set of generators for E, where a_{i}, b_{j} are elements of K, and where p(\underline{f}, \underline{s}) is a polynomial in the f_{i}, s_{j} without constant or linear term. Let α′′_{K,0} be the π-point of G^{0} defined by

α′′_{K,0}(t) = ∑_{i=1}^{m} a_{i}f_{i}

and α′′_{K,1} be the π-point of τ defined by

α′′_{K,1}(t) = ∑_{j=1}^{r} b_{j}s_{j}.

Finally, let

α′′_{K} = α′′_{K,0} + α′′_{K,1}

and let E′ ⊂ E equal the minimal subgroup such that ∑_{j=1}^{r} b_{j}s_{j} ∈ KE′. By [10, 2.2] and a simple base change argument, α′′_{K} is equivalent to α′_{K} as a π-point of (G^{0})^{E} × E.

To complete the proof, it suffices to show that α_{K} is not equivalent (as a π-point of G) to some π-point factoring through (G^{0})^{E′′} × E′′ for some strictly smaller subgroup E′′ ⊂ E′. This is immediate if E′ is trivial. Thus, we may assume that E′ is non-trivial.

Suppose there exists a π-point

\xymatrix{γ_{L} = i′ ∘ γ′_{L} : L[t]/t^{p} \ar[r]^{γ′_{L}} & L((G^{0})^{E′′} × E′′) \ar[r]^{i′} & LG}

equivalent to α_{K} which factors through (G^{0})^{E′′} × E′′ with E′′ ⊂ E′. We may assume that E′′ is a minimal such subgroup for γ_{L}. By passing to a field extension if necessary, we may assume L = K. As for α_{K}, let γ′′_{K} = γ′′_{K,0} + γ′′_{K,1} be the “linear” part of γ′_{K} with respect to the same set of generators s_{i} of KE. If N is a (finite dimensional) τ-module, let p^{∗}N denote the kG-module with action given by projecting kG to kτ. Observe that for any finite-dimensional kG-module M, (i∘α′′_{K})^{∗}(M_{K}) is free if and only if (i′ ∘γ′′_{K})^{∗}(M_{K}) is free since i∘α′′_{K} ∼ α_{K} ∼ γ_{K} ∼ i′ ∘ γ′′_{K}. Since the action of G^{0} on modules of the form p^{∗}N for a τ-module N is trivial, we conclude that (i_{E} ∘ α′′_{K,1})^{∗}(N_{K}) ≃ (i ∘ α′′_{K})^{∗}(p^{∗}N_{K}), where i_{E} : E ↪ τ is the embedding of E into τ. Similarly, (i_{E′′} ∘ γ′′_{K,1})^{∗}(N_{K}) ≃ (i ∘ γ′′_{K})^{∗}(p^{∗}N_{K}), where i_{E′′} : E′′ ↪ τ is the embedding of E′′ into τ. Since N can be taken to be an arbitrary τ-module, we conclude that i_{E} ∘ α′′_{K,1} ∼ i_{E′′} ∘ γ′′_{K,1} as π-points of τ.

By the minimality assumption on E′ for α′′_{K}, we conclude that [α′′_{K,1}] ∈ Π_{0}(E′). Similarly, [γ′′_{K,1}] ∈ Π_{0}(E′′). If E′′ were a proper subgroup of E′, then the “Quillen decomposition” for the finite group τ (see (4.0.1)) would imply that Π_{0}(E′)/N_{τ} (E′)∩ Π_{0}(E′′)/N_{τ} (E′′) = ∅ in Π(τ). This contradicts the fact that i_{E} ∘ α′′_{K,1} ∼ i_{E′′} ∘ γ′′_{K,1}, so that we conclude that E′′ = E′ and thus that E′ is minimal as required. □

The following theorem is the culmination of special cases proved earlier as Theorems 2.7, 2.9, and 3.4.

Theorem 4.2. Let G be a finite group scheme over k and let

α_{K} : K[t]/t^{p} → KG, β_{L} : L[t]/t^{p} → LG

be equivalent generic π-points of G. Then for any finite dimensional kG-module M, α_{K}^{∗}(M_{K}) has the same Jordan type as β_{L}^{∗}(M_{L}).

Proof. As seen in [12, 4.2], we may find a finite field extension F/k such that G_{F} satisfies the hypotheses of Proposition 4.1 that G_{F} splits as a semi-direct product G_{F}^{0} ⋊ τ, with τ = π_{0}(G_{F}) constant and G_{F}^{0} geometrically connected. Thus, appealing to Lemma 2.8, we may assume that G itself satisfies these conditions.

Let [α_{K}] ∈ Π(G) be a generic point and choose some representative

\xymatrix{α_{K} = i_{C} ∘ α̃_{K} : K[t]/t^{p} \ar[r]^{α̃_{K}} & KC_{K} \ar[r]^{i_{C}} & KG}

with C_{K} ⊂ G_{K} a unipotent abelian subgroup scheme. Let E ⊂ π_{0}(C_{K}) be the maximal elementary abelian p-subgroup and observe that Π(C_{K}^{0} × E) → Π(C_{K}) is a homeomorphism. Thus, we may choose \xymatrix{α̃′_{K} : K[t]/t^{p} \ar[r] & K(C_{K}^{0} × E)} such that when composed with the embedding K(C_{K}^{0} ×E) ↪ KC_{K} this map becomes equivalent to α̃_{K} as a π-point of C_{K}. Let \xymatrix{α′_{K} = i ∘ α̃′_{K} : K[t]/t^{p} \ar[r] & K((G^{0})^{E} × E)} where i : K(C_{K}^{0} × E) ↪ K((G^{0})^{E} × E) is the embedding. By construction, the composition of α′_{K} with the embedding K((G^{0})^{E} × E) ↪ KG is equivalent to α_{K} as a π-point of G. In view of (4.0.1), the fact that α_{K} is generic implies that α′_{K} is a generic π-point of (G^{0})^{E} × E. By Proposition 4.1, we can find

\xymatrix{α′′_{K} : K[t]/t^{p} \ar[r] & K((G^{0})^{E} × E′)}

where E′ is a subgroup of E and such that [α′′_{K}] ∈ Π_{0}((G^{0})^{E′} × E′) and α′′_{K} ∼ α′_{K} as π-points of (G^{0})^{E} × E. The latter implies that α′′_{K} is generic for (G^{0})^{E} × E. Since k((G^{0})^{E} × E) ≃ k((G^{0})^{E} × 𝔾_{a(1)}^{×r}) where r is the rank of E, we may apply Theorem 3.4 to (G^{0})^{E} × E. Thus,

α_{K}^{∗}(M_{K}) ≃ α_{K}^{′′∗}(i^{′∗}M_{K})

where i′ : (G^{0})^{E} × E′ ↪ G. Repeating this argument for β_{L}, we conclude that

β_{L}^{∗}(M_{L}) ≃ β_{L}^{′′∗}(j^{′∗}M_{L})

where \xymatrix{β′′_{L} : L[t]/t^{p} \ar[r] & L((G^{0})^{F} × F′)} is a generic π-point of (G^{0})^{F} × F′ with

F′ ⊂ F elementary abelian subgroups of τ, [β′′_{L}] ∈ Π_{0}((G^{0})^{F′} ×F′), and j′∘β′′_{L} ∼ β_{L}. Here, j′ : (G^{0})^{F} × F′ ↪ G.

We therefore obtain [α′′_{K}] ∈ Π_{0}((G^{0})^{E′} × E′), [β′′_{L}] ∈ Π_{0}((G^{0})^{F′} × F′) such that α′′_{K}, β′′_{L} become equivalent (generic) π-points of G after composing with the embeddings of the corresponding subgroup schemes into G. This implies that E′ and F′ are conjugate by an element of τ; otherwise

Π_{0}((G^{0})^{E′} × E′)/N_{τ} (E′) ∩ Π_{0}((G^{0})^{F′} × F′)/N_{τ} (F′)

would be empty, which would contradict the Quillen decomposition (4.0.1). Thus, after replacing β′′_{L} by a conjugate, which does not affect the Jordan type, we may assume that α′′_{K}, β′′_{L} are equivalent generic π-points of (G^{0})^{E′} × E′. Applying Theorem 3.4 once again (using the isomorphism k((G^{0})^{E′} ×E′) ≃ k((G^{0})^{E′} ×𝔾_{a(1)}^{×r′}), where r′ = rank E′, we conclude that the Jordan type of α_{K}^{′′∗}(i^{′∗}M_{K}) equals that of β_{L}^{′′∗}(j^{′∗}M_{L}), and hence the Jordan type of α_{K}^{∗}(M_{K}) equals that of β_{L}^{∗}(M_{L}). □

We emphasize the conclusion of Theorem 4.2 with the following definition. As we shall see in examples given at the end of this section, a given kG-module M can have different Jordan types at different (equivalence classes of) generic π-points; moreover, at some generic points, the Jordan type need not be maximal.

Definition 4.3. Let G be a finite group scheme and M be a kG-module. A generic Jordan type of M is the Jordan type of α_{K}^{∗}(M_{K}) for some generic π-point α_{K} of G. We denote the isomorphism class of α^{∗}(M_{K}) (as a K[t]/t^{p}-module) by

[α_{K}]^{∗}(M_{K}).

For the purposes of the following proposition, we utilize the coproduct on the algebra k[t]/t^{p} given by the formula ∇(t) = t ⊗ 1 + 1 ⊗ t. We point out here that this is a temporary convention: as it will be shown in Corollary 4.5 and more generally in Remark 4.6, the Jordan type of a tensor product of finite dimensional k[t]/t^{p}-modules does not depend on the Hopf algebra structure given to k[t]/t^{p}.

Since a π-point is not necessarily a Hopf algebra map and not even always equivalent to one, the following proposition is somewhat striking. The conclusion of this proposition does not always hold for non-generic π-points.

Proposition 4.4. Let C be a unipotent abelian finite group scheme over k and let α_{K} : K[t]/t^{p} → KC be a generic π-point of C. Then for any pair of finite dimensional kC-modules M, N,

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

Proof. Because we may base change to the perfect closure of k without changing the conclusion, we may assume that k is perfect. Thus, we may assume that kC ≃ k[t_{1}, …, t_{r}]/(t_{1}^{p^{e_{1}}} , …, t_{r}^{p^{er}}) (see [21, 14.4]). Set T_{i} = t_{i}^{p^{e_{i}−1}}. Then any π-point β : k[t]/t^{p} → kC sends t to a polynomial in {t_{i}} whose p-th power is trivial and thus each of whose monomials is divisible by some T_{i}; the flatness of β is equivalent to the conditon that some monomial constituting β(t) is a non-zero multiple of some T_{i}.

Let ρ_{M} : kC → End(M) be the map defined by the representation M of C, and similarly define ρ_{N} and ρ_{M⊗N}. Since α_{K} is generic, Theorem 2.9 implies that ρ_{M}(α_{K}(t)) is maximal among the images under ρ_{M} of all p-nilpotent elements of the commutative algebra kC, and similarly for N. Proposition 1.3 implies the equivalences

(4.4.1)  ρ_{M}(α_{K}(t)) ∼ ρ_{M}(α_{K}(t)) + s_{1}ρ_{M}(T_{1}) + ⋯ + s_{r}ρ_{M}(T_{r})

and

(4.4.2)  ρ_{N}(α_{K}(t)) ∼ ρ_{N}(α_{K}(t)) + s_{r+1}ρ_{N}(T_{1}) + ⋯ + s_{2r}ρ_{N}(T_{r})

where s_{i} are indeterminants. Therefore,

(4.4.3) ρ_{M}(α_{K}(t)) ⊗ 1 + 1 ⊗ ρ_{N}(α_{K}(t)) ∼ ρ_{M}(α_{K}(t)) ⊗ 1 + s_{1}ρ_{M}(T_{1}) ⊗ 1 + ⋯ +
s_{r}ρ_{M}(T_{r}) ⊗ 1 + 1 ⊗ ρ_{N}(α_{K}(t)) + 1 ⊗ s_{r+1}ρ_{N}(T_{1}) + ⋯ + 1 ⊗ s_{2r}ρ_{N}(T_{r}).

Indeed, if A is the matrix which makes the endomorphisms of 4.4.1 similar, and B is such a matrix for 4.4.2, then A ⊗ B makes the endomorphisms in 4.4.3 similar. Proposition 1.3 implies that ρ_{M}(α_{K}(t)) ⊗ 1 + 1 ⊗ ρ_{N}(α_{K}(t)) is maximal with respect to {ρ_{M}(T_{i}) ⊗ 1, 1 ⊗ ρ_{N}(T_{j}); 1 ≤ i, j ≤ r}. Thus, Theorem 1.12 implies that ρ_{M}(α_{K}(t)) ⊗ 1 + 1 ⊗ ρ_{N}(α_{K}(t)) is maximal with respect to

(4.4.4) \{ρ_{M}(p(\underline{t}))⊗1, 1⊗ρ_{N}(q(\underline{t})); each monomial of p, q divisible by some T_{i}\}.

Let ∇ : kC → kC ⊗ kC be the coproduct on kC, and let I be the augmentation ideal of kC. Since ∇(α_{K}(t)) − α_{K}(t) ⊗ 1 − 1 ⊗ α_{K}(t) ∈ I ⊗ I (see [15, I.2.4]) and is p-nilpotent, this difference is a sum of terms each of which is either a product of the form (p(\underline{t}) ⊗ 1) · (1 ⊗ f(\underline{t})) with each monomial of p(\underline{t}) divisible by some T_{i} and with f having no constant term or a product of the form (g(\underline{t}) ⊗ 1) · (1 ⊗ q(\underline{t})) with g having no constant term and each monomial of q(\underline{t}) divisible by some T_{i}. Theorem 1.12 and the maximality of ρ_{M}(α_{K}(t)) ⊗ 1 + 1 ⊗ ρ_{N}(α_{K}(t)) with respect to (4.4.4) imply

ρ_{M}(α_{K}(t)) ⊗ 1 + 1 ⊗ ρ_{N}(α_{K}(t)) ∼ ((ρ_{M} ⊗ ρ_{N}) ∘ ∇)(α_{K}(t)) = ρ_{M⊗N}(α_{K}(t)).

Thus, α_{K}^{∗}(M) ⊗_{K} α_{K}^{∗}(N) and α_{K}^{∗}(M_{K} ⊗_{K} N_{K}) have the same Jordan type.

□

The following is an interesting, though very special case of Proposition 4.4 in which α_{K} : K[t]/t^{p} → KC is the identity map id : k[t]/t^{p} → k[t]/t^{p}.

Corollary 4.5. Let C be a unipotent abelian finite group scheme with kC = k[t]/t^{p} and let M, N be finite dimensional kC-modules. Then the tensor product of M, N as kC-modules is isomorphic as a k[t]/t^{p}-module with the tensor product M ⊗ N using the coproduct ∇(t) = t ⊗ 1 + 1 ⊗ t.

Remark 4.6. The proof of Proposition 4.4 does not use the fact that kC is the Hopf algebra of a group scheme: namely, we do not need cocommutativity of the coproduct. The only fact about the coproduct which is needed is that ∇(α_{K}(t)) − α_{K}(t)⊗1−1⊗α_{K}(t) ∈ I ⊗I, a property which holds for the coproduct of any Hopf algebra (see, for example, ex.3 on p.19, [21]). With this in mind, we can strengthen the statement of the Corollary 4.5 as follows: Let M, N be finite dimensional k[t]/t^{p}-modules, let M ⊗N denote the k[t]/t^{p}-module determined by the coproduct ∇(t) = t ⊗ 1 + 1 ⊗ t. Then for any other coproduct ∇′ : k[t]/t^{p} → k[t]/t^{p} ⊗ k[t]/t^{p} associated to a Hopf algebra structure on k[t]/t^{p}, the resulting k[t]/t^{p}-module is isomorphic to M ⊗ N.

Proposition 4.4 provides the key verification which enables us to prove the following pleasing properties of generic Jordan type. Recall that the Heller shift of a kG-module M, denoted ΩM, is defined to be the kernel of the surjection \xymatrix{P (M) \ar@{->>}[r] & M} , where P (M) is the minimal projective cover of M.

Proposition 4.7. Let G be a finite group scheme over a field k and let [α_{K}] ∈ Π(G) be a generic point for some K/k. Let M, N be finite dimensional kG-modules.

• [α_{K}]^{∗}(M_{K} ⊕ N_{K}) ≃ [α_{K}]^{∗}(M_{K}) ⊕ [α_{K}]^{∗}(N_{K}).

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

• [α_{K}]^{∗}(Ω(M_{K})) = Ω([α_{K}]^{∗}(M_{K})) in the stable module category of finite

dimensional K[t]/t^{p}-modules, stmod(K[t]/t^{p}).

Thus, [α_{K}]^{∗} : (kG − mod) → (K[t]/t^{p} − mod) induces a functor on tensor triangulated categories

[α_{K}]^{∗} : (stmod(kG)) → (stmod(K[t]/t^{p})).

Proof. The first property follows from the observation that α_{K}^{∗} commutes with direct sums.

Let α_{K} : K[t]/t^{p} → KG be a generic π-point and let C_{K} ⊂ G_{K} be a unipotent abelian subgroup scheme through which α_{K} factors. Write α_{K} = i∘α′_{K} : K[t]/t^{p} → KC_{K} → KG. If we replace α′_{K} by a generic π-point α′′_{L} : L[t]/t^{p} → LC_{L} of C_{K}, the new composition i ∘ α′′_{L} represents the same generic point of Π(G). Thus, [α_{K}]^{∗} = [i ∘ α′′_{L}]^{∗}. The second property now follows from the observation that i is a Hopf algebra map and that Proposition 4.4 applies to α′′_{L}.

The third property follows from the exactness of α_{K}^{∗} and the fact that α_{K}^{∗} sends projectives to projectives.

□

For an arbitrary finite group scheme G with Π(G) reducible, the following construction produces a finite dimensional kG-module M which has at least two distinct generic Jordan types. We implicitly use Proposition 4.7 in the justification of this example.

Example 4.8. Let G be any finite group scheme with Π(G) reducible; for example, G any finite group with at least two distinct conjugacy classes of maximal elementary abelian p-subgroups. Write Π(G) = X ∪ Y with X, Y proper closed subsets and choose generic points [α_{K}] ∈ X, [β_{L}] ∈ Y. Extending scalars, if necessary, we can assume L = K. Choose ζ ∈ H^{2i}(G, k), ξ ∈ H^{2j}(G, k) such that

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

(Here, α_{K}^{∗} is the map in cohomology H^{∗}(G, K) → H^{∗}(K[t]/t^{p}, K) induced by α_{K}.) Let L_{ζ}, L_{ξ} be the “Carlson modules” associated to ζ, ξ (see [2, II.5.9]). By definition of L_{ζ}, we have a short exact sequence

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

where the map ζ̃ : Ω^{2i}k → k is given by ζ via the isomorphism \underline{Hom}(Ω^{2i}k, k) = H^{2i}(G, k). Extending scalars to K and then pulling back the short exact sequence via the map α_{K}, we get a short exact sequence of K[t]/t^{p} - modules

\xymatrix{0 \ar[r] & α_{K}^{∗}(L_{ζ,K}) \ar[r] & Ω^{2i}K \ar[r]^{(ζ̃_{K})↓_{α_{K}}} & K \ar[r] & 0.}

Since α_{K}^{∗}(ζ_{K}) = 0, the map \xymatrix{(ζ̃_{K}) ↓_{α_{K}}: Ω^{2i}K \ar[r] & K} factors through a projective

K[t]/t^{p} - module. Since Ω^{2i}K = K ⊕ proj as K[t]/t^{p} - modules, we conclude that the kernel α_{K}^{∗}(L_{ζ,K}) = K ⊕ ΩK ⊕ proj . In other words, the Jordan type of α_{K}^{∗}(L_{ζ,K}) is 1[1] + 1[p − 1] + m[p], for some m ≥ 0. Here, we denote by [i] an indecomposable K[t]/t^{p} - module of dimension i. Arguing similarly, we get that β_{K}^{∗}(L_{ζ,K}) and α_{K}^{∗}(L_{ξ,K}) are projective whereas β_{K}^{∗}(L_{ξ,K}) has Jordan form 1[1] + 1[p − 1] + n[p], for some n ≥ 0.

Let M = L_{ζ} ⊕ L_{ξ}^{⊕2}. Then the generic Jordan types [α_{K}]^{∗}(M_{K}), [β_{K}]^{∗}(M_{K}) are different. Indeed, it is enough to show that they stably different, i.e. different ignoring projective summands. The stable Jordan type of [α_{K}]^{∗}(M_{K}) equals that of [α_{K}]^{∗}(L_{ζ,K}) which is 1[1] + 1[p − 1], whereas the stable Jordan type of [β_{K}]^{∗}(M_{K}) equals that of [β_{K}]^{∗}(L_{ξ,K}^{⊕2}) which is 2[1] + 2[p − 1]. In particular, M_{K} has the absolute Jordan type [α_{K}]^{∗}(M_{K}).

The following explicit example reflects the same behavior of generic Jordan types as the examples constructed in Example 4.8.

Example 4.9. Let G = GL(3, 𝔽_{p}) with p > 2. The irreducible components of Π(G) are indexed by the conjugacy classes of maximal elementary p-subgroups of G which are represented by subgroups of the unipotent subgroup U(3, 𝔽_{p}) of strictly upper triangular matrices. There are 3 conjugacy classes, represented by the following subgroups:

\{( \begin{matrix}1 & a & b \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix} ) a, b ∈ 𝔽_{p}\} \{( \begin{matrix}1 & a & b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) a, b ∈ 𝔽_{p}\} \{( \begin{matrix}1 & 0 & b \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix} ) a, b ∈ 𝔽_{p}\} .

Let M be the standard 3-dimensional (rational) representation of G. Then the generic Jordan type of M indexed by the first of these maximal elementary abelian subgroups of G is a single block of size 3, whereas the generic Jordan types indexed by each of the other conjugacy classes of maximal elementary abelian subgroups are strictly smaller.

We now present the extension of Theorem 3.5 to arbitrary finite group schemes.

Theorem 4.10. Let G be a finite group scheme over k and M a finite dimensional kG-module. Let α_{K} : K[t]/t^{p} → KG be a π-point of G which has maximal Jordan type on M. Then for any π-point β_{L} : L[t]/t^{p} → LG which specializes to α_{K}, the Jordan type of α_{K}^{∗}(M_{K}) equals the Jordan type of β_{L}^{∗}(M_{L}); in particular, if α_{K} ∼ β_{L}, then the Jordan type of α_{K}^{∗}(M_{K}) equals the Jordan type of β_{L}^{∗}(M_{L}).

Proof. We first consider the special case in which α_{K} ∼ β_{L}, and verify that β_{L}^{∗}(M_{L}) has the same (maximal for M) Jordan type as α_{K}^{∗}(M_{K}). We proceed as in the proof of Theorem 4.2. Write

\xymatrix{α_{K} = i_{C} ∘ α′_{K} : K[t]/t^{p} \ar[r]^{α′_{K}} & KC_{K} \ar[r]^{i_{C}} & KG}

with C_{K} ⊂ G_{K} a unipotent abelian subgroup scheme. By [10, 4.2], we may find some subgroup scheme j : C_{K}^{0} × E ⊂ C_{K} for some elementary abelian p-subgroup E ⊂ π_{0}(C_{K}) and a π-point α′′_{Ω} : Ω[t]/t^{p} → Ω(C_{K}^{0} × E) for some field extension Ω/K such that

\xymatrix{j ∘ α′′_{Ω} : Ω[t]/t^{p} \ar[r]^{α′′_{Ω}} & Ω(C_{K}^{0} × E) \ar[r]^{j} & ΩC_{K}}

is equivalent to α′_{K} as a π-point of C_{K}, extending scalars if necessary. Since α_{K} has maximal Jordan type on M, the π-point α′_{K} of C_{K} has maximal Jordan type on the C_{K}-module i_{C}^{∗}(M_{K}) (see Remark 2.6). Hence, Theorem 2.9 implies that (j∘α′′_{Ω})^{∗}(i_{C}^{∗}(M_{Ω})) has the same (maximal for i_{C}^{∗}(M_{Ω})) Jordan type as α_{K}^{′∗}(i_{C}^{∗}(M_{K})), and, hence, as α_{K}^{∗}(M_{K}). Extending scalars to have K = Ω, we may now replace α_{K} by the equivalent π-point (i_{C} ∘ j) ∘ α′′_{K} which has the same maximal Jordan type on M as α_{K}. Hence, we may assume that α_{K} factors as

\xymatrix{α_{K} = i ∘ α′_{K} : K[t]/t^{p} \ar[r]^{α′_{K}} & K(C_{K}^{0} × E) \ar[r]^{i} & KG.}

Decompose \xymatrix{i : K(C_{K}^{0} × E) \ar[r] & KG} as

\xymatrix{i = i_{E} ∘ i_{C,E} : K(C_{K}^{0} × E) \ar[r]^{i_{C,E}} & K((G^{0})^{E} × E) \ar[r]^{i_{E}} & KG,}

and let α̃_{K} be the restriction of α_{K} to the subgroup scheme (G^{0})^{E} × E of G:

\xymatrix{α̃_{K} = i_{C,E} ∘ α′_{K} : K[t]/t^{p} \ar[r]^{α′_{K}} & K(C_{K}^{0} × E) \ar[r]^{i_{C,E}} & K((G^{0})^{E} × E).}

By Proposition 4.1, we may find \xymatrix{α′_{K} : K[t]/t^{p} \ar[r] & K((G^{0})^{E} × E′)} equivalent to

α̃_{K} as a π-point of (G^{0})^{E} ×E with the additional property that [α′_{K}] ∈ Π_{0}((G^{0})^{E′} × E′) for some E′ ⊂ E. Since k((G^{0})^{E} × E) ≃ k((G^{0})^{E} × 𝔾_{a(1)}^{×r}) (where r = rank(E)), we may apply Theorem 3.5 to (G^{0})^{E} × E. We conclude that α′_{K} has the same (maximal) Jordan type on i_{E,E′}^{∗}(M_{K}) as α̃_{K} has on i_{E}^{∗}(M_{K}), where i_{E,E′} : K((G^{0})^{E} × E′) ↪ KG. Since α_{K} = i_{E} ∘ α̃_{K}, we further conclude that the Jordan type of (i_{E,E′} ∘ α′_{K})^{∗}(M_{K}) is the same as of α_{K}^{∗}(M_{K}). Hence, replacing α_{K} with the equivalent π-point i_{E,E′} ∘ α′_{K} which has the same (maximal) Jordan type on M, we may assume that α_{K} factors as

\xymatrix{α_{K} = i_{E} ∘ α̃_{K} : K[t]/t^{p} \ar[r]^{α̃_{K}} & K((G^{0})^{E} × E) \ar[r]^{i_{E}} & KG}

with the additional property [α̃_{K}] ∈ Π_{0}((G^{0})^{E} × E) for some elementary abelian p-group E ⊂ π_{0}(G).

Performing the same series of operations on β_{L}, we find a π-point β′_{L} equivalent to β_{L} (possibly after some extensions of scalars) which factors as

(4.10.1)  \xymatrix{β′_{L} = i_{F} ∘ β̃_{L} : L[t]/t^{p} \ar[r]^{β̃_{L}} & L((G^{0})^{F} × F) \ar[r]^{i_{F}} & LG}

with the additional property [β̃_{L}] ∈ Π_{0}((G^{0})^{F} × F) for some elementary abelian p-group F ⊂ π_{0}(G). (Observe that we do not claim yet that β′_{L} has the same Jordan type on M as β_{L}.)

Using (4.0.1) as in the last paragraph of the proof of Theorem 4.2 we conclude that some π_{0}(G)-conjugate of β̃_{L} is equivalent to α̃_{K} as a π-point of (G^{0})^{E} × E. Thus, Theorem 3.5 implies that β̃_{L}^{∗}(i_{F}^{∗} (M_{L})) has the same (maximal for M) Jordan type as α_{K}^{∗}(i_{E}^{∗}(M_{K})) ≃ α_{K}^{∗}(M_{K}). Consequently, the preceding argument given for α_{K} now “run backwards” and applied to β_{L} implies that β_{L} has the same maximal Jordan type on M as α_{K}.

More generally, assume that β_{L} specializes to α_{K} as a π-point of G and construct β′_{L} = i_{F} ∘ β̃_{L} as in (4.10.1). Since the image of Π((G^{0})^{F} × F) in Π(G) is closed, and since β′_{L} ∼ β_{L} specializes to α_{K}, we can find γ_{K} : K[t]/t^{p} → K((G^{0})^{F} × F) (possibly after a scalar extension) such that i_{F} ∘ β̃_{L} specializes to i_{F} ∘ γ_{K}, and i_{F} ∘ γ_{K} ∼ α_{K} as π-points of G.

The Quillen decomposition (4.0.1) implies that β̃_{L} specializes to a π_{0}(G)-conjugate of γ_{K} as a π-point of (G^{0})^{F} × F. Since i_{F} ∘ γ_{K} ∼ α_{K}, the just proved special case (in which β_{L} ∼ α_{K}) implies that γ_{K}^{∗}(i_{F}^{∗} (M_{K})) has the same maximal Jordan type for M as α_{K}^{∗}(M_{K}). Hence, γ_{K} has maximal Jordan type on i_{F}^{∗} (M) by Remark 2.6. Thereom 3.5 applied to (G^{0})^{F} × F implies that (β′_{L})^{∗}(M_{L}) ≃ β̃_{L}(i_{F}^{∗} (M_{L})) has the same (maximal for i_{F}^{∗} (M_{L})) Jordan type as γ_{K}^{∗}(i_{F}^{∗} (M_{K})) which, in turn, is the same as the Jordan type of α_{K}^{∗}(M_{K}). Since β_{L} ∼ β′_{L}, we conclude by another application of the case of equivalent π-points that β_{L}^{∗}(M_{L}) has the same (maximal for M) Jordan type as α_{K}^{∗}(M_{K}). □

The following corollary follows immediately from Theorem 4.10.

Corollary 4.11. Let G be a finite group scheme and M a finite dimensional kGmodule. Then each maximal Jordan type of M can be realized as the Jordan type of α_{K}^{∗}(M_{K}) for some generic point [α_{K}] ∈ Π(G).

In particular, if Π(G) is irreducible, then for any finite dimensional kG-module M the Jordan type of α_{K}^{∗}(M) is the absolute maximal type for M whenever α_{K} represents the generic point of Π(G).

In order to consider examples obtained by induction from subgroup schemes, we shall employ the following well known property of support varieties. For lack of a good reference, we provide a simple proof of this result. Recall that if H ⊂ G is a closed subgroup scheme and N an H-module, then the induced module Ind_{H}^{G}(N) is the G-module with underlying vector space (N ⊗ k[G])^{H}, with G action given by the left regular action of G of k[G] and with H-invariants taken with respect to the given action on N and the right regular action on k[G].

Lemma 4.12. Let G be a finite group scheme and i : H ⊂ G a closed subgroup scheme. Let N be a finite dimensional kH-module, and let M = Ind_{H}^{G}(N). Then the natural map i : Π(H) → Π(G) satisfies the property

Π(G)_{M} ⊂ i(Π(H)_{N}).

Proof. Observe that the action of H^{•}(G, k) on Ext_{G}^{∗}(M, M) factors as the composition of the restriction H^{•}(G, k) → H^{•}(H, k) followed by the natural action of H^{•}(H, k) on Ext_{H}^{∗}(N, M ↓_{H}) = Ext_{G}^{∗}(M, M). Hence, a homogeneous prime ideal of H^{•}(G, k) contains the annihilator of Ext_{G}^{∗}(M, M) (and hence lies in Proj |G|_{M} = Π(G)_{M}) if and only if it is the pre-image of a homogeneous prime ideal of H^{•}(H, k) containing the annihilator of Ext_{H}^{∗}(N, M ↓_{H}) and the latter are among the homogeneous prime ideals of H^{•}(H, k) containing the annihilator of Ext_{H}^{∗}(N, N) (i.e., an element of Proj |H|_{N} = Π(H)_{N}) since the action of H^{∗}(H, k) on Ext_{H}^{∗}(N, M ↓_{H}) factors through Ext_{H}^{∗}(N, N). □

To complement Corollary 4.11, we give an example of a finite dimensional kGmodule for which there is no absolute maximal Jordan type.

Example 4.13. 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 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 conjugacy classes of maximal elementary abelian p-subgroups represented by E = (ℤ/pℤ)^{×p} (which is normal) and F = (ℤ/pℤ × ℤ/pℤ × ⋯ × ℤ/pℤ)^{ℤ/pℤ} × ℤ/pℤ.

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. As usual, we may assume L = K after scalar extension.

Let n be a positive integer, let M = Ind_{E}^{G}(Ω_{E}k), and let N = L_{ξ}^{⊕n} where L_{ξ} is a Carlson module such that [β_{K}] ∈ Π(G)_{L_{ξ}} but [α_{K}] ∉ Π(G)_{L_{ξ}}. This can be achieved by choosing ξ ∈ H^{2N}(G, k) with the property that the restriction of ξ to H^{2N}(F, k) is 0 and to H^{2N}(E, k) is non-zero.

Since E is normal in G, the double coset formula implies that M ↓_{E} ≃ (Ω_{E}k)^{⊕f}. Since [α_{K}] is a generic π-point of E, a representative α_{K} can be chosen to factor through kE. Hence, the Jordan type of α_{K}^{∗}(Ω_{E}k) is

1[p − 1] + \frac{e − p}{p}[p]

and the dimension is e − 1. Hence, the Jordan type of α_{K}^{∗}(M_{K}) is

f[p − 1] + f · \frac{e − p}{p}[p].

Moreover, β_{K}^{∗}(M_{K}) is projective (of dimension is (e − 1)f) by Lemma 4.12.

Let ℓ = \frac{dim L_{ξ}−p}{p}. Arguing as in Example 4.8, we get that α_{K}^{∗}(L_{ξ,_{K}}) is projective (of dimension p(ℓ + 1)), and the Jordan type of β_{K}^{∗}(L_{ξ,_{K}}) is 1[1] + 1[p − 1] + ℓ[p]. Hence, α_{K}^{∗}(N_{K}) is projective of dimension n(ℓ+1)p, and the Jordan type of β_{K}^{∗}(N_{K}) is

n[1] + n[p − 1] + nℓ[p].

We now compute the Jordan types of the module M ⊕ N at the generic points α_{K} and β_{L}. We have

α_{K}^{∗}((M ⊕ N)_{K}) ≃ α_{K}^{∗}(M_{K}) ⊕ α_{K}^{∗}(N_{K}) ≃ f[p − 1] + f · \frac{e − p}{p}[p] + n(ℓ + 1)[p] ≃

f[p − 1] + (\frac{e − p}{p}f + n(ℓ + 1))[p]

and

β_{K}^{∗}((M ⊕ N)_{K}) ≃ β_{K}^{∗}(M_{K}) ⊕ β_{K}^{∗}(N_{K}) ≃ \frac{(e − 1)f}{p} [p] + n[1] + +n[p − 1] + nℓ[p] ≃

n[1] + n[p − 1] + (\frac{(e − 1)f}{p} + nℓ)[p].

Observe that for such two Jordan types to be incomparable it suffices for α_{K}^{∗}((M ⊕ N)_{K}) to have fewer blocks of size [p] and fewer blocks altogether than β_{K}^{∗}((M ⊕ N)_{K}). This amounts to two inequalities

\frac{(e − p)f}{p} + n(ℓ + 1) < \frac{(e − 1)f}{p} + nl

and

f + \frac{(e − p)f}{p} + n(ℓ + 1) < 2n + \frac{(e − 1)f}{p} + nℓ.

This reduces to

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

Since p > 3 divides f, there exists n satisfying the inequalities 4.13.1. For such n, the corresponding module M ⊕ N has incomparable generic Jordan types at the points α_{K} and β_{K}.

The conclusion of Theorem 4.10 is not valid for generic Jordan types. Namely, as the following example shows, one can find a generic π-point α_{K} of a finite group scheme G, a π-point β which is a specialization of α_{K} and a G-module M such that the Jordan type of β^{∗}(M) is strictly greater than that of α_{K}^{∗}(M_{K}). According to Corollary 4.11, in any such example Π(G) must be reducible.

Example 4.14. We return to the example G = (ℤ/pℤ)^{p} ⋊ ℤ/pℤ of Example 4.13 and retain the same notation Π(G) = X ∪ Y, E, F, e, f. Let M = Ind_{E}^{G} k. Then the double coset formula implies that M ↓_{E} is trivial, so that Π(G)_{M} = X by Lemma 4.12. Let {g_{1}, …, g_{p}} be generators of E, t_{i} = g_{i} − 1 and K = k(s_{1}, …, s_{p}). Let α_{K} : K[t]/t^{p} → KE → KG be a generic π-point of G corresponding to the component X defined by

α_{K}(t) = s_{1}t_{1} + ⋯ + s_{p}t_{p}.

The Jordan type of α_{K}^{∗}(M_{K}) is trivial. Let β be a k-rational π-point of G representing the point on the intersection X ∩ Y given by

β(t) = t_{1} + t_{2} + ⋯ + t_{p}.

Since E acts trivially on M, the Jordan type of β^{∗}(M) is trivial.

Let γ : k[t]/t^{p} → kF → kG be a k-rational π-point of G such that [γ] ∈ Y, [γ] ∉ X. Since Π(G)_{M} ⊂ X, we conclude that γ^{∗}(M) is projective. Since both γ and β factor through kF, β(t) and γ(t) commute. Define a new π-point β̃ by

β̃(t) = β(t) + γ^{2}(t).

By Proposition 2.2, [β] = [β̃] in Π(G) and, hence, β̃ is a specialization of α_{K}. On the other hand, since γ^{∗}(M) is projective and p > 2, β̃^{∗}(M) is non-trivial. Thus, the Jordan type of β̃^{∗}(M) is greater than that of α_{K}^{∗}(M_{K}).

As we see in the following example, the maximal Jordan type can sometimes be easily determined as a familiar invariant associated to a given kG-module M.

Example 4.15. Let G be a reductive algebraic group over k with Lie algebra g. Let G = G_{(1)}, so that kG ≃ u(g), the restricted enveloping algebra of g = Lie(G). Then the maximal Jordan type of a finite dimensional kG-module M is the Jordan type of any regular nilpotent element X ∈ g acting upon M provided that every nilpotent element of g is p-nilpotent.

Consider the special case G = SL_{n(1)} and let M be the standard n-dimensional representation of SL_{n} restricted to G. We make no assumption on the size of p with respect to n (i.e., we do not assume that every nilpotent element of sl_{n} is p-nilpotent). Each p-nilpotent X ∈ sl_{n} determines a 1-parameter subgroup μ_{X} : 𝔾_{a(1)} → G. The Jordan type of (μ_{X})^{∗}(M) is merely the Jordan type of X

itself. The maximal Jordan type for M is that given by the partition given by [\frac{n}{p}]

blocks of size p and one block of size n − p [\frac{n}{p}].

More generally, let G = SL_{n(r)}, r ≥ 1, with M once again the standard representation of SL_{n} restricted to G. Since any π-point of SL_{n(1)} determines a πpoint of SL_{n(r)} by composition with the closed (flat) embedding SL_{n(1)} ⊂ SL_{n(r)}, the maximal Jordan type for M ↓_{SL_{n(1)}} is at most the maximal Jordan type for M ↓_{SL_{n(r)}}. On the other hand, the maximal Jordan type for M ↓_{SL_{n(1)}} is maximal possible of any module of dimension n. Thus, the maximal Jordan type for M as SL_{n(r)}-module remains the partition with [\frac{n}{p}] blocks of size p and one block of size

n − p [\frac{n}{p}].

The following most elementary example of an infinitesimal group scheme of height bigger than 1 gives a first indication of the behavior of generic Jordan type for infinitesimal group schemes.

Example 4.16. Let G = 𝔾_{a(r)} for some r > s > 1, and let u_{1}, …, u_{r} be the standard algebraic generators of kG as described in §3. Consider a finite dimensional k𝔾_{(r−s)}-module M and the pull-back p^{∗}M = M^{(s)} via the projection p : 𝔾_{a(r)} → 𝔾_{a(r)}/𝔾_{a(s)} ≃ 𝔾_{a(r−s)} sending u_{i} ∈ k𝔾_{a(r)} to u_{i−s} ∈ k𝔾_{a(r−s)} for i > s and 0 otherwise. Observe that the action of u_{i} ∈ k𝔾_{a(r)} on M^{(s)} can be identified with the action of u_{i−s} ∈ k𝔾_{a(r−s)} on M. We readily verify that the generic Jordan type of the k𝔾_{(r−s)}-module M is equal to the generic Jordan type of the k𝔾_{a(r)}- module M^{(s)}. Namely, the generic type of M is determined by the action of a generic linear combination of the generators u_{i−s} ∈ k𝔾_{a(r−s)}, and the generic type of M^{(s)} is determined by the action of a generic linear combination of u_{i} ∈ k𝔾_{a(r)}. But since u_{i} ∈ k𝔾_{a(r)}, i < s acts trivially on M^{(s)} and u_{i} ∈ k𝔾_{a(r)}, i ≥ s, acts as u_{i−s} ∈ k𝔾_{a(r−s)} acts on M, these generic linear combinations clearly yield the same Jordan type.

5. Non-maximal support varieties

In this section we refine the construction of support varieties for modules over a finite group scheme. Cohomological support varieties were first introduced in the case of finite groups by J. Carlson [3], and subsequently extended by various authors, initially to restricted Lie algebras, then to infinitesimal group schemes, and finally to arbitrary finite group schemes. Carlson conjectured and G. Avrunin and L. Scott proved [1] that for elementary abelian p-groups the cohomological support variety of a module had a representation theoretic interpretation as the rank variety. This was subsequently generalized, with the ultimate formulation being the assertion of the existence of the naturally constructed homeomorphism (2.1.1) restricting to a homeomorphism of support varieties

Π(G)_{M} ≃ Proj |G|_{M}

for any finite group scheme G and any finite dimensional kG-module M [12].

If the dimension of M is not divisible by p, then the support variety Π(G)_{M} is necessarily equal to Π(G). In this section, we introduce a refinement Γ(G)_{M} ⊂ Π(G) of Π(G)_{M} ⊂ Π(G) which does give information about a kG-module M even when the dimension of M is not divisible by p. On the other hand, if the (absolute) maximal Jordan type of M is projective, then the absolute maximal Jordan type for M is the type of a projective K[t]/t^{p}-module so that the non-maximal support variety of M equals the support variety of M: Γ(G)_{M} = Π(G)_{M}.

Definition 5.1. Let G be a finite group scheme over k and M be a finite dimensional kG-module. We define

Γ(G)_{M} ⊂ Π(G)

to be the set of those equivalence classes [α_{K}] ∈ Π(G) such that α_{K} does not have the maximal Jordan type on M for some choice α_{K} : K[t]/t^{p} → KG of representative of [α_{K}].

Observe that Theorem 4.10 verifies that [α_{K}] ∈ Γ(G)_{M} if and only if for every choice of representative α_{K} of [α_{K}] the Jordan type of α_{K}^{∗}(M) is not maximal for M.

Theorem 5.2. Let G be a finite group scheme over k and M be a finite dimensional kG-module. Then Γ(G)_{M} is a closed subspace of Π(G), the “non-maximal support variety” of M.

Proof. We first consider the special case in which G is infinitesimal of height r. Let A denote the coordinate algebra k[V_{r}(G)] and consider the universal 1-paramter subgroup (3.0.1),

\xymatrix{u : 𝔾_{a(r),A} \ar[r] & G_{A},}

which provides M_{A} with a canonical rational 𝔾_{a(r),A}-module structure. Let X ⊂ V_{r}(G) be the subset of 1-parameter subgroups μ : 𝔾_{a(r),K} → G_{K} such that the π-point μ ∘ ϵ does not have maximal Jordan type on M. Then X consists of those points x ∈ V_{r}(G) for which the Jordan type of ϵ^{∗}(u^{∗}(M_{A}) ⊗_{A} k(x)) is not maximal for M.

Let \xymatrix{f_{A} : A[t]/t^{p} \ar[r] & End_{A}(M)} be the composition

\xymatrix{f_{A} : A[t]/t^{p} \ar[r]^{ϵ} & A𝔾_{a(r)} \ar[r]^{u} & AG \ar[r]^{ρ_{M}} & End_{A}(M).}

Then x ∈ X if and only if the Jordan type of (f_{A} ⊗_{A} k(x))(t) ∈ End_{k(x)}(M ⊗ k(x)) is not maximal for M. This is described by a set of equations on the ranks of powers of f_{A}(t). Thus, X is the locus of points of V_{r}(G) = Spec A for which these equations admit a solution.

For a general finite group scheme G, we extend scalars to be able to assume that the Quillen decomposition (4.0.1) applies to G. Then Π(G) is the finite union of the images of the closed maps Π((G^{0})^{E} × E) → Π(G) as E ⊂ τ = π_{0}(G) varies over conjugacy classes of elementary abelian p-subgroups of τ. Thus, for a given finite dimensional kG-module M, Γ(G)_{M} consists of the union of the images of Π((G^{0})^{E} × E) indexed by those conjugacy classes such that the Jordan type of M at the generic point of (G^{0})^{E}×E is not maximal and the images of Γ((G^{0})^{E}×E)_{M} ⊂ Π((G^{0})^{E} × E) indexed by those conjugacy class such that the Jordan type of M at the generic point of (G^{0})^{E} × E is maximal. Consequently, it suffices to prove that each Γ((G^{0})^{E} × E)_{M} ⊂ Π((G^{0})^{E} × E) is closed. Since k((G^{0})^{E} × E)) ≃ k((G^{0})^{E} × 𝔾_{a(1)}^{×r}) where r is the rank of E, this follows from the verification in the preceding paragraph for infinitesimal group schemes (applied to (G^{0})^{E} ×𝔾_{a(1)}^{×r}). □

Proposition 5.3. Let G be a finite group scheme over k and M, N be finite dimensional kG-modules, and f : H → G be a flat map of finite group schemes Then

(1) Γ(G)_{M⊕N} ⊂ Γ(G)_{M} ∪ Γ(G)_{N}. Moreover, if Π(G) is irreducible, then equality holds.

(2) Γ(H)_{f^{∗}M} ⊂ (f_{∗})^{−1}(Γ(G)_{M}) where f_{∗} : Π(H) → Π(G) is the map of schemes induced by f.

Proof. Let α_{K} : K[t]/t^{p} → KG be a π-point with the property that α_{K}^{∗}(M_{K}) has maximal Jordan type for M and α_{K}^{∗}(N_{K}) has maximal Jordan type for N, that is [α_{K}] ∉ Γ(G)_{M} ∪ Γ(G)_{N}. Suppose, α_{K}^{∗}(M_{K} ⊕ N_{K}) does not have maximal Jordan type for M_{K} ⊕ N_{K}. Then there exists a π-point β_{L} such that α_{K}^{∗}(M_{K} ⊕ N_{K}) < β_{L}^{∗}(M_{L} ⊕ N_{L}). This in turn implies that either α_{K}^{∗}(M_{K}) < β_{L}^{∗}(M_{L}) or α_{K}^{∗}(N_{K}) < β_{L}^{∗}(N_{L}) which contradicts maximality of Jordan types of α_{K}^{∗}(M_{K}) and α_{K}^{∗}(N_{K}). We conclude that the Jordan type of α_{K}^{∗}(M_{K} ⊕ N_{K}) is maximal for M ⊕ N which implies that [α_{K}] ∉ Γ(G)_{M⊕N}.

Assume Π(G) is irreducible with generic π-point β_{L}. To prove that some [α_{K}] ∈ Γ(G)_{M} ∪ Γ(G)_{N} is an element of Γ(G)_{M⊕N}, it suffices to assume that α_{K}^{∗}(M_{K}) is not maximal for M, that is α_{K}^{∗}(M_{K}) < β_{L}^{∗}(M_{L}). We have α_{K}^{∗}(N_{K}) ≤ β_{L}^{∗}(N_{L}) since β_{L} is generic. The maximality condition on ranks which defines the partial ordering (1.1.1) implies that α_{K}^{∗}(M_{K} ⊕ N_{K}) < β_{L}^{∗}(M_{L} ⊕ N_{L}).

To prove the second assertion, assume that the equivalence class of a π-point α_{K} of H is not in f_{∗}^{−1}(Γ(G)_{M}). Equivalently, f ∘ α_{K} has maximal Jordan type on M.

This implies that the Jordan type of (f ∘ α_{K})^{∗}(M_{K}) ≃ α_{K}^{∗}(f^{∗}M)_{K} is maximal for M among all π-points of G. In particular, it is maximal for M among all π-points of the form f ∘ β_{L} where β_{L} is a π-point of H. Hence, α_{K} has maximal Jordan type on the H-module f^{∗}M. We conclude that [α_{K}] ∉ Γ(H)_{f^{∗}M}, which implies the second assertion.

□

The following example demonstrates that the containment of statement (1) of Proposition 5.3 can not be sharpened to an equality.

Example 5.4. Let G be as in Example 4.8, M_{1} = L_{ζ}, M_{2} = L_{ξ}^{⊕2}. Then Γ(G)_{M_{1}} = Π(G)_{M_{1}}, Γ(G)_{M_{2}} = Π(G)_{M_{2}}, so that Γ(G)_{M_{1}}∪Γ(G)_{M_{2}} = Π(G), whereas Γ(G)_{M_{1}⊕M_{2}} ≠ Π(G). Thus,

Γ(G)_{M_{1}⊕M_{2}} ≠ Γ(G)_{M_{1}} ∪ Γ(G)_{M_{2}}.

On the other hand, if both M and N have maximal Jordan types which are projective, then

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

The following examples involve the consideration of the standard n-dimensional representation of SL_{n}. For these examples, determination of the non-maximal support variety is particularly easy, but the behavior of these varieties is illustrative.

Example 5.5. Let G = SL_{n(1)} and let M be the standard n-dimensional representation of SL_{n} restricted to G. Recall that Π(G_{(1)}) = Proj(N_{p}), the result of applying “proj” to the homogeneous closed subvariety N_{p} ⊂ sℓ_{n} of p-nilpotent matrices ([9], [20]). By Example 4.15, the non-maximal support variety is homeomorphic to the subset of Proj(N_{p}) consisting of equivalence classes of p-nilpotent matrices whose Jordan type is less than the type consisting of [\frac{n}{p}] blocks of size

p and one block of size n − p [\frac{n}{p}]. This agrees with Π(G)_{M} ⊂ Π(G) if and only if n is divisible by p. For p ≥ n, the non-maximal support variety is the complement inside the nullcone N (of all nilpotent matrices) of the open subset of regular nilpotent matrices.

Example 5.6. As in Example 4.9, let G be GL(3, 𝔽_{p}) with p > 3 and M be the standard 3-dimensional GL_{n}-module M restricted to GL(n, 𝔽_{p}). By the computation of Example 4.9, the non-maximal support variety is the union of two irreducible components.

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Department of Mathematics, Northwestern University, Evanston, IL 60208-2730

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

Department of Mathematics, University of Washington, Seattle, WA 98195-4350

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

Department of Mathematics, Northwestern University, Evanston, IL 60208-2730

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