CONSTRUCTIONS FOR INFINITESIMAL GROUP SCHEMES

ERIC M. FRIEDLANDER^{∗} AND JULIA PEVTSOVA^{∗∗}

Abstract. Let G be an infinitesimal group scheme over a field k of characteristic p > 0. We introduce the global p-nilpotent operator Θ_{G} : k[G] → k[V (G)], where V (G) is the scheme which represents 1-parameter subgroups of G. This operator Θ_{G} applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain kGmodules M (including those of constant Jordan type), we employ Θ_{G} to associate various algebraic vector bundles on ℙ(G), the projectivization of V (G). These vector bundles not only distinguish certain representations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on ℙ(G).

Contents

0. Introduction 1 1. Infinitesimal group schemes 4 2. Global p-nilpotent operators 11 3. θ_{v} and local Jordan type 21 4. π-points and ℙ(G) 28 5. Vector bundles for modules of constant j-rank 34 6. Examples and calculations with bundles 45 References 55

0. Introduction

In [25], [26], the foundations of a theory of support varieties were established for an infinitesimal group scheme G over a field k of characteristic p > 0, extending earlier work for elementary abelian p-groups and p-restricted finite dimensional Lie algebras ([6], [13]). These foundations relied upon cohomological calculations to identify cohomological support varieties and introduced 1-parameter subgroups to provide an alternate, representation-theoretic perspective. In contrast to the situation for finite groups, the cohomological variety for G infinitesimal is of considerable geometric complexity; partly for this reason, computations of explicit examples are challenging. In this present paper, we build upon this earlier work as well as more recent work of the authors ([15], [17]) to initiate a more detailed investigation of representations of G. Although representations of infinitesimal group schemes are less familiar than representations of finite groups, their importance is evident: for example, the representation theories of the family of all infinitesimal kernels G = 𝔊_{(r)} of a smooth connected algebraic group 𝔊 is essentially equivalent to the rational representation theory of 𝔊.

Date: February 22, 2010.

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

∗ partially supported by the NSF grant #0300525.

∗∗ partially supported by the NSF grant #0629156.

An important structure associated to an infinitesimal group scheme G of height ≤ r is the scheme V (G) of one-parameter subgroups 𝔾_{a(r)} → G (see [25]). In this paper we observe that the representability of V (G) leads to a p-nilpotent element Θ_{G} in kG ⊗ k[V (G)], where kG is the group algebra of G. For any kG-module M, Θ_{G} determines a global p-nilpotent operator on M. The operator Θ_{G} encodes the local Jordan type of a kG-module M which in turn determines the support variety of M. Even though the scheme V (G) was generalized to all finite group schemes in [15], [17] via the notion of π-points, the construction of Θ_{G} does not appear to extend to arbitrary finite groups.

The homogeneity of our global operator Θ_{G} enables us to associate to a kGmodule M of constant Jordan type a collection of vector bundles on ℙ(G) = Proj k[V (G)], which we view as a family of global invariants of M. Certain modules with the same local Jordan type can be distinguished by these global invariants.

The vector bundles that we investigate are constructed directly and explicitly from kG-modules. Hence, we offer an important new method to create interesting examples of algebraic vector bundles on varieties of the form ℙ(G). Varieties of this form include projective spaces, weighted projective spaces, and various singular varieties associated to algebraic groups. For example, for G = GL_{n(r)}, ℙ(G) is the projectivization of the variety of r-tuples of pairwise commuting p-nilpotent matrices. We expect that our technique of constructing algebraic vector bundles on such singular, geometrically interesting varieties will lead to insights into their algebraic K-theory.

The reader might find it instructive to contrast our use of representations of G to construct vector bundles on ℙ(G) with the Borel-Weil construction which employs bundles on flag varieties for an algebraic group 𝔊 to construct representations of 𝔊. Our construction of vector bundles plays a role in the forthcoming papers by the first author, Jon Carlson, and Andrei Suslin [9] and by the second author and David Benson [5].

In this paper, we also attempt to address the lack of specific examples in the representation theory of infinitesimal groups schemes (other than those of height 1). Throughout this paper, we work with the following four fundamental, yet concrete, classes of examples.

i.) p-restricted Lie algebras, 𝔤;

ii.) infinitesimal additive group schemes, 𝔾_{a(r)};

iii.) infinitesimal general linear groups, GL_{n(r)};

iv.) the height 2, infinitesimal special linear group, SL_{2(2)}.

We consistently endeavor to make our general results more concrete by applying them to our examples.

In Section 1, we recall some of the highlights from [25], [26] concerning the cohomology and theory of supports of finite dimensional kG-modules for an infinitesimal group scheme G. A key result summarized in Theorem 1.16 is the close relationship between the spectrum Spec H^{•}(G, k) of the cohomology of G and the scheme V (G) representing (infinitesimal) 1-parameter subgroups of an infinitesimal group scheme G.

In the second section, we define the global p-nilpotent operator Θ_{G} : k[G] ⟶ k[V (G)] for an infinitesimal group scheme G. For any finite dimensional kG-module M, Θ_{G} determines a p-nilpotent endomorphism of the free k[V (G)]-module M ⊗ k[V (G)]. We establish in Proposition 2.11 that Θ_{G} is homogeneous, where k[V (G)] is equipped with its natural grading. We also verify that Θ_{G} is natural with respect to change of group.

In the third section, we verify that specializations θ_{v} of Θ_{G} at points v ∈ V (G) determine the local Jordan type of a finite dimensional kG-module M. Theorem 3.7 can be viewed as providing an algorithm for obtaining the local Jordan type in terms of the representation G → GL_{N} defining the kG-module M. We utilize Θ_{G} and its specializations to establish constraints for a kG-module M to be of constant rank (and thus of constant Jordan type). We also establish the relationship between the local Jordan type of a module and its Frobenius twists.

We envision that some of our constructions for infinitesimal group schemes may lead to analogues for a general finite group scheme. With this in mind, we begin the fourth section with a dictionary between 1-parameter subgroups for infinitesimal group schemes and π-points for general finite group schemes. Given a finite dimensional kG-module M, we consider the projectivization of the operator Θ_{G},

Θ̃_{G} : M ⊗ O_{ℙ(G)} → M ⊗ O_{ℙ(G)}(p^{r−1}),

a p-nilpotent operator on the free, coherent sheaf M ⊗ O_{ℙ(G)} on ℙ(G). We verify in Proposition 4.8 that Θ̃_{G} determines via base change the local Jordan type of a kG-module M at any 1-parameter subgroup μ_{v} : 𝔾_{a(r),k(v)} → G_{k(v)}. Theorem 4.13 shows that the condition that M be of constant j-rank is equivalent to the condition that the coherent sheaf Im Θ̃_{G}^{j} be locally free.

In the fifth section, we initiate an investigation of the algebraic vector bundles Ker{Θ̃_{G}^{j}, M}, Im{Θ̃_{G}^{j}, M} on ℙ(G) associated to kG-modules of constant Jordan type and more generally of constant j-rank. We give examples of such kG-modules in each of our four representative examples, and investigate the associated vector bundles. As we see, taking kernels of powers of the global p-nilpotent power operator sends modules of constant Jordan type to vector bundles. We also obtain vector bundles by taking kernels modulo images (as inspired by a construction of M. Duflo and V. Serganova for Lie superalgebras in [11]). As an application, we prove in Proposition 5.17 a geometric characterization of endotrivial modules.

Finally, in the last section, we provide numerous explicit examples. These include the infinitesimal group scheme G = 𝔾_{a(1)}×𝔾_{a(1)}, which has the same representation theory as the elementary abelian p-group ℤ/p × ℤ/p, as well as the first Frobenius kernel of the reductive group SL_{2}. One intriguing comparison which we investigate in particularly simple examples is the relationship between the Grothendieck group of projective kG-modules and the Grothendieck group of algebraic vector bundles on ℙ(G). Combined with our explicit calculations, Proposition 6.12 can be viewed both as a means to distinguish certain non-isomorphic projective kG-modules and as a means of constructing non-isomorphic algebraic vector bundles on ℙ(G).

Throughout, k will denote an arbitrary field of characteristic p > 0. Unless explicit mention is made to the contrary, G will denote an infinitesimal group scheme over k. If M is a kG-module and K/k is a field extension, then we denote by M_{K} the KG-module obtained by base extension.

The authors are grateful to the University of Bielefeld and to MSRI for their hospitality. We thank Dan Grayson for pointing out the occurrence of weighted projective spaces and Paul Smith for numerous useful conversations.

1. Infinitesimal group schemes

The purpose of this first section is to summarize the important role played by (infinitesimal) 1-parameter subgroups of an infinitesimal group scheme as presented in [25]. The four representative examples of Example 1.5, (\underline{𝔤}, 𝔾_{a(r)}, GL_{n(r)}, SL_{2(2)}), and their associated schemes of 1-parameter subgroups discussed in Example 1.12 will serve as explicit models to which we will frequently return.

Definition 1.1. A finite group scheme G over k is a group scheme over k whose coordinate algebra k[G] is finite dimensional over k.

Equivalently, G is a functor from commutative k-algebras to groups, R ↦ G(R), represented by a finite dimensional commutative k-algebra, the coordinate algebra k[G] of G.

Associated to G, we have its group algebra kG = Hom_{k}(k[G], k); more generally, for any commutative k-algebra R, we have the R-group algebra RG = Hom_{k}(k[G], R).

Notation 1.2. If f : G → H is a map of finite group schemes, we denote by

f^{∗} : k[H] → k[G] and f_{∗} : kG → kH

the induced maps on coordinate and group algebras respectively.

Observe that the R-group algebra of G consists of all k-linear homomorphisms, whereas G(R) = Hom_{k−alg}(k[G], R) is the subgroup of RG^{×} consisting of k-algebra homomorphisms.

Definition 1.3. Let G be a finite group scheme over k and M a k-vector space. Then a (left) kG-module structure on M is given by one of the following equivalent sets of data (see, for example, [22]):

• The structure M → M ⊗ k[G] of a (right) k[G]-comodule on M.

• The structure kG ⊗ M → M of a kG-module on M.

• A functorial (with respect to R) group action G(R) × (R ⊗ M) → (R ⊗ M).

For most of this paper we shall restrict our consideration to infinitesimal group schemes, a special class of finite group schemes which we now define.

Definition 1.4. An infinitesimal group scheme G (over k) of height ≤ r is a finite group scheme whose coordinate algebra k[G] is a local algebra with maximal ideal 𝔪 such that x^{p^{r}} = 0 for all x ∈ 𝔪.

Example 1.5. We shall frequently consider the following four examples.

(1) A finite dimensional p-restricted Lie algebra 𝔤 corresponds naturally with a height 1 infinitesimal group scheme which we denote \underline{𝔤} ([22, I.8.5]). The group algebra of \underline{𝔤} is the restricted enveloping algebra 𝔲(𝔤) of 𝔤. If 𝔤 is the Lie algebra of a group scheme 𝔊, then the coordinate algebra of \underline{𝔤} is given by k[𝔊]/(x^{p}, x ∈ 𝔪), where 𝔪 is the maximal ideal of k[𝔊] at the identity of 𝔊.

(2) Let 𝔾_{a} denote the additive group, so that k[𝔾_{a}] = k[T] with coproduct defined by ∇(T) = T ⊗ 1 + 1 ⊗ T. As a functor, 𝔾_{a} : (comm k − alg) → (grps) sends an algebra R to its underlying abelian group. For any r ≥ 1, we consider the r^{th} Frobenius kernel of 𝔾_{a},

𝔾_{a(r)} ≡ Ker{F^{r} : 𝔾_{a} → 𝔾_{a}}.

Here F : 𝔾_{a} → 𝔾_{a} is the (geometric) Frobenius specified by its map on coordinate algebras k[T] → k[T] given as the k-linear map sending T to T^{p}. The coordinate algebra of 𝔾_{a(r)} is given by k[𝔾_{a(r)}] = k[T]/T^{p^{r}}, whereas the group algebra of 𝔾_{a(r)} is given by

(1.5.1)  k𝔾_{a(r)} ≃ k[𝔾_{a(r)}]^{#} ≃ k[u_{0}, …, u_{r−1}]/(u_{0}^{p}, …, u_{r−1}^{p}),

where u_{i} is a linear dual to T^{p^{i}}, 0 ≤ i ≤ r − 1.

(3) Let GL_{n} denote the general linear group, the representable functor sending a commutative algebra R to the group GL_{n}(R). For any r ≥ 1, we consider the r^{th} Frobenius kernel of GL_{n},

GL_{n(r)} ≡ Ker{F^{r} : GL_{n} → GL_{n}},

where the geometric Frobenius

F : GL_{n}(R) → GL_{n}(R)

is defined by raising each matrix entry to the p^{th} power. The coordinate algebra of GL_{n(r)} is given by

k[GL_{n(r)}] = \frac{k[X_{i,j}]}{(X_{i,j}^{p^{r}} − δ_{i,j})}_{1≤i,j≤n},

whereas the group algebra of GL_{n(r)} is given as

k GL_{n(r)} = Hom_{k}(k[GL_{n(r)}], k),

the k-space of linear functionals k[GL_{n(r)}] to k. The coproduct

∇ : k[GL_{n(r)}] → k[GL_{n(r)}] ⊗ k[GL_{n(r)}]

is given by sending X_{i,j} to ∑_{k} X_{ik} ⊗ X_{kj}.

(4) The height 2 infinitesimal group scheme SL_{2(2)} is essentially a special case of GL_{n(r)}. This is once again defined as the kernel of the second iterate of Frobenius

SL_{2(2)} ≡ Ker{F^{2} : SL_{2} → SL_{2}}.

The coordinate algebra of SL_{2(2)} is given by

k[SL_{2(2)}] = \frac{k[X_{1,1}, X_{1,2}, X_{2,1}, X_{2,2}]}{(X_{1,1}X_{2,2} − X_{1,2}X_{2,1} − 1, X_{i,j}^{p^{2}} − δ_{i,j})}

whereas the group algebra of SL_{2(2)} is given as

k SL_{2(2)} = k⟨e, f, h, e^{(p)}, f^{(p)}, h^{(p)}⟩/⟨relations⟩

with e, f, h, e^{(p)}, f^{(p)}, h^{(p)} the dual basis vectors to X_{1,2}, X_{2,1}, X_{1,1} − 1, X_{1,2}^{p}, X_{2,1}^{p}, (X_{1,1} − 1)^{p} respectively.

We denote by 𝔾_{a(r),R} the base extension of 𝔾_{a(r)} to a commutative k-algebra R.

Definition 1.6. A (infinitesimal) 1-parameter subgroup of height r of an affine group scheme G_{R} over a commutative k-algebra R is a homomorphism of R-group schemes 𝔾_{a(r),R} → G_{R}.

We recall the description of height r 1-parameter subgroups of GL_{n} given in [25].

Proposition 1.7. [25, 1.2] If G = GL_{n} and if R is a commutative k-algebra, then a 1-parameter subgroup of GL_{n,R} of height r, f : 𝔾_{a(r),R} → GL_{n,R}, is naturally (with respect to R) equivalent to a comodule map

Δ_{f} : R^{n} → R[T]/T^{p^{r}} ⊗_{R} R^{n}, Δ_{f}(v) = ∑_{j=0}^{p^{r−1}} T^{j} ⊗ β_{j}(v), β_{j} ∈ M_{n}(R)

satisfying the constraints of being counital and coassociative. This in turn is equivalent to specifying an r-tuple of matrices α_{0} = β_{0}, α_{1} = β_{p}, …, α_{r−1} = β_{p^{r−1}} in M_{n}(R) such that each α_{i} has p^{th} power 0 and such that the α_{i}’s pairwise commute. The other coefficient matrices β_{j} are given by the formula

(1.7.1)  β_{j} = \frac{α_{0}^{j_{0}} ⋯ α_{r−1}^{j_{r−1}}}{(j_{0})! ⋯ (j_{r−1})!} ∈ M_{n}(R), j = ∑_{i=0}^{r−1} j_{i}p^{i} with 0 ≤ j_{i} < p.

As shown in [25], Proposition 1.7 implies the following representability of the functor of 1-parameter subgroups of height r.

Theorem 1.8. [25, 1.5] For any affine group scheme G, the functor from commutative k-algebras to sets

R ↦ Hom_{grp sch}(𝔾_{a(r),R}, G_{R})

is representable by an affine scheme V_{r}(G) = Spec k[V_{r}(G)]. Namely, this functor is naturally isomorphic to the functor

R ↦ Hom_{k−alg}(k[V_{r}(G)], R).

By varying r, we can associate a family of affine schemes to an affine group scheme G. In the following remark we make explicit the relationship between various V_{r}(G) for the same G and varying r’s.

Remark 1.9. For r > s ≥ 1, let p_{r,s} : 𝔾_{a(r)} → 𝔾_{a(s)} be the canonical projection given by the natural embedding of the coordinate algebras

\xymatrix{p_{r,s}^{∗} : k[𝔾_{a(s)}] = k[T]/T^{p^{s}} \ar[r]^{T →T^{p^{r−s}}} & k[T]/T^{p^{r}} = k[𝔾_{a(r)}].}

The corresponding map on group algebras

\xymatrix{k𝔾_{a(r)} ≃ k[u_{0}, …, u_{r−1}]/(u_{0}^{p}, …, u_{r−1}^{p}) \ar[r]^{p_{r,s,∗}} & k𝔾_{a(s)} ≃ k[v_{0}, …, v_{s−1}]/(v_{0}^{p}, …, v_{s−1}^{p})}

sends {u_{0}, …, u_{r−s−1}} to {0, …, 0}, and {u_{r−s}, …, u_{r−1}} to {v_{0}, …, v_{s−1}}. Precomposition with p_{r,s} determines a canonical embedding of affine schemes

\xymatrix{i_{s,r} : V_{s}(G) \ar@{^{(}->}[r] & V_{r}(G)} ,

where a one-parameter subgroup μ : 𝔾_{a(s),R} → G_{R} of height s is sent to the oneparameter subgroup μ∘p_{r,s} : 𝔾_{a(r),R} → 𝔾_{a(s),R} → G_{R} of height r. The construction is transitive, that is, we have i_{s,r} = i_{s′,r} ∘ i_{s,s′} for s ≤ s′ ≤ r. Hence, we have an inductive system

V_{1}(G) ⊂ V_{2}(G) ⊂ … ⊂ V_{r}(G) ⊂ …

Conversely, any one-parameter subgroup 𝔾_{a(s′),R} → G_{R} can be decomposed as

\xymatrix{𝔾_{a(s′),R} \ar[r]^{p_{s′,s}} & 𝔾_{a(s),R} \ar@{^{(}->}[r] & G_{R}}

for some s ≤ s′. If G is an infinitesimal group scheme of height ≤ r then, we may choose s ≤ r. This justifies the following definition

Definition 1.10. Let G be an infinitesimal group scheme. Then the closed immersion i_{r,r′} : V_{r}(G) ↪ V_{r′}(G) for r′ > r is an isomorphism provided the height of G is ≤ r. We denote by V (G) the stable value of V_{r}(G),

V (G) ≡ \underrightarrow{lim}_{r} V_{r}(G).

We next make explicit the construction of 1-parameter subgroups for GL_{n} as in Proposition 1.7. This construction can be applied to any affine group scheme of exponential type (see [25, §1] and also [23] for an extended list of groups of exponential type). We define the homomorphism

exp\underline{α} : 𝔾_{a(r),R} → GL_{n,R}

of R-group schemes corresponding to an r-tuple \underline{α} = (α_{0}, …, α_{r−1}) ∈ M_{n}(R)^{×r} of pairwise commuting p-nilpotent matrices to be the natural transformation of groupvalued functors on commutative R-algebras S sending any s ∈ S with s^{p^{r}} = 0 to

(1.10.1)  exp(sα_{0}) · exp(s^{p}α_{1}) · · · · · exp(s^{p^{r−1}}α_{r−1}) ∈ GL_{n}(S),

where for any p-nilpotent matrix A ∈ GL_{n}(S) we set

exp(A) = 1 + A + \frac{A^{2}}{2} + … + \frac{A^{p−1}}{(p − 1)!}.

The following proposition proved in [25] identifies the functor of 1-parameter subgroups in the case of infinitesimal general linear groups.

Proposition 1.11. [25, 1.2] The scheme of one-parameter subgroups V_{r}(GL_{n}) is isomorphic to the scheme of r-tuples of pairwise commuting p-nilpotent n × n matrices N_{p}^{[r]}(gl_{n}); the identification is given by sending \underline{α} = (α_{0}, …, α_{r−1}) ∈ N_{p}^{[r]}(gl_{n})(R) to the one-parameter subgroup exp\underline{α} : 𝔾_{a(r),R} → GL_{n,R}.

Example 1.12. We describe V (G) in each of the four examples of Example 1.5.

(1) V (\underline{𝔤}) ≃ N_{p}(𝔤), the closed subvariety of the affine space underlying 𝔤 consisting of p-nilpotent elements x ∈ 𝔤 (that is, x^{[p]} = 0). Let 𝔤_{a} be the Lie algebra of the additive group 𝔾_{a}. Note that 𝔤_{a} is a one-dimensional restricted Lie algebra with trivial p-restriction. Each p-nilpotent element x ∈ 𝔤_{R} = 𝔤 ⊗_{k} R determines a map of p-restricted Lie algebras over R where R is a commutative k-algebra: 𝔤_{a,R} → 𝔤_{R}. The corresponding map of height 1 infinitesimal group schemes 𝔾_{a(1),R} → \underline{𝔤}_{R} is the associated 1-parameter subgroup of \underline{𝔤}.

(2) V (𝔾_{a(r)}) ≃ 𝔸^{r}. The r-tuple \underline{a} = (a_{0}, …, a_{r−1}) ∈ R^{×r} = 𝔸^{r}(R) corresponds to the 1-parameter subgroup μ\underline{a} : 𝔾_{a(r),R} → 𝔾_{a(r),R} whose map on coordinate algebras R[T]/T^{p^{r}} → R[T]/T^{p^{r}} sends T to ∑_{i} a_{i}T^{p^{i}} ([25, 1.10]).

(3) By Proposition 1.11, V (GL_{n(r)}) = N_{p}^{[r]}(gl_{n}), the variety of r-tuples of pairwise commuting, p-nilpotent n × n matrices. The embedding i_{r,r+1} : V_{r}(GL_{n}) ≃ N_{p}^{[r]}(gl_{n}) ⊂ V_{r+1}(GL_{n}) ≃ N_{p}^{[r+1]}(gl_{n}) described in Remark 1.9 is given by sending an r-tuple (α_{0}, …, α_{r−1}) to the (r + 1)-tuple (0, α_{0}, …, α_{r−1}).

Let X_{i,j} be the coordinate functions of R[GL_{n(r)}] ≃ R[X_{i,j}]/(X_{i,j}^{p^{r}} − δ_{i,j}). Then exp∗ ; \underline{α} : R[GL_{n(r)}] → R[𝔾_{a(r)}] is given by sending X_{i,j} for some 1 ≤ i, j ≤ n to the (i, j)-entry of the polynomial p\underline{α}(t) with matrix coefficients whose coefficient of t^{d} is computed as the multiple of s^{d} in the (i, j)-entry of the matrix (1.10.1).

Upon performing the indicated multiplication in (1.10.1), the coefficient of p\underline{α}(t) multiplying s^{p^{ℓ}} is α_{ℓ} for 0 ≤ ℓ < r, whereas coefficients of p\underline{α}(t) multiplying s^{n} for n not a power of p are determined as in formula (1.7.1). Consequently, we conclude that exp∗ ; \underline{α}(X_{i,j}) is a polynomial in t whose coefficient multiplying T^{p^{ℓ}} is (α_{ℓ})_{i,j} for 0 ≤ ℓ < r.

(4) Since SL_{2(2)} is a group scheme with an embedding of exponential type (see [25, 1.8]), its variety admits a description similar to the one of GL_{n(r)}. Namely, V (SL_{2(2)}) is the variety of pairs of p-nilpotent proportional 2 × 2 matrices \underline{α} = (α_{0}, α_{1}). This variety is given explicitly as the affine scheme with coordinate algebra

k[V (SL_{2(2)})] = k[x_{0}, y_{0}, z_{0}, x_{1}, y_{1}, z_{1}]/(x_{i}y_{i}−z_{i}^{2}, x_{0}y_{1}−x_{1}y_{1}, z_{0}y_{1}−z_{1}y_{0}, x_{0}z_{1}−x_{1}z_{0}).

We give an explicit description of the map on coordinate algebras

exp∗ ; \underline{α} : R[SL_{2(2)}] → R[𝔾_{a(2)}] ≃ R[T]/T^{p^{2}}

induced by the one-parameter subgroup exp\underline{α} : 𝔾_{a(2),R} → SL_{2(2),R}. This description follows immediately from the general discussion in the previous example. Let \underline{α} =

([ \begin{matrix}c_{0} & a_{0} \\ b_{0} & −c_{0} \end{matrix} ] , [ \begin{matrix}c_{1} & a_{1} \\ b_{1} & −c_{1} \end{matrix} ]) ∈ N^{[2]}(sl_{2}). Then exp∗ ; \underline{α} is determined by the formulae

X_{1,1} ↦ 1 + c_{0}T + c_{1}T^{p}, X_{1,2} ↦ a_{0}T + a_{1}T^{p}
X_{2,1} ↦ b_{0}T + b_{1}T^{p}, X_{2,2} ↦ 1 − c_{0}T − c_{1}T^{p},

where X_{i,j} are the standard polynomial generators of k[SL_{2(2)}] ≃ _{\frac{k[X_{1,1},X_{1,2},X_{2,1},X_{2,2}]}{(det −1,X_{i,j}^{p^{2}}−δ_{i,j})}}.

Remark 1.13. If k(v) denotes the field of definition of the point v ∈ V (G) for an infinitesimal group scheme G (see [26, p.743] for a discussion of the field of definition), then we have a naturally associated map Spec k(v) → V (G) and, hence, an associated group scheme homomorphism over k(v) (for r sufficiently large):

\xymatrix{μ_{v} : 𝔾_{a(r),k(v)} \ar[r] & G_{k(v)}} .

Note that if K/k is a field extension and μ : 𝔾_{a(r),K} → G_{K} is a group scheme homomorphism, then this data defines a point v ∈ V (G) and a field embedding k(v) ↪ K such that μ is obtained from μ_{v} via scalar extension from k(v) to K.

We next recall the rank variety and cohomological support variety of a kG-module of an infinitesimal group scheme. We denote by

H^{•}(G, k) = \begin{cases}H^{∗}(G, k), & if p = 2, \\ H^{ev}(G, k) & if p > 2. \end{cases}

Definition 1.14. Let G be a finite group scheme and M a finite dimensional kG-module. We define the cohomological support variety for M to be

|G|_{M} ≡ V (ann_{H^{•}(G,k)} Ext_{kG}^{∗}(M, M)),

the reduced closed subscheme of |G| = Spec H^{•}(G, k)_{red} given as the variety of the annihilator ideal of Ext_{kG}^{∗}(M, M).

The map of R-algebras (but not of Hopf algebras for r > 1),

(1.14.1)  \xymatrix{ϵ : R[u]/u^{p} \ar[r]^{u↦u_{r−1}} & R[u_{0}, …, u_{r−1}]/(u_{i}^{p}) ≃ R𝔾_{a(r)}} ,

makes its first appearance in the following definition and will recur throughout this paper.

Definition 1.15. Let G be an infinitesimal group scheme and M a finite dimensional kG-module. We define the rank variety for M to be the reduced closed subscheme V (G)_{M} whose points are given as follows:

V (G)_{M} = {v ∈ V (G) : (μ_{v,∗} ∘ ϵ)^{∗}(M_{k(v)}) is not free as a k[u]/u^{p} - module }.

Proposition [26, 6.2] asserts that V (G)_{M} is a closed subvariety of V (G). A key result of [26] is the following theorem relating the scheme of 1-parameter subgroups V (G) to the cohomology of G.

Theorem 1.16. ([26, 5.2, 6.8, 7.5]) Let G be an infinitesimal group scheme of height ≤ r. There is a natural homomorphism of k-algebras

ψ : H^{•}(G, k) → k[V (G)]

with nilpotent kernel whose image contains the p^{r}-th power of each element of k[V (G)]. Hence, the associated morphism of schemes

Ψ : V (G) → Spec H^{•}(G, k)

is a p-isogeny.

If M is a finite dimensional kG-module, then Ψ restricts to a homeomorphism

\xymatrix{Ψ_{M} : V (G)_{M} \ar[r]^{∼} & |G|_{M}.}

Furthermore, every closed conical subspace of V (G) is of the form V (G)_{M} for some finite dimensional kG-module M.

In the special case of of G = GL_{n(r)} the isogeny Ψ has an explicitly constructed inverse.

Theorem 1.17. ([25, 5.2]) There exists a homomorphism of k-algebras

ϕ : k[V (GL_{n(r)})] → H^{•}(GL_{n(r)}, k)

such that ψ∘ϕ is the r^{th} iterate of the k-linear Frobenius map. Hence, the associated morphisms of schemes

Ψ : V (GL_{n(r)}) → Spec H^{•}(GL_{n(r)}, k), Φ : Spec H^{•}(GL_{n(r)}, k) → V (GL_{n(r)})

are mutually inverse homeomorphisms.

Example 1.18. We investigate V (G)_{M} for the four examples of Example 1.5.

(1) Let M be a p-restricted 𝔤-module of dimension m, given by the map of prestricted Lie algebras ρ : 𝔤 → End_{k}(M) ≃ 𝔤𝔩_{m}. Then V (\underline{𝔤})_{M} ⊂ V (\underline{𝔤𝔩_{m}}) consists of those p-nilpotent elements of 𝔤 whose Jordan type (as an m × m-matrix in 𝔤𝔩_{m}) has at least one block of size < p (see [13]).

(2) For G = 𝔾_{a(r)}, kG ≃ kE where E is an elementary abelian p-group of rank r. The rank variety of a kE-module was first investigated in [6].

We consider directly the rank variety V (𝔾_{a(r)})_{M} of a finite dimensional k𝔾_{a(r)}- module M. The data of such a module is the choice of r p-nilpotent, pair-wise commuting endomorphisms ũ_{0}, …, ũ_{r−1} ∈ End_{k}(M), given as the image of the distinguished generators of k𝔾_{a(r)} as in (1.5.1). A 1-parameter subgroup of 𝔾_{a(r)} has the form μ\underline{a} : 𝔾_{a(r),K} → 𝔾_{a(r),K} for some r-tuple \underline{a} = (a_{0}, …, a_{r−1}) of K-rational points as in Example 1.12(2). The condition that μ\underline{a} be a point of V (𝔾_{a(r)})_{M} is the condition that (μ\underline{a} ∘ ϵ)^{∗}(M_{K}) is not free as a K[u]/u^{p}-module, which is equivalent to the condition that M_{K} is not free as a K[ũ]/u^{p}-module where ũ = a_{r−1}ũ_{0} + a_{r−2}^{p}ũ_{1} ⋯ + a_{0}^{p^{r−1}} ũ_{r−1} ∈ End_{K}(M_{K}) (see [26, 6.5]).

(3) Let M be a finite dimensional kG-module with G = GL_{n(r)}. By Theorem 1.16, V (GL_{n(r)})_{M} ⊂ V (GL_{n(r)}) is the closed subvariety whose set of points in a field K/k are 1-parameter subgroups : exp\underline{α} : 𝔾_{a(r),K} → GL_{n(r),K} indexed by rtuples \underline{α} = (α_{0}, …, α_{r−1}) ∈ M_{n}(K) of p-nilpotent, pairwise commuting matrices such that (exp_{\underline{α},∗} ∘ ϵ)^{∗}(M_{K}) is not a free as a K[u]/u^{p}-module. The action of u on M_{K} is determined utilizing Example 1.12(3). Namely, the action of u is given by composing the coproduct M_{K} → K[GL_{n(r)}] ⊗ M_{K} defining the GL_{n(r)}-module structure on M_{K} with the linear functional

\xymatrix{u_{r−1} ∘ exp∗ ; \underline{α} : K[GL_{n(r)}] \ar[r]^{^{exp∗ ; \underline{α}}} & K[𝔾_{a(r)}] \ar[r]^{u_{r−1}} & K} .

In §3, we shall investigate this case in more detail by considering some concrete examples.

(4) A complete description of support varieties for simple modules for SL_{2(r)} can be found in [26, §7]. We describe the situation for G = SL_{2(2)}. Let S_{λ} be irreducible modules of highest weight λ, where 0 ≤ λ ≤ p^{2} − 1. For λ < p − 1, the module S_{λ} has dimension less than p and thus V (G)_{S_{λ}} = V (G)_{S_{λ}^{(1)}} = V (G). Here, S_{λ}^{(i)} is the

i^{th} Frobenius twist of S_{λ}. For λ = p − 1, the restriction of S_{p−1} to SL_{2(1)} ⊂ SL_{2(2)} is projective (the Steinberg module for SL_{2(1)}) but S_{p−1} is not itself projective. Hence, V (G)_{S_{p−1}} is a proper non-trivial subvariety of V (G). Using the notation introduced in Example 1.12(4), we have

V (G)_{S_{p−1}} = {(α_{0}, 0) | α_{0} ∈ N(sl_{2})} ⊂ V (G),

and

V (G)_{S_{p−1}^{(1)}} = {(0, α_{1}) | α_{1} ∈ N(sl_{2})} ⊂ V (G)

(see [26, 6.10]). V (G)_{S_{p−1}} can be described as the subscheme of V (G) defined by the equations x_{1} = y_{1} = z_{1} = 0. For λ = λ_{0} + λ_{1}p where λ_{0}, λ_{1} ≤ p − 1 we have S_{λ} ≃ S_{λ_{0}} ⊗ S_{λ_{1}}^{(1)} by the Steinderg tensor product theorem. Hence, we can compute the support variety of S_{λ} using the tensor product property of support varieties.

For λ = p^{2} − 1, S_{λ} is the Steinberg module for SL_{2(2)}, it is projective and, hence, V (G)_{S_{p^{2}−1}} = {0}. Overall, we get

V (G)_{S_{λ}} = \begin{cases}N^{[2]}(sl_{2}), & if λ_{0}, λ_{1} ≠ p − 1, \\ \{(α_{0}, 0) | α_{0} ∈ N(sl_{2})\} & if λ_{0} = p − 1, λ_{1} ≠ p − 1, \\ \{(0, α_{1}) | α_{1} ∈ N(sl_{2})\} & if λ_{0} ≠ p − 1, λ_{1} = p − 1, \\ 0 & if λ = p^{2} − 1. \end{cases}

2. Global p-nilpotent operators

In this section, we introduce in Definition 2.1 and study the global p-nilpotent operator

Θ_{G} : k[G] → k[V (G)],

a k-linear but not multiplicative map defined for any infinitesimal group scheme G. This operator, when viewed as an element of kG⊗k[V (G)], encodes all 1-parameter subgroups of G: any 1-parameter subgroup μ : 𝔾_{a(r),K} → G_{K} corresponds to a K-valued point of V (G), and μ_{∗}(u_{r−1}) equals the specialization in KG of Θ_{G} ∈ kG ⊗ k[V (G)] at this point.

If M is a rational G-module, then Θ_{G} determines the k[V (G)]-linear operator

Θ_{M} : M ⊗ k[V (G)] → M ⊗ k[V (G)]

as formalized in Definition 2.3.

Before giving definitions, we mention as motivation the example G = 𝔾_{a(1)}^{×2}. In this case, the group algebra kG equals k[x, y]/(x^{p}, y^{p}), the scheme of 1-parameter subgroups equals V (G) = 𝔸^{2}, and k[V (G)] = k[𝔸^{2}] = k[s, t]. In this special case, Θ_{G} takes the form

Θ_{G} = x ⊗ s + y ⊗ t ∈ k[x, y]/(x^{p}, y^{p}) ⊗ k[s, t].

If M is a kG-module, then the k[s, t]-linear operator Θ_{M} is given by

Θ_{M} : M ⊗ k[s, t] → M ⊗ k[s, t],
m ⊗ 1 ↦ xm ⊗ s + ym ⊗ t.

“Specializing” Θ_{M} at some (a, b) ∈ K^{2} for some field extension K/k yields the action of ax + by on M_{K}.

To construct our global operator, we proceed as follows. Let G be an algebraic affine group scheme over k (that is, G is an affine group scheme such that the coordinate algebra k[G] is finitely generated over k ([27, 3.3])) and let A = k[V_{r}(G)]. The natural isomorphism of covariant functors on commutative k-algebras R

Hom_{grp sch}(𝔾_{a(r),R}, G_{R}) ≃ Hom_{k−alg}(A, R),

given in Theorem 1.8 implies the existence of a universal 1-parameter subgroup of height r

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

the subgroup corresponding to the identity map on A. The subgroup U_{G,r} induces a map on coordinate algebras

\xymatrix{U_{G,r}^{∗} : A ⊗ k[G] \ar[r] & A ⊗ k[𝔾_{a(r)}].}

Recall that k𝔾_{a(r)} ≃ k[u_{0}, …, u_{r−1}]/(u_{0}^{p}, …, u_{r−1}^{p}), where u_{r−1}, the dual element to T^{p^{r−1}}, is a linear functional u_{r−1} : k[𝔾_{a(r)}] → k.

Definition 2.1. Let G be an algebraic affine group scheme over k. We define

\xymatrix{Θ_{G,r} : k[G] \ar[r] & k[V_{r}(G)]}

to be the k-linear, p-nilpotent functional defined by the composition (2.1.1)

\xymatrix{k[G] \ar[r]^{1⊗id} & k[V_{r}(G)] ⊗ k[G] \ar[r]^{U_{G,r}^{∗}} & k[V_{r}(G)] ⊗ k[𝔾_{a(r)}] \ar[r]^{id ⊗u_{r−1}} & k[V_{r}(G)].}

As an element of Hom(k[G], k[V_{r}(G)]) ≡ kG⊗k[V_{r}(G)], Θ_{G,r} can be equivalently defined as

(2.1.2)  Θ_{G,r} = U_{G,r,∗}(u_{r−1}) ∈ kG ⊗ k[V_{r}(G)],

where

U_{G,r,∗} : k𝔾_{a(r)} ⊗ k[V_{r}(G)] → kG ⊗ k[V_{r}(G)].

Thus, Θ_{G,r} as given in (2.1.1) satisfies the property that its composition with k[V_{r}(G)] → K corresponding to the K-rational point μ : 𝔾_{a(r),K} → G_{K} equals μ_{∗}(u_{r−1}).

The following proposition justifies using the simplified notation

(2.1.3)  Θ_{G} : k[G] → k[V (G)],

where V (G) = \underrightarrow{lim}_{r} V_{r}(G) as in Definition 1.10. Namely, Θ_{G} is defined to be isomorphic to Θ_{G,r} provided that G is infinitesimal of height ≤ r.

Recall the canonical projection p_{r′,r} : 𝔾_{a(r′)} ↠ 𝔾_{a(r)}, and the induced closed embedding i_{r,r′} : V_{r}(G) ↪ V_{r′}(G), introduced in Remark 1.9, for r′ ≥ r.

Proposition 2.2. Let G be an an infinitesimal group scheme and let r′ ≥ r. Let A_{r} = k[V_{r}(G)], A_{r′} = k[V_{r′}(G)], and let

ϕ : A_{r′} ↠ A_{r}

be the surjective homomorphism corresponding to the canonical embedding i_{r,r′} : V_{r}(G) ↪ V_{r′}(G). Consider A_{r} as an A_{r′}-module via ϕ. Then

Θ_{G,r} = Θ_{G,r′} ⊗_{A_{r′}} 1 ∈ kG ⊗ A_{r′} ⊗_{A_{r′}} A_{r} ≃ kG ⊗ A_{r}.

Moreover, if G is an infinitesimal group scheme of height ≤ r, then Θ_{G,r} is thereby naturally identified with Θ_{G,r′}.

Proof. Consider the composition

(2.2.1)  \xymatrix{𝔾_{a(r′),A_{r}} \ar[dr]_{p_{r′,r}} \ar[rr]^{U_{G,r}∘p_{r′,r}} &  & G_{A_{r}} \\  & 𝔾_{a(r),A_{r}} \ar[ur]_{U_{G,r}} & .}

Since U_{G,r} ∈ V_{r}(G)(A_{r}) ≃ Hom(A_{r}, A_{r}) corresponds to the identity map on A_{r}, and p_{r′,r} is the map that induces ϕ : A_{r′} → A_{r}, we conclude that the composition U_{G,r} ∘ p_{r′,r} ∈ V_{r′}(G) ≃ Hom(A_{r′}, A_{r}) corresponds to ϕ. Hence, the universality of U_{G,r′} implies that U_{G,r} ∘ p_{r′,r} is obtained by pushing down the universal oneparameter subgroup U_{G,r′} via ϕ : A_{r′} → A_{r}. Therefore, we conclude

(2.2.2)  U_{G,r} ∘ p_{r′,r} = U_{G,r′} ⊗_{A_{r′}} A_{r}

which implies the equality of maps of group algebras

(2.2.3)  U_{G,r′,∗} ⊗_{A_{r′}} A_{r} = U_{G,r,∗} ∘ p_{r′,r,∗} : k𝔾_{a(r′)} ⊗ A_{r} → kG ⊗ A_{r}.

Since p_{r′,r,∗}(u_{r′−1}) = u_{r−1} ∈ k𝔾_{a(r)}, we conclude (U_{G,r,∗}∘p_{r′,r,∗})(u_{r′−1}) = U_{G,r,∗}(u_{r−1}) = Θ_{G,r}, whereas (U_{G,r′,∗} ⊗_{A_{r′}} A_{r})(u_{r′−1}) = U_{G,r′,∗}(u_{r′−1}) ⊗_{A_{r′−1}} 1 = Θ_{G,r′} ⊗_{A_{r′}} 1.

The second statement follows immediately from the fact that for G of height ≤ r, the map ϕ : A_{r′} → A_{r} is an isomorphism as shown in Remark 1.9. □

Let G be an affine group scheme over k, M be a kG-module, and ∇_{M} : M → M ⊗ k[G] be the corresponding co-action. A k-linear functional with values in a commutative k-algebra A, Θ : k[G] → A, determines an action of Θ on M ⊗ A which is the A-linear extension

(2.2.4)  Θ_{M} : M ⊗ A → M ⊗ A

of the map

\xymatrix{M \ar[r]^{∇_{M}} & M ⊗ k[G] \ar[r]^{id ⊗Θ_{M}} & M ⊗ A} .

If G is finite, we may view Θ ∈ Hom_{k}(k[G], A) as an element of kG ⊗ A (which we also denote by Θ). From this point of view, the action (2.2.4) is simply multiplication by Θ.

We now define the global p-nilpotent operator on a G-module M.

Definition 2.3. Let G be an infinitesimal group scheme. For any k[G]-comodule M with co-action ∇_{M} : M → M ⊗ k[G], we define the p-nilpotent operator

Θ_{M} : M ⊗ k[V (G)] → M ⊗ k[V (G)]

to be the k[V (G)]-linear extension of the map (id_{M} ⊗Θ_{G})∘∇_{M} : M → M ⊗k[V (G)].

Remark 2.4. The fact that Θ_{M} is p-nilpotent follows immediately from (2.1.2) since u_{r−1}^{p} = 0.

Slightly abusing notation, we shall often refer to Θ_{G} itself as the global pnilpotent operator.

We reformulate the pairing (2.2.4) in a more geometric fashion as follows.

Proposition 2.5. Let G be a group scheme over k, V be an affine k-scheme, and let M be a finite dimensional G-module. Then a k-linear functional Θ : k[G] → k[V] determines the pairing of k-schemes

(2.5.1)  V × M → M.

As a pairing of representable functors of commutative k-algebras A, this pairing sends (v : k[V] → A, m = ∑_{i} a_{i} ⊗ m_{i}) to ∑_{i} a_{i}(∑_{j} v(Θ(f_{i,j}))m_{j}) where ∇(m_{i}) = ∑_{j} f_{i,j} ⊗ m_{j}.

Example 2.6. We describe the global p-nilpotent operator Θ_{G} in each of the four examples of Example 1.5.

(1) Let G = GL_{m(1)} ≡ \underline{𝔤𝔩}_{m}, with group algebra k\underline{𝔤𝔩}_{m} = u(𝔤𝔩_{m}). Then the composition of

Θ_{G} : k[G] = k[X_{i,j}]/(X_{i,j}^{p} − δ_{i,j}) → k[N_{p}(𝔤𝔩_{m})]

with some K-rational point x ∈ N_{p}(𝔤𝔩_{m}) is the evaluation of the matrix coordinate functions X_{i,j} on x. In other words,

Θ_{\underline{𝔤𝔩}_{m}} = ∑_{1≤i,j≤m} x_{i,j} ⊗ \overline{X}_{i,j} ∈ 𝔲(𝔤𝔩_{m}) ⊗ k[N_{p}(𝔤𝔩_{m})],

where \overline{X}_{i,j} is the image of the i, j-matrix coefficient on 𝔤𝔩_{m}.

For a general p-restricted Lie algebra 𝔤, we have

(2.6.1)  Θ\underline{𝔤} = ∑ x ⊗ \overline{X} ∈ 𝔲(𝔤) ⊗ k[N_{p}(𝔤)]

where the sum is over basis elements X ∈ 𝔤^{#} with image \overline{X} ∈ k[N_{p}(𝔤)] and with dual x ∈ 𝔤.

We record an explicit formula for the universal p-nilpotent operator in the case of 𝔤 = sl_{2} for future reference. We have k[N_{p}(sl_{2})] ≃ k[x, y, z]/(xy + z^{2}). Let e, f, h be the standard basis of the p-restricted Lie algebra sl_{2}. Then

Θ_{\underline{s}l_{2}} = xe + yf + zh.

Observe that this formula agrees with the presentation of a “generic” π-point for u(sl_{2}) as given in [17, 2.5].

(2) Take G = 𝔾_{a(r)}. Then k[𝔾_{a(r)}] ≃ k[T]/T^{p^{r}}, and k[V (𝔾_{a(r)})] ≃ k[x_{0}, …, x_{r−1}] is graded in such a way that x_{i} has degree p^{i} (see Proposition 2.11 below). We compute \xymatrix{Θ_{𝔾_{a(r)}} : k[𝔾_{a(r)}] \ar[r] & k[V (𝔾_{a(r)})]} explicitly in this case (see also [26, 6.5.1]).

One-parameter subgroups of 𝔾_{a(r),K} are in one-to-one correspondence with the additive polynomials in K[T]/T^{p^{r}}, that is, polynomials of the form p(T) = a_{0}T + a_{1}T^{p} + … + a_{r−1}T^{p^{r−1}} (see [25, 1.10]). The map on coordinate algebras induced by the universal one-parameter subgroup U : 𝔾_{a(r),k[V (G)]} → 𝔾_{a(r),k[V (G)]} is given by the “generic” additive polynomial:

\xymatrix{U^{∗} : k[x_{0}, …, x_{r−1}][T]/T^{p^{r}} \ar[r] & k[x_{0}, …, x_{r−1}][T]/T^{p^{r}}} .

T ↦ x_{0}T + x_{1}T^{p} + … + x_{r−1}T^{p^{r−1}}.

To determine the linear functional

\xymatrix{Θ_{𝔾_{a(r)}} = u_{r−1} ∘ U^{∗} : k[T]/T^{p^{r}} \ar[r] & k[x_{0}…. , x_{r−1}]} ,

it suffices to determine the values of Θ_{𝔾_{a(r)}} on the linear generators {T^{i}}, 0 ≤ i ≤ p^{r} − 1. Since u_{r−1} is the dual to T^{p^{r−1}}, this further reduces to determining the coefficient by T^{p^{r−1}} in U^{∗}(T^{i}) = (x_{0}T + x_{1}T^{p} + … + x_{r−1}T^{p^{r−1}})^{i}. Computing this coefficient, we conclude that Θ_{𝔾_{a(r)}} is given explicitly on the basis elements of k[𝔾_{a(r)}] ≃ k[T]/T^{p^{r}} by

(2.6.2)  T^{i} ↦ ∑_{i_{0}+i_{1}p+···+i_{r−1}p^{r−1}=p^{r−1}^{i_{0}+i_{1}+···+i_{r−1}=i}} \binom{i}{i_{0}, i_{1}, …, i_{r−1}}x_{0}^{i_{0}} … x_{r−1}^{i_{r−1}},

where \binom{i}{i_{0},i_{1},...,i_{r−1}} = \frac{i!}{i_{0}!i_{1}!...i_{r−1}!} is the multinomial coefficient. Let \{v_{0}, …, v_{p^{r}−1}\} be the linear basis of k𝔾_{a(r)} dual to {1, T, …, T^{p^{r}−1}}. Dualizing (2.6.2), we conclude that Θ_{𝔾_{a(r)}} as an element of k𝔾_{a(r)} ⊗ k[V (𝔾_{a(r)})]) has the following form:

(2.6.3) Θ_{𝔾_{a(r)}} = ∑_{i=0}^{p^{r}−1} v_{i} [  ∑_{i_{0}+i_{1}p+···+i_{r−1}p^{r−1}=p^{r−1}^{i_{0}+i_{1}+···+i_{r−1}=i}} \binom{i}{i_{0}, i_{1}, …, i_{r−1}}x_{0}^{i_{0}} … x_{r−1}^{i_{r−1}} ] .

By [25, 1.4], v_{i} can expressed in terms of the algebraic generators u_{j} of k𝔾_{a(r)} via the following formulae

v_{i} = \frac{u_{0}^{j_{0}} … u_{r−1}^{j_{r−1}}}{j_{0}! … j_{r−1}!} ,

where i = j_{0} + j_{1}p + … + j_{r−1}p^{r−1} (0 ≤ j_{ℓ} ≤ p − 1) is the p-adic expansion of i.

Using these formulae, it is straightforward to calculate the term of Θ_{𝔾_{a(r)}} which is linear with respect to u_{i} (and homogeneous of degree p^{r−1} with respect to the grading of k[V (𝔾_{a(r)})]:

u_{0}x_{r−1} + u_{1}x_{r−2}^{p} + … + u_{r−1}x_{0}^{p^{r−1}}

The “linear” term gives the entire operator Θ_{𝔾_{a(r)}} for r = 1, 2, but for r ≥ 3, the non-linear terms start to appear.

(3) Let G = GL_{n(r)}. Recall that V (GL_{n(r)}) is the k-scheme of r-tuples of pnilpotent, pair-wise commuting matrices. For notational convenience, let A denote k[V (GL_{n(r)})] = k[M_{n}^{×r}]/I, a quotient of the coordinate algebra of the variety M_{n}^{×r} of r-tuples of n × n matrices. Then U_{GL_{n(r)}} : 𝔾_{a(r),A} → GL_{n(r),A} is specified by the A-linear map on coordinate algebras

(2.6.4)  U_{GL_{n(r)}}^{∗} : A[GL_{n(r)}] → A[T]/T^{p^{r}}, X_{i,j} ↦ ∑_{ℓ=0}^{p^{r}−1}(β_{ℓ})_{i,j}T^{j}

where {X_{i,j}; 1 ≤ i, j ≤ n} are the matrix coordinate functions of GL_{n}, where β_{ℓ} is given as in formula (1.7.1) in terms of the matrices α_{0}, …, α_{r−1} ∈ M_{n}(A), and α_{i} = β_{p^{i}} have matrix coordinate functions which generate A. (Indeed, the n^{2}r entries of α_{0}, …, α_{r−1} viewed as variables generate A, with relations given by the conditions that these matrices must be p-nilpotent and pairwise commuting.)

The p-nilpotent operator

Θ_{GL_{n(r)}} : k[GL_{n(r)}] → k[V (GL_{n(r)})]

is given by the k-linear functional sending a polynomial in the matrix coefficients P (X_{i,j}) ∈ k[GL_{n(r)}] to the coefficient of T^{p^{r−1}} of the sum of products corresponding to the polynomial P given by replacing each X_{i,j} by ∑_{ℓ=0}^{p^{r}−1}(β_{ℓ})_{i,j}T^{ℓ} (when taking products of matrix coefficients, one uses the usual rule for matrix multiplication);

P (X_{i,j}) ↦ P ( ∑_{ℓ=0}^{p^{r}−1}(β_{ℓ})_{i,j}T^{ℓ}) ↦ coeff of T^{p^{r−1}}.

For example, the coaction k^{n} → k[GL_{n}]⊗k^{n} corresponding to the natural representation of GL_{n} on k^{n} determines an action of Hom_{k}(k[GL_{n(r)}], A) ⊂ Hom_{k}(k[GL_{n}], A) on A^{n}, so that Θ_{GL_{n(r)}} : A^{n} → A^{n} is given in matrix form by (Θ_{GL_{n(r)}}(X_{i,j})).

(4) We consider G = SL_{2(2)}, and assume notations and conventions of Example 1.12(4). Let A = k[SL_{2(2)}]. Using the general discussion in (2.6(3)) above (also compare to (1.12(4))), one readily computes that the map on coordinate algebras U_{SL_{2(2)}}^{∗} : A[SL_{2(2)}] → A[𝔾_{a(2)}] ≃ A[T]/T^{p^{2}} is given by

X_{1,1} ↦ 1 + z_{0}T + z_{1}T^{p}, X_{1,2} ↦ x_{0}T + x_{1}T^{p}
X_{2,1} ↦ y_{0}T + y_{1}T^{p}, X_{2,2} ↦ 1 − z_{0} − z_{1}T^{p}.

By (2.1.2), Θ_{SL_{2(2)}} = U_{SL_{2(2)} ∗}(u_{1}), the functional given by “reading off the coefficient of T^{p}”.

Let e, f, h, e^{(p)}, f^{(p)}, h^{(p)} ∈ k SL_{2(2)} denote respectively the linear duals of the functions X_{1,2}, X_{2,1}, X_{1,1} − 1, X_{1,2}^{p}, X_{2,1}^{p}, (X_{1,1} − 1)^{p} on SL_{2(2)}, and set

e^{(i)} = \frac{e^{i}}{i!} , f^{(i)} = \frac{f^{i}}{i!} , \binom{h}{i} = \frac{h(h − 1)(h − 2) … (h − i + 1)}{i!}

for i < p. Fix the linear basis of k[SL_{2(2)}] given by powers of X_{1,2}, X_{2,1}, X_{1,1} − 1 (in this fixed order). Then the element of k SL_{2(2)} dual to X_{1,2}^{i}X_{2,1}^{j}(X_{1,1} − 1)^{ℓ} for i + j + ℓ ≤ p is given by

(X_{1,2}^{i}X_{2,1}^{j}(X_{1,1} − 1)^{ℓ})^{#} = e^{(i)}f^{(j)}\binom{h}{ℓ}

(where \binom{h}{p} is identified with h^{(p)} by definition). With these conventions Θ_{SL_{2(2)}} ∈ k SL_{2(2)} ⊗k[V (SL_{2(2)})] equals

(2.6.5) x_{1}e + y_{1}f + z_{1}h + x_{0}^{p}e^{(p)} + y_{0}^{p}f^{(p)} + z_{0}^{p}h^{(p)} + ∑_{i,j,ℓ<p^{i+j+ℓ=p}} x_{0}^{i}y_{0}^{j}z_{0}^{ℓ}e^{(i)}f^{(j)}\binom{h}{ℓ}.

Our motivational example for G = 𝔾_{a(1)} × 𝔾_{a(1)} from the beginning of this section is a special case of (2.6(1)).

Example 2.7. Let G = 𝔾_{a(1)}^{×r}. Then G corresponds to the abelian p-nilpotent Lie algebra g_{a}^{⊕r}, and kG = k[u_{0}, …, u_{r−1}]/(u_{0}^{p}, …, u_{r−1}^{p}). We have V (G) ≃ 𝔸^{r}, and k[V (G)] ≃ k[X_{0}, …, X_{r−1}] where all generators are in degree one. Then Θ_{G} ∈ kG ⊗ k[V (G)] is given by the simple formula

Θ_{G} = u_{0}X_{0} + ⋯ + u_{r−1}X_{r−1}.

It is useful to contrast this formula with a much more complicated result for G = 𝔾_{a(r)} in (2.6(2)).

To complement Example 2.6, we make explicit the action of Θ_{G} on some representation of G for each of the four types of finite group schemes we have been considering in examples.

Example 2.8. (1) Let G = \underline{𝔤} and let M = 𝔤^{ad} denote the adjoint representation of the p-restricted Lie algebra 𝔤; let \{x_{i}\} be a basis for 𝔤. We identify Θ\underline{𝔤} as the k[N_{p}(𝔤)]-linear endomorphism

Θ\underline{𝔤} : 𝔤^{ad} ⊗ k[N_{p}(𝔤)] → 𝔤^{ad} ⊗ k[N_{p}(𝔤)], x ⊗ 1 ↦ ∑_{i}[x_{i}, x] ⊗ X_{i},

where X_{i} is the image under the projection S^{∗}(𝔤^{#}) → k[N_{p}(𝔤)] of the dual basis element to x_{i}.

(2) Let M denote the cyclic k𝔾_{a(r)}-module

M = k[u_{0}, …, u_{r−1}]/(u_{0}, u_{1}^{p}, …, u_{r−1}^{p}) ≃ k[u_{1}, …, u_{r−1}]/(u_{1}^{p}, …, u_{r−1}^{p}).

As recalled in Example 2.6(2), k[V (𝔾_{a(r)})] = k[𝔸^{r}] = k[a_{0}, …, a_{r−1}], k𝔾_{a(r)} = k[u_{0}, …, u_{r−1}]/(u_{i}^{p}), and

Θ_{𝔾_{a(r)}} ∈ A[u_{0}, …, u_{r−1}]/(u_{i}^{p})

is given by the complicated, but explicit formula (2.6.3). We conclude that

Θ_{𝔾_{a(r)}} : M ⊗ A → M ⊗ A

is the A-linear endomorphism sending u_{i} to \overline{Θ}_{𝔾_{a(r)}} · u_{i}, where \overline{Θ}_{𝔾_{a(r)}} is the image of Θ_{𝔾_{a(r)}} under the projection A[u_{0}, …, u_{r−1}]/(u_{i}^{p}) → A[u_{1}, …, u_{r−1}]/(u_{i}^{p}).

(3) Let M be the restriction to GL_{n(r)} of the canonical n-dimensional rational GL_{n}-module V_{n}. By Example 1.12(3), A = k[V (GL_{n(r)}] is the quotient of k[𝔤𝔩_{n}]^{⊗r} by the ideal generated by the equations satisfied by an r-tuple of n × n-matrices with the property that each matrix is p-nilpotent and that the matrices pair-wise commute. The complexity of the map

Θ_{GL_{n(r)}} : V_{n} ⊗ A → V_{n} ⊗ A

is revealed even in the case n = 2 which is worked out explicitly below.

(4) Let M be the restriction to SL_{2(2)} of the rational GL_{2}–representation V_{2}. Then Example 1.12(4) gives an explicit description of A = k[V (SL_{2(2)})] as a quotient of k[x_{0}, y_{0}, z_{0}, x_{1}, y_{1}, z_{1}] and (2.6.5) gives Θ_{SL_{2(2)}} explicitly. Since V_{2} is a homogeneous polynomial representation of GL_{2} of degree 1, the divided powers e^{(p)}, f^{(p)} and h^{(p)} as well as all products of the form e^{(i)}f^{(j)}\binom{h}{ℓ} act trivially on M. Hence, the map

Θ_{SL_{2(2)}} : M ⊗ A → M ⊗ A

is given by the matrix

\xymatrix{ & ^{[ \begin{matrix}z_{1} & x_{1} \\ y_{1} & −z_{1} \end{matrix} ]} \ar[dr] \\ A^{2} &  & A^{2}.}

When viewing group schemes as functors, it is often convenient to think of G_{k[V (G)]} as G × V (G) (i.e., G × V (G) = Spec(k[V (G)] ⊗ k[G])). From this point of view, U_{G} has the form

\xymatrix{U_{G} : 𝔾_{a(r)} × V (G) \ar[r] & G × V (G).}

The following naturality property of Θ_{G} will prove useful when we consider M ⊗ k[V (G)] as a free, coherent sheaf on V (G) and restrict this sheaf to V (H) ⊂ V (G) equipped with its action of H.

Proposition 2.9. Choose r > 0 and consider a closed embedding i : H ↪ G of algebraic affine group schemes over k. Let ϕ : V_{r}(H) → V_{r}(G) be the closed embedding of affine schemes induced by i, with associated surjective map ϕ^{∗} : k[V_{r}(G)] → k[V_{r}(H)] on coordinate algebras. Then the following square commutes

(2.9.1)  \xymatrix{𝔾_{a(r)} × V_{r}(H) \ar[d]_{id×ϕ} \ar[r]^{U_{H,r}} & H × V_{r}(H) \ar[d]^{i×ϕ} \\ 𝔾_{a(r)} × V_{r}(G) \ar[r]^{U_{G,r}} & G × V_{r}(G).}

Consequently, the following square of k-linear maps commutes:

(2.9.2)  \xymatrix{k[G] \ar[d]^{i^{∗}} \ar[r]^{Θ_{G,r}} & k[V_{r}(G)] \ar[d]^{ϕ^{∗}} \\ k[H] \ar[r]^{Θ_{H,r}} & k[V_{r}(H)].}

Thus, for any rational G-module M we have a compatibility of coactions on M:

(2.9.3)  \xymatrix{M \ar[r]^{∇_{M}} \ar[dr]_{∇_{M↓_{H}}} & M ⊗ k[G] \ar[d]^{id ⊗i^{∗}} \ar[r]^{id ⊗Θ_{G,r}} & M ⊗ k[V_{r}(G)] \ar[d]^{id ⊗ϕ^{∗}} \\  & M ⊗ k[H] \ar[r]^{id ⊗Θ_{H,r}} & M ⊗ k[V_{r}(H)].}

Proof. The fact that ϕ : V_{r}(H) → V_{r}(G) induced by the closed embedding i : H ↪ G is itself a closed embedding is given by [25, 1.5]. By universality of U_{G,r}, the composition (i×id)∘U_{H,r} : 𝔾_{a(r)} ×V_{r}(H) → G×V_{r}(H) is obtained by pull-back of U_{G,r} via some morphism V_{r}(H) → V_{r}(G). By comparing maps on R-valued points, we verify that this morphism must be ϕ. This implies the commutativity of (2.9.1).

The commutative square (2.9.1) gives a commutative square on coordinate algebras:

(2.9.4)  \xymatrix{k[V_{r}(G)] ⊗ k[G] \ar[d]^{ϕ^{∗}⊗ i^{∗}} \ar[r]^{U_{G,r}^{∗}} & k[V_{r}(G)] ⊗ k[𝔾_{a(r)}] \ar[d]^{ϕ^{∗}⊗ id} \\ k[V_{r}(H)] ⊗ k[H] \ar[r]^{U_{H,r}^{∗}} & k[V_{r}(H)] ⊗ k[𝔾_{a(r)}].}

Concatenating (2.9.4) on the right with the commutative square of linear maps

\xymatrix{k[V_{r}(G)] ⊗ k[𝔾_{a(r)}] \ar[d]^{ϕ^{∗}⊗id} \ar[r]^{u_{r−1}} & k[V_{r}(G)] \ar[d]^{ϕ^{∗}} \\ k[V_{r}(H)] ⊗ k[𝔾_{a(r)}] \ar[r]^{u_{r−1}} & k[V_{r}(H)]}

and with the inclusions k[G] → k[V_{r}(G)] ⊗ k[G] and k[H] → k[V_{r}(H)] ⊗ k[H] on the left, we obtain a commutative diagram:

\xymatrix{k[G] \ar[d]^{i^{∗}} \ar@{^{(}->}[r] & k[V_{r}(G)] ⊗ k[G] \ar[d]^{ϕ^{∗}⊗ i^{∗}} \ar[r]^{U_{G,r}^{∗}} & k[V_{r}(G)] ⊗ k[𝔾_{a(r)}] \ar[d]^{ϕ^{∗}⊗ id} \ar[r]^{u_{r−1}} & k[V_{r}(G)] \ar[d]^{ϕ^{∗}} \\ k[H] \ar@{^{(}->}[r] & k[V_{r}(H)] ⊗ k[H] \ar[r]^{U_{H,r}^{∗}} & k[V_{r}(H)] ⊗ k[𝔾_{a(r)}] \ar[r]^{u_{r−1}} & k[V_{r}(H)].}

Eliminating the middle square, we obtain the square (2.9.2). Hence, it is commutative.

Finally, the commutativity of (2.9.3) follows immediately from the commutativity of (2.9.2). □

Pre-composition determines a natural action V_{r}(G) × V_{r}(𝔾_{a(r)}) → V_{r}(G) for any algebraic affine group scheme G. Recall from [25, 1.10] that V_{r}(𝔾_{a(r)}) ≃ 𝔸^{r}: morphisms of group schemes 𝔾_{a(r),A} → 𝔾_{a(r),A} over A have associated maps on coordinate algebras A[T]/T^{p^{r}} → A[T]/T^{p^{r}} given by additive polynomials, that is polynomials of the form a_{0}T + a_{1}T^{p} + ⋯ a_{r−1}T^{p^{r−1}}. Restricting the action V_{r}(G)×V_{r}(𝔾_{a(r)}) → V_{r}(G) to linear polynomials 𝔸^{1} ⊂ V_{r}(𝔾_{a(r)}) ≃ 𝔸^{r} determines a natural action

(2.9.5)  V_{r}(G) × 𝔸^{1} → V_{r}(G),

which is equivalent by [25, 1.11] to a functorial grading on k[V_{r}(G)].

Proposition 2.10. Let G be an algebraic affine group scheme over k. Then the coordinate algebra k[V_{r}(G)] of V_{r}(G) is a graded algebra generated by homogeneous generators of degrees p^{i}, 0 ≤ i < r.

Proof. The coordinate algebra k[V_{r}(G)] is graded by [25, 1.12]. If G = GL_{N}, then an R-valued point of V_{r}(GL_{N}) is given by an r-tuple of N × N pair-wise commuting, p-nilpotent matrices with entries in R, (α_{0}, …, α_{r−1}). The action of c ∈ V (𝔾_{a(1)})(R) on (α_{0}, α_{1}, …, α_{r−1}) ∈ V (GL_{N(r)})(R) is given by the formula

(α_{0}, α_{1}, …, α_{r−1}) × c = (cα_{0}, c^{p}α_{1}, …, c^{p^{r−1}}α_{r−1}).

Hence, the coordinate functions of the matrix α_{i} have grading p^{i} and, therefore, k[V_{r}(GL_{N})] is generated by homogeneous elements of degree p^{i}, 0 ≤ i < r.

Let i : G → GL_{N} be a closed embedding of a finite group scheme G into some GL_{N}. The naturality of the grading (see [25, 1.12]) implies that the surjective map ϕ : k[V_{r}(GL_{N})] → k[V_{r}(G)] is a map of graded algebras. □

Proposition 2.11. For any algebraic affine group scheme G and integer r > 0, the k-linear map

Θ_{G,r} : k[G] → k[V_{r}(G)]

has image contained in the homogeneous summand of k[V_{r}(G)] of degree p^{r−1}. If G is infinitesimal, then this is equivalent to the following:

U_{G,r∗} : k𝔾_{a(r)} ⊗ k[V_{r}(G)] → kG ⊗ k[V_{r}(G)]

sends u_{r−1} ⊗ 1 ∈ k𝔾_{a(r)} ⊗ k[V_{r}(G)] to some ∑ x_{i} ⊗ a_{i} ∈ kG ⊗ k[V_{r}(G)] with each a_{i} ∈ k[V_{r}(G)] homogeneous of degree p^{r−1}.

Proof. Let A = k[V_{r}(G)]. Since Θ_{G,r} factors through the r^{th} Frobenius kernel of G, we may assume G is infinitesimal. Let ⟨λ_{i}⟩ be a set of linear generators of k[G], and ⟨λ̌_{i}⟩ be the dual set of linear generators of kG. Then U_{G,r,∗}(u_{r−1}) = ∑ λ̌_{i} ⊗ f_{i} if and only if u_{r−1}(U_{G}^{∗}(λ_{i})) = f_{i} if and only if U_{G,r}^{∗}(λ_{i}) = …+f_{i}T^{p^{r−1}} +. . .. Hence, the assertion that Θ_{G,r} is homogeneous of degree p^{r−1} is equivalent to showing that the map k[G] → A defined by reading off the coefficient of

U_{G,r}^{∗} : k[G] → A ⊗ k[G] → A ⊗ k[𝔾_{a(r)}] → A[T]/T^{p^{r}}

of the monomial T^{p^{r−1}} is homogeneous of degree p^{r−1}.

The coordinate algebra k[𝔾_{a(r)}] ≃ k[T]/T^{p^{r}} has a natural grading with T assigned degree 1. This grading corresponds to the monoidal action of 𝔸^{1} on 𝔾_{a(r)} by multiplication:

\xymatrix{𝔾_{a(r)} × 𝔸^{1} \ar[r]^{s×a↦sa} & 𝔾_{a(r)}} .

We proceed to prove that this action is compatible with the action of 𝔸^{1} on V_{r}(G) which defines the grading on A in the sense that the following diagram commutes:

(2.11.1)  \xymatrix{𝔾_{a(r)} × 𝔸^{1} × V_{r}(G) \ar[d]^{action×id} \ar[r]^{id ×action} & 𝔾_{a(r)} × V_{r}(G) \ar[r]^{U_{G,r}} & G × V_{r}(G) \ar[d]^{pr_{G}} \\ 𝔾_{a(r)} × V_{r}(G) \ar[r]^{U_{G,r}} & G × V_{r}(G) \ar[r]^{pr_{G}} & G.}

Commutativity of (2.11.1) is equivalent to the commutativity of the corresponding diagram of S-valued points for any choice of finitely generated commutative k-algebras S and element a ∈ S:

(2.11.2)

\xymatrix{𝔾_{a(r)}(S) × V_{r}(G)(S) \ar[d]^{a×1} \ar[r]^{1×a} & 𝔾_{a(r)}(S) × V_{r}(G)(S) \ar[r]^{U_{G,r}(S)} & G(S) × V_{r}(G)(S) \ar[d]^{pr_{G}} \\ 𝔾_{a(r)}(S) × V_{r}(G)(S) \ar[r]^{U_{G,r}(S)} & G(S) × V_{r}(G)(S) \ar[r]^{pr_{G}} & G(S).}

Choose an embedding of G into some GL_{N(r)}. Using Proposition 2.9 and the naturality with respect to change of G of the action of 𝔸^{1} on V_{r}(G), we can compare the diagram (2.11.2) for G and for GL_{N(r)}. The injectivity of G(S) → GL_{N(r)}(S) implies that it suffices to assume that G = GL_{N(r)}. Let s ∈ 𝔾_{a(r)}(S), \underline{α} = (α_{0}, …, α_{r−1}) ∈ V_{r}(GL_{N})(S). Then a ∘ \underline{α} = (aα_{0}, a^{p}α_{1}, …, a^{p^{r−1}}α_{r−1}), so exp\underline{α}(s) = exp(sα_{0}) exp(s^{p}α_{1}) … exp(s^{p^{r−1}}α_{p−1}) ∈ GL_{N(r)}(S) by (1.10.1). Thus, restricted to the point (s, \underline{α}) ∈ (𝔾_{a(r)} × V_{r}(GL_{N}))(S), (2.11.2) becomes

(2.11.3)  \xymatrix{(s, \underline{α}) \ar[d]^{a×1} \ar[r]^{1×a} & (s, a ∘ \underline{α}) \ar[r]^{U_{G,r}} & (exp_{a∘\underline{α}}(s), a ∘ \underline{α}) \ar[d] \\ (as, \underline{α}) \ar[r]^{U_{G,r}} & (exp\underline{α}(as), \underline{α}) \ar[r] & exp\underline{α}(as).}

Commutativity of (2.11.3) is implied by the evident equality exp_{a∘\underline{α}}(s) = exp\underline{α}(as).

Consequently, we have a commutative diagram on coordinate algebras corresponding to (2.11.1):

(2.11.4)  \xymatrix{A ⊗ k[t] ⊗ k[𝔾_{a(r)}] & A ⊗ k[𝔾_{a(r)}] \ar[l]_{act^{∗}⊗id} & A ⊗ k[G] \ar[l]_{U_{G,r}^{∗}} \\ A ⊗ k[𝔾_{a(r)}] \ar[u]_{id ⊗act^{∗}} & A ⊗ k[G] \ar[l]_{U_{G,r}^{∗}} & k[G]. \ar@{^{(}->}[u] \ar@{^{(}->}[l]}

The map \xymatrix{act^{∗} : A \ar[r] & A ⊗ k[t] = A ⊗ k[𝔸^{1}]} of the upper horizontal arrow is the map on coordinate algebras which corresponds to the grading on A. The left vertical map corresponds to the grading on k[𝔾_{a(r)}] ≃ k[T]/T^{p^{r}} and is given explicitly by T ↦ t ⊗ T.

For λ ∈ k[G], write U_{G,r}^{∗}(1 ⊗ λ) = ∑ f_{i} ⊗ c_{i}T^{i} ∈ A ⊗ k[𝔾_{a(r)}]. The composition of the lower horizontal and left vertical maps of (2.11.4) sends λ to ∑ f_{i} ⊗ t^{i} ⊗ c_{i}T^{i}. On the other hand, the composition of the right vertical and upper horizontal maps of (2.11.4) sends λ to ∑ act^{∗}(f_{i}) ⊗ c_{i}T^{i}. We conclude that

f_{i} ⊗ t^{i} = act^{∗}(f_{i}),

so that f_{i} is homogeneous of degree i. □

As a corollary (of the proof of) Proposition 2.11, we see why for G infinitesimal of height ≤ r the homogeneous degree of Θ_{G,r} ∈ kG ⊗ k[V_{r}(G)] is p^{r−1} whereas the homogeneous degree of Θ_{G,r+1} ∈ kG ⊗ k[V_{r+1}(G)] is p^{r}.

Corollary 2.12. Let G be an infinitesimal group of height ≤ r. Then the map i^{∗} : k[V_{r+1}(G)] → k[V_{r}(G)] of Proposition 2.2 is a graded isomorphism which divides degrees by p.

Proof. Let π^{∗} : k[V_{r}(G)] → k[V_{r+1}(G)] be the inverse of i^{∗}. The commutativity of (2.11.1) implies that we may compute the effect on degree of π^{∗} by identifying the effect on degree of the map p^{∗} : k[𝔾_{a(r)}] = k[t]/t^{p^{r}} → k[t]/t^{p^{r+1}} = k[𝔾_{a(r+1)}]. Yet this map clearly multiplies degree by p. □

3. θ_{v} and local Jordan type

The purpose of this section is to exploit our universal p-nilpotent operator Θ_{G} to investigate the local Jordan type of a finite dimensional kG-module M. The local Jordan type of M gives much more detailed information about a kG-module M than the information which can be obtained from the support variety (or, rank variety) of M. In this section, we work through various examples, give an algorithm for computing local Jordan types, and understand the effect of Frobenius twists. Moreover, we establish restrictions on the rank and dimension of kG-modules of constant Jordan type.

Definition 3.1. Let G be an infinitesimal group scheme and v ∈ V (G). Let k(v) denote the residue field of V (G) at v, and let

μ_{v} = U_{G} ⊗_{k[V (G)]} k(v) : 𝔾_{a(r),k(v)} → G_{k(v)}

be the associated 1-parameter subgroup (for r ≥ ht(G)). We define the local pnilpotent operator at v, θ_{v} , to be

θ_{v} = Θ_{G} ⊗_{k[V (G)]} k(v) = μ_{v∗}(u_{r−1}) ∈ k(v)G.

Equivalently, for a k(v)-rational point v : Spec k(v) → V (G), we define θ_{v} = Θ_{G}(v), the evaluation of Θ_{G} at v:

\xymatrix{θ_{v} : k[G] \ar[r]^{Θ_{G}} & k[V (G)] \ar[r] & k(v)}

where the second map corresponds to the point v.

In the special case that G = GL_{n(r)} for some n > 0, we use the alternate notation θ\underline{α} for the local p-nilpotent operator at \underline{α} = (α_{0}, …, α_{r−1}) ∈ V (GL_{n(r)}) ≃ N_{p}^{[r]}(gl_{n}):

(3.1.1)  θ\underline{α} = exp_{\underline{α},∗}(u_{r−1}) ∈ k(\underline{α}) GL_{n(r)},

where k(\underline{α}) is the residue field of \underline{α} ∈ V (GL_{n(r)}).

Let K be a field. Then a finite dimensional K[u]/u^{p}-module M is a direct sum of cyclic modules of dimension ranging from 1 to p. We may thus write M ≃ a_{p}[p]+⋯+a_{1}[1], where [i] is the cyclic K[u]/u^{p}-module K[u]/u^{i} of dimension i. We refer to the p-tuple

(3.1.2)  JType(M, u) = (a_{p}, …, a_{1})

as the Jordan type of the K[u]/u^{p}-module M. We also refer to JType(M, u) as the Jordan type of the p-nilpotent operator u on M.

For simplicity, we introduce the following notation.

Definition 3.2. With notation as in Definition 3.1, we set

JType(M, θ_{v}) ≡ JType((μ_{v,∗})^{∗}(M_{k(v)}), u_{r−1}).

We refer to this Jordan type as the local Jordan type of M at v ∈ V (G).

Remark 3.3. Essentially by definition, the rank variety V (G)_{M} of a finite dimensional kG-module for an infinitesimal group scheme G is the closed, reduced subscheme of V (G) consisting of those points v ∈ V (G) at which the local Jordan type of M has some Jordan block of size < p; in other words, those v ∈ V (G) for which JType(M, θ_{v}) ≠ \frac{dim M}{p} [p].

The following proposition will enable us to make more concrete and explicit the local Jordan type of a kG-module M at a given 1-parameter subgroup of G.

Proposition 3.4. Let \underline{α} = (α_{0}, …, α_{r−1}) ∈ V (GL_{n(r)}) be an r-tuple of p-nilpotent pair-wise commuting matrices. Let M be a k GL_{(r)}-module of dimension N, and let ρ : GL_{m(r)} → GL_{N} be the associated structure map. The (i, j)-matrix entry of the action of the local p-nilpotent operator θ\underline{α} ∈ k(\underline{α}) GL_{n(r)} of (3.1.1) on M equals the coefficient of T^{p^{r−1}} of

(exp\underline{α})^{∗}(ρ^{∗}X_{i,j}) ∈ k(\underline{α})[𝔾_{a(r)}] ≃ k(\underline{α})[T]/T^{p^{r}},

where {X_{i,j}, 1 ≤ i, j ≤ N} are the matrix coordinate functions of GL_{N}.

Proof. Let ⟨m_{i}⟩_{1≤i≤N} be the basis of M corresponding to the structure map ρ. The structure of M as a comodule for k[GL_{n(r)}] is given by

M → M ⊗ k[GL_{n(r)}], m_{j} ↦ ∑_{i} m_{i} ⊗ ρ^{∗}X_{i,j},

and thus the comodule structure of M_{k(\underline{α})} for k(\underline{α})[𝔾_{a(r)}] is given by

M → M ⊗ k(\underline{α})[𝔾_{a(r)}], m_{j} ↦ ∑_{i} m_{i} ⊗ exp∗ ; \underline{α}(ρ^{∗}X_{i,j}).

The proposition follows from the fact that u_{r−1} : k(\underline{α})[𝔾_{a(r)}] → k(\underline{α}) is given by reading off the coefficient of T^{p^{r−1}} ∈ k(\underline{α})[𝔾_{a(r)}]. □

Example 3.5. We investigate the local Jordan type of the various representations considered in Example 2.8.

(1) Consider the adjoint representation M = 𝔤^{ad} of a p-restricted Lie algebra 𝔤 and a 1-parameter subgroup

μ_{X} : 𝔾_{a(1),K} → \underline{𝔤}_{K}, inducing K[u]/u^{p} → 𝔲(𝔤_{K})

sending u to some p-nilpotent X ∈ 𝔤_{K}. The local Jordan type of 𝔤^{ad} at μ_{X} is simply the Jordan type of the endomorphism ad_{X} : 𝔤_{K}^{ad} → 𝔤_{K}^{ad},

JType(𝔤^{ad}, θ_{X}) = JType(X).

(2) Let M = k[𝔾_{a(r)}]/(u_{0}) ≃ k[u_{1}, …, u_{r−1}]/(u_{1}^{p}, …, u_{p−1}^{p}) be a cyclic k𝔾_{a(r)} = k[u_{0}, …, u_{r−1}]/(u_{0}^{p}, …, u_{r−1}^{p})-module, and let μ\underline{a} : 𝔾_{a(r)} → 𝔾_{a(r)} be a 1-parameter subgroup for some \underline{a} of V (𝔾_{a(r)}) = 𝔸^{r}. Then

JType(M, θ\underline{a}) = \{ \begin{matrix}p^{r−2}[p], ∃i > 0, a_{i} ≠ 0 \\ p^{r−1}[1], otherwise. \end{matrix}

(3) Let G = GL_{n(r)}, and let V_{n} be the canonical n-dimensional rational representation of GL_{n(r)}. We apply Proposition 3.4, observing that ρ for V_{n} is simply the natural inclusion GL_{n(r)} ⊂ GL_{n}. Since

(3.5.1)  exp∗ ; \underline{α}(X_{i,j}) = ∑_{ℓ=0}^{p^{r}−1}[β_{ℓ}]_{i,j}t^{ℓ},

where β_{ℓ} are matrices determined by α_{i} as in Proposition 1.7, we conclude

JType(V_{n}, θ\underline{α}) = JType(α_{r−1}).

Specializing to r = 2,

JType(V_{n}, θ_{(α_{0},α_{1})}) = α_{1}.

(4) “Specializing” to G = SL_{2(2)}, consider \underline{α} = ([ \begin{matrix}c_{0} & a_{0} \\ b_{0} & −c_{0} \end{matrix} ] , [ \begin{matrix}c_{1} & a_{1} \\ b_{1} & −c_{1} \end{matrix} ]).

Then JType(V_{2}, θ\underline{α}) equals the Jordan type of the matrix [ \begin{matrix}c_{1} & a_{1} \\ b_{1} & −c_{1} \end{matrix} ].

We extend Example 3.5(3) by considering tensor powers V_{n}^{⊗d} of the canonical rational representation of GL_{n} restricted to GL_{n(2)}. In this example, the role of both entries of the pair \underline{α} = (α_{0}, α_{1}) is non-trivial.

Example 3.6. Consider the N = n^{d}-dimensional rational GL_{n}-module M = V_{n}^{⊗d} where V_{n} is the canonical n-dimensional rational GL_{n}-module. Let ρ : GL_{n(r)} → GL_{N} be the representation of M restricted to GL_{n(r)}. A basis of M is {e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{d}}; 1 ≤ i_{j} ≤ n}, where {e_{i}; 1 ≤ i ≤ n} is a basis for V_{n}. Let {X_{i_{1},j_{1};...,i_{d},j_{d}}, 1 ≤ i_{t}, j_{t} ≤ n} denote the matrix coordinate functions on GL_{N}, and let {Y_{s,t}, 1 ≤ s, t ≤ n} denote the matrix coordinate functions of GL_{n}.

Then ρ^{∗} : k[GL_{N}] → k[GL_{n(r)}] is given by

X_{i_{1},j_{1};...;i_{d},j_{d}} ↦ Y_{i_{1},j_{1}} ⋯ Y_{i_{d},j_{d}}.

Thus,

(exp\underline{α})^{∗}(ρ^{∗}(X_{i_{1},j_{1};...,i_{d},j_{d}})) = (exp\underline{α})^{∗}(Y_{i_{1},j_{1}}) ⋯ (exp\underline{α})^{∗}(Y_{i_{d},j_{d}}).

Now, specialize to r = 2 so that we can make this more explicit. Then the coefficient of T^{p} of (exp_{(α_{0},α_{1})})^{∗}(ρ^{∗}(X_{i_{1},j_{1};...,i_{d},j_{d}})) is

(3.6.1)  ∑_{k=1}^{d}(α_{1})_{i_{k},j_{k}} + ∑_{f_{1}+···+f_{d}=p^{0≤f_{k}<p}} \frac{1}{f_{1}!} ⋯ \frac{1}{f_{d}!}((α_{0})^{f_{1}})_{i_{1},j_{1}} ⋯ ((α_{0})^{f_{d}})_{i_{d},j_{d}}.

This gives the action of θ_{(α_{0},α_{1})} on M.

To simplify matters even further, consider the special case (α_{0})^{2} = 0. For 1 ≤ d < p, θ_{(α_{0},α_{1})} on M is given by the N × N-matrix

(i_{1}, j_{1}; …; i_{d}, j_{d}) ↦ (∑_{k=1}^{d}(α_{1})_{i_{k},j_{k}}).

For d = p, the action of θ_{(α_{0},α_{1})} on M is given by the N × N-matrix

(i_{1}, j_{1}; …; i_{p}, j_{p}) ↦ (∑_{k=1}^{p}(α_{1})_{i_{k},j_{k}} + (α_{0})_{i_{1},j_{1}} ⋯ (α_{0})_{i_{p},j_{p}}).

An analogous calculation applies to the the d-fold symmetric product S^{d}(V_{n}) and d-fold exterior product Λ^{d}(V_{n}) of the canonical n-dimensional rational GL_{n}-module V_{n}.

The proof of Proposition 3.4 applies equally well to prove the following straightforward generalization, which one may view as an algorithmic method of computing the “local Jordan type” of a kG-module M of dimension N. The required input is an explicit description of the map on coordinate algebras ρ^{∗} given by ρ : G → GL_{N} determining the kG-module M.

Theorem 3.7. Let G be an infinitesimal group scheme of height ≤ r, and let ρ : G → GL_{N} be a representation of G on a vector space M of dimension N. Consider some v ∈ V (G), and let μ_{v} : 𝔾_{a(r),k(v)} → G_{k(v)} be the corresponding 1-parameter subgroup of height r. Then the (i, j)-matrix entry of the action of θ_{v} ∈ k(v)G on M equals the coefficient of T^{p^{r−1}} of

(μ_{v})^{∗}(ρ^{∗}X_{i,j}) ∈ k(v)[𝔾_{a(r)}],

where {X_{i,j}, 1 ≤ i, j ≤ N} are the matrix coordinate functions of GL_{N}.

As a simple corollary of Theorem 3.7, we give a criterion for the local Jordan type of the kG-module M to be trivial (i.e., equal to (dim M)[1]) at a 1-parameter subgroup μ_{v}, v ∈ V (G).

Corollary 3.8. With the hypotheses and notation of Theorem 3.7,

JType(M, θ_{v}) = JType((μ_{v,∗})^{∗}(M_{k(v)}), u_{r−1}) = (dim M)[1]

if deg (ρ ∘ μ_{v})^{∗}(X_{i,j}) < p^{r−1} for all 1 ≤ i, j ≤ N.

One means of constructing kG-modules is by applying Frobenius twists to known kG-modules. Our next objective is to establish (in Proposition 3.10) a simple relationship between the p-nilpotent operator θ\underline{α} on a k GL_{n(r)}-module M and θ\underline{α} on the s-th Frobenius twist M^{(s)} of M for any 0 ≠ v ∈ V (GL_{n(r)}).

Before formulating this relationship, we make explicit the definition of the Frobenius map for an arbitrary affine group scheme over k. Let G be an affine group scheme over k and define for any s > 0 the s^{th} Frobenius map F^{s} : G → G^{(s)} given by the k-linear algebra homomorphism

(3.8.1)  F^{s∗} : k[G^{(s)}] = k ⊗_{p^{s}} k[G] → k[G], a ⊗ f ↦ a · f^{p^{s}},

where k ⊗_{p^{s}} k[G] is the base change of k[G] along the p^{s}-power map k → k (an isomorphism only for k perfect). If G is defined over 𝔽_{p^{s}} (for example, if G = GL_{n}), then we have a natural isomorphism

\xymatrix{k[G] = k ⊗_{𝔽_{p^{s}}} 𝔽_{p^{s}}[G] \ar[r]^{∼} & k ⊗_{p^{s}} k ⊗_{𝔽_{p^{s}}} 𝔽_{p^{s}}[G] = k[G^{(s)}]}

so that F^{s} can be viewed as a self-map of G.

Definition 3.9. If M is a k[G]-comodule, then the s^{th} Frobenius twist M^{(s)} of M is the k-vector space k ⊗_{p^{s}} M equipped with the comodule structure

F^{s∗} ∘ (k ⊗_{p^{s}} ∇_{M}) : M^{(s)} → M^{(s)} ⊗ k[G^{(s)}] → M^{(s)} ⊗ k[G].

If G is a finite group scheme, then we shall view M^{(s)} as a kG-module via the map F_{∗}^{s} : kG → kG^{(s)} dual to (3.8.1).

To be more explicit, suppose the N-dimensional kG-module M is given by ρ : G → GL_{N} (so that M = ρ^{∗}(V_{N}), where V_{N} is the canonical N-dimensional GL_{N}- module) and assume that G is defined over 𝔽_{p^{s}}. Let μ_{v} : 𝔾_{a(r),K} → G_{K} be a 1-parameter subgroup, corresponding to some v ∈ V (G). Then the identification of M^{(s)} with (ρ ∘ F^{s})^{∗}(V_{N}) implies that

(3.9.1)  JType(M^{(s)}, θ_{v}) = JType(M, θ_{F^{s}(v)})

where θ_{F^{s}(v)} = (F^{s} ∘ μ_{v})_{∗}(u_{r−1}).

Let G = GL_{n(r)}, and let R be a finitely generated commutative k-algebra. The Frobenius self-map is given explicitly on the R-values of GL_{n(r)} by the formula

F : α ↦ ϕ(α),

where ϕ applied to α ∈ M_{n}(R) raises each entry of α to the p-th power. For t in 𝔾_{a(r)}(R), we compute

(F ∘ exp_{(α_{0},...,α_{r−1})})(t) = F (exp(tα_{0}) exp(t^{p}α_{1}) … exp(t^{p^{r−1}}α_{r−1})) =

exp(t^{p}ϕ(α_{0})) exp(t^{p^{2}}ϕ(α_{1})) … exp(t^{p^{r−1}}ϕ(α_{r−2})) = exp_{(0,ϕ(α_{0}),...,ϕ(α_{r−2}))}(t).

Iterating s times, we obtain the following formula for G = GL_{n(r)}:

(3.9.2)  F^{s} ∘ exp_{(α_{0},...,α_{r−1})} = exp_{(0,0,...,0,ϕ^{s}(α_{0}),...,ϕ^{s}(α_{r−1−s}))}

where the first non-zero entry on the right happens at the (s + 1)-st place.

For G = 𝔾_{a(r)}, the Frobenius map F : 𝔾_{a(r)} → 𝔾_{a(r)} is given by raising an element a ∈ 𝔾_{a(r)}(R) to the p-th power. Let \underline{a} = (a_{0}, …, a_{r−1}) be a point in V (𝔾_{a(r)}) ≃ 𝔸^{r}, and let μ\underline{a} : 𝔾_{a(r)} → 𝔾_{a(r)} be the corresponding 1-parameter subgroup. For t ∈ 𝔾_{a(r)}(R), we have μ(t) = a_{0} + a_{1}t + ⋯ + a_{r−1}t^{p^{r−1}} (see [25, §1]). The following formula is now immediate:

(3.9.3)  F^{s} ∘ μ_{(a_{0},...,a_{r−1})} = μ_{(0,...,0,a_{0}^{p^{s}} ,...,a_{r−1−s}^{p^{s}})}.

Combining (3.9.1) and (3.9.2), we derive the following proposition.

Proposition 3.10. Let M be a finite dimensional representation of GL_{n(r)} and let \underline{α} = (α_{0}, …, α_{r−1}) be a point in V (GL_{n(r)}). Then

JType(M^{(s)}, θ\underline{α}) = JType(M, θ_{F^{s}∘\underline{α}}),

where F^{s} ∘ \underline{α} = (0, …, 0, ϕ^{s}(α_{1}), …, ϕ^{s}(α_{r−1−s})).

If M is a finite dimensional k𝔾_{a(r)}-module, and \underline{a} = (a_{0}, …, a_{r−1}) is a point in V (𝔾_{a(r)}) ≃ 𝔸^{r}, then

JType(M^{(s)}, θ\underline{a}) = JType(M, θ_{F^{s}∘\underline{a}}),

where F^{s} ∘ \underline{a} = (0, …, 0, a_{0}^{p^{s}}, …, a_{r−1−s}^{p^{s}}).

Proposition 3.10 has the following immediate corollary (see Remark 3.3).

Corollary 3.11. Let M be a finite dimensional representation of GL_{n(r)}. Then \underline{α} = (α_{0}, …, α_{r−1}) ∈ V (GL_{n(r)})_{M} if and only if F^{s} ∘ α = (0, …, 0, ϕ^{s}(α_{0}), …, ϕ^{s}(α_{r−1−s})) ∈ V (GL_{n(r)})_{M^{(s)}}.

The following definition introduces interesting classes of kG-modules which have special local behavior.

Definition 3.12. Let G be an infinitesimal group scheme and j a positive integer less than p. A finite dimensional kG-module M is said to be of constant j-rank if and only if

rk(M, θ_{v}^{j}) ≡ rk{θ_{v}^{j} : M_{k(v)} → M_{k(v)}}

is independent of v ∈ V (G) − {0}, where θ_{v} is the local p-nilpotent operator at v as introduced in Definition 3.1.

M is said to be of constant Jordan type if and only if it is of constant j-rank for all j, 1 ≤ j < p. M is said to be of constant rank if it is of constant 1-rank.

As we see in the following example, one can have rational GL_{n}-modules of constant Jordan type when restricted to GL_{n(r)} of arbitrarily high degree d. This should be contrasted with Corollary 3.17.

Example 3.13. Consider the rational GL_{n}-module M = det^{⊗d}, the d^{th} power of the determinant representation for some d > 0. This is a polynomial representation of degree n^{d}. The restriction of M to any Frobenius kernel GL_{n(r)} has (trivial) constant Jordan type, for the further restriction of M to any abelian unipotent subgroup of GL_{n} has trivial action.

We shall see below that kG-modules of constant j-rank lead to interesting constructions of vector bundles (see Theorem 5.1). We conclude this section by establishing two constraints, Propositions 3.15 and 3.19, on kG-modules to be modules of constant rank.

We first need the following elementary lemma.

Lemma 3.14. Let M be a 𝔾_{a(r)}-module such that the local Jordan type at every v ∈ V (𝔾_{a(r)}) is trivial. Then M is trivial as a k𝔾_{a(r)}-module.

Proof. The action of 𝔾_{a(r)} on M is given by the action of r commuting p-nilpotent operators ũ_{i}, 0 ≤ i < p on M. Moreover, for \underline{a} = (0, …, 1, 0, …, 0), with 1 at the i^{th} spot,

JType(M, θ\underline{a}) = JType(M, ũ_{i}),

as follows from the explicit description of Θ_{𝔾_{a(r)}} in Example 2.6(2). Thus, if the local Jordan type of M is trivial at each \underline{a} = (0, …, 1, 0, …, 0), then each ũ_{i} must act trivially on M and M is therefore a trivial 𝔾_{a(r)}-module. □

Proposition 3.15. Let 𝔊 be an algebraic group generated by 1-parameter subgroups i : 𝔾_{a} ⊂ 𝔊. Let ρ : 𝔊 → GL_{N} determine a non-trivial rational representation M of 𝔊. Let D_{i} be the minimum of the degrees of (ρ ∘ i)^{∗}(X_{s,t}) ∈ k[𝔾_{a}] = k[T] as X_{s,t} ranges over the matrix coordinate functions of GL_{N}. Let D = max{D_{i}, i : 𝔾_{a} ⊂ 𝔊}.

If r > log_{p} D + 1, then M is not of constant Jordan type as a 𝔊_{(r)}-module.

Proof. Because M is non-trivial and 𝔊 is generated by its 1-parameter subgroups, we conclude that i^{∗}M is a non-trivial rational 𝔾_{a} representation for some 1-parameter subgroup i : 𝔾_{a} ⊂ 𝔊. The condition r > log_{p} D ≥ log_{p} D_{i} implies that i^{∗}M is not r-twisted (i.e., of the form N^{(r)}). Lemma 3.14 implies that the local Jordan type of i^{∗}M at some 1-parameter subgroup μ_{v} : 𝔾_{a(r),k(v)} → G_{k(v)} is non-trivial. On the other hand, Corollary 3.8 implies that the Jordan type of i^{∗}M at the identity 1-parameter subgroup id : 𝔾_{a(r)} → 𝔾_{a(r)} is trivial provided that r−1 > log_{p} D. □

As an immediate corollary, we conclude the following.

Corollary 3.16. Let 𝔊 be an algebraic group generated by 1-parameter subgroups and M a non-trivial rational representation of 𝔊. Then for r >> 0, M is not of constant Jordan type as a k𝔊_{(r)}-module.

As an explicit example of Proposition 3.15, we obtain the following corollary (which should be contrasted with Example 3.13).

Corollary 3.17. Let M be a non-trivial polynomial representation of SL_{n} of degree D. If r > log_{p} D + 1, then M is not a k SL_{n(r)}-module of constant rank.

The following lemma, which is a straightforward application of the Generalized Principal Ideal Theorem (see [12, 10.9]), shows that the dimension of a non-trivial module of constant rank of 𝔾_{a(r)} cannot be “too small” compared to r.

Lemma 3.18. Let M be a finite dimensional 𝔾_{a(r)}-module. If M is a non-trivial 𝔾_{a(r)}-module of constant rank, then the following inequality holds:

(3.18.1)  dim_{k} M ≥ \sqrt{r}

Proof. By extending scalars if necessary we may assume that k is algebraically closed. Let m = dim_{k} M. Let k𝔾_{a(r)} = k[u_{0}, ….u_{r−1}]/(u_{0}^{p}, …, u_{r−1}^{p}), let K = k(s_{0}, …, s_{r−1}) where s_{i} are independent variables, and let α_{K} : K[t]/t^{p} → K𝔾_{a(r)} be a map of K–algebras defined by α_{K}(t) = s_{0}u_{0}+⋯+s_{r−1}u_{r−1}. Choose a k-linear basis of M, and let A(s_{0}, …, s_{r−1}) be a nilpotent matrix in M_{m}(k[s_{0}, …, s_{r−1}]) representing the action of α_{K}(t) on M_{K}. Let I_{n}(A(s_{0}, …, s_{r−1})) denote the ideal generated by all n × n minors of A(s_{0}, …, s_{r−1}). By [12, 10.9], the codimension of any minimal prime over I_{n}(A(s_{0}, …, s_{r−1})) is at most (m − n + 1)^{2}.

Assume that (3.18.1) does not hold, that is, m < \sqrt{r}. Hence, (m − n + 1)^{2} < r for any 1 ≤ n ≤ m. The variety of I_{n}(A(s_{0}, …, s_{r−1})) is a subvariety inside Spec k[s_{0}, …, s_{r−1}] ≃ 𝔸^{r} which has dimension r. Since the codimension of the variety of I_{n}(A(s_{0}, …, s_{r−1})) is at most (m−n+1)^{2}, we conlude that the dimension is at least r−(m−n+1)^{2} ≥ 1. Hence, the minors of dimension n×n have a common non-trivial zero. Taking n = 1, we conclude that A(b_{0}, …, b_{r−1}) is a zero matrix for some non-zero specialization b_{0}, …, b_{r−1} of s_{0}, …, s_{r−1}. Consequently, M is trivial at the π-point of 𝔾_{a(r)} corresponding to b_{0}, …, b_{r−1}. Since M is non-trivial, Lemma 3.14 implies that M is not a module of constant rank. □

As an immediate corollary, we provide an additional necessary condition for a k𝔊_{(r)}- module to have constant rank.

Proposition 3.19. Let 𝔊 be a (reduced) affine algebraic group and M be a rational representation of 𝔊. Assume that 𝔊 admits a 1-parameter subgroup μ : 𝔾_{a(r)} → 𝔊 such that μ^{∗}(M) is a non-trivial k𝔾_{a(r)}-module. If r ≥ (dim M)^{2} + 1, then M is not a k𝔊_{(r)}-module of constant rank.

4. π-points and ℙ(G)

In a series of earlier papers, we have considered π-points for a finite group scheme G (as recalled in Definition 4.1) and investigated finite dimensional kG-modules M using the “Jordan type of M” at various π-points. In particular, in [18], we verified that this Jordan type is independent of the equivalence class of the π-point provided that either the π-point is generic or the Jordan type of M at some representative of the equivalence class is maximal.

As we recall below, whenever G is an infinitesimal group scheme, then the πpoint space Π(G) of equivalence classes of π-points is essentially the projectivization of V (G). The purpose of the first half of this section is to relate the discussion of the previous section concerning the local Jordan type of a finite kG-module to our earlier work formulated in terms of π-points for general finite group schemes.

One special aspect of an infinitesimal group scheme G is that equivalence classes of π-points of G have canonical (up to scalar multiple) representatives.

Unless otherwise specified (as in Definition 4.1 immediately below), G will denote an infinitesimal group scheme over k, and V (G) will denote V_{r}(G) for some r ≥ ht(G). Throughout this section we assume that dim V (G) ≥ 1, and work with ℙ(G) = Proj k[V (G)]. We note that if dim V (G) = 0, then Theorem 1.16 implies that the projective resolution of the trivial module k is finite. Since kG is selfinjective, this further implies that k is projective. Hence, kG is semi-simple, and does not have any π-points (see, for example, [15, §2]).

Definition 4.1. (see [17]) Let G be a finite group scheme.

(1) A π-point of G is a (left) flat map of K-algebras α_{K} : K[t]/t^{p} → KG for some field extension K/k with the property that there exists a unipotent abelian closed subgroup scheme i : C_{K} ⊂ G_{K} defined over K such that α_{K} factors through i_{∗} : KC_{K} → KG_{K} = KG.

(2) If β_{L} : L[t]/t^{p} → LG is another π-point of G, then α_{K} is said to be a specialization of β_{L} , written β_{L} ↓ α_{K}, provided that for any finite dimensional kG-module M, α_{K}^{∗}(M_{K}) being free as K[t]/t^{p}-module implies that β_{L}^{∗}(M_{L}) is free as L[t]/t^{p}-module.

(3) Two π-points α_{K} : K[t]/t^{p} → KG, β_{L} : L[t]/t^{p} → LG are said to be equivalent, written α_{K} ∼ β_{L}, if α_{K} ↓ β_{L} and β_{L} ↓ α_{K}.

(4) A π-point of G, α_{K} : K[t]/t^{p} → KG, is said to be generic if there does not exist another π-point β_{L} : L[t]/t^{p} → LG which specializes to α_{K} but is not equivalent to α_{K}.

(5) If M is a finite dimensional kG-module and α_{K} : K[t]/t^{p} → KG a π-point of G, then the Jordan type of M at α_{K} is by definition the Jordan type of α_{K}^{∗}(M_{K}) as K[t]/t^{p}-module.

Because the group algebra of a finite group scheme is always faithfully flat over the group algebra of a subgroup scheme (see [27, 14.1]), the condition on a flat map α_{K} : K[t]/t^{p} → KG is equivalent to the existence of a factorization i_{∗} ∘ α′_{K} with α′_{K} : K[t]/t^{p} → KC_{K} flat.

Definition 4.2. Let G be an infinitesimal group scheme, and let v ∈ V (G) be the point associated to the 1-parameter subgroup μ_{v} : 𝔾_{a(r),k(v)} → G_{k(v)}. Then the π-point of G associated to v is

μ_{v,∗} ∘ ϵ : k(v)[u]/u^{p} → k(v)G,

where ϵ : k(v)[u]/u^{p} → k(v)𝔾_{a(r),k(v)} is as defined in (1.14.1).

The following theorem is a complement to Theorem 1.16, revealing that spaces of (equivalence) classes of π-points are very closely related to (cohomological) support varieties.

Theorem 4.3. ([17, 7.5]) Let G be an finite group scheme. Then the set of equivalence classes of π-points, denoted Π(G), can be given a scheme structure, which is defined in terms of the representation theory of G. Moreover, there is an isomorphism of schemes

Proj H^{•}(G, k) ≃ Π(G).

If G is an infinitesimal group scheme so that H^{•}(G, k) is related to k[V (G)] as in Theorem 1.16, then the resulting homeomorphism

(4.3.1)  \xymatrix{ℙ(G) ≡ Proj k[V (G)] \ar[r] & Proj H^{•}(G, k) ≃ Π(G)}

is given on points by sending x ∈ ℙ(G) to the equivalence class of the π-point μ_{v,∗} ∘ ϵ for any v ∈ V (G)\{0} projecting to x. In particular, equivalence classes of generic π-points of G are represented by (μ_{v,∗} ∘ ϵ) as v ∈ V (G) runs through the (scheme-theoretic) generic points of V (G).

Furthermore, for any finite dimensional kG-module M, (4.3.1) restricts to a homeomorphism of subvarieties

ℙ(G)_{M} ≃ Π(G)_{M},

where ℙ(G)_{M} = Proj(k[V (G)_{M}]), and Π(G)_{M} consists of those equivalence classes of π-points α_{K} of G such that α_{K}^{∗}(M_{K}) is not free (as a K[u]/u^{p}-module).

Generic π-points are particularly important when developing invariants of representations. The following corollary of Theorem 4.3 gives an explicit set of representatives of equivalence classes of generic π-points of G.

Proposition 4.4. Let G be an infinitesimal group scheme with universal 1-parameter subgroup \xymatrix{U_{G} : 𝔾_{a(r),k[V (G)]} \ar[r] & G_{k[V (G)]}} . For each minimal prime ideal P_{i} of k[V (G)], let K_{i} denote the field of fractions of k[V (G)]/P_{i}. Then the compositions

(U_{G,∗} ⊗_{k[V (G)]} K_{i}) ∘ ϵ : K_{i}[u]/u^{p} → K_{i}G

(sending u to θ_{K_{i}}) are non-equivalent representatives of the equivalence classes of generic π-points of G.

To obtain vector bundles, we require the following wll known, elementary observation about commutative graded algebras.

Lemma 4.5. Let A be a finitely generated commutative, graded k-algebra with homogeneous generators whose degrees divide d and let X = Proj A. Then O_{X}(d) is a locally free sheaf of rank 1 on X.

Proof. Let {f_{i}} be a finite set of homogeneous generators of A, set d_{i} equal to the degree of f_{i}, and choose d such that each d_{i} divides d. Set U_{i} = X −Z(f_{i}), an affine open subset of X with coordinate algebra (A_{f_{i}})_{0}, the degree zero subalgebra of the localization of A obtained by inverting f_{i}. Then multiplication by f_{i}^{d/d_{i}} induces an isomorphism (O_{X})_{|U_{i}} →^{→} (O_{X}(d))_{|U_{i}}. Thus the restriction of O_{X}(d) to each U_{i} of the open covering {U_{i}} is free of rank 1. □

Let G be an infinitesimal group scheme of height ≤ r and recall from Proposition 2.11 that Θ_{G,r} ∈ kG ⊗ k[V_{r}(G)] is homogeneous of degree p^{r−1}.

Definition 4.6. Let G be an infinitesimal group scheme of height ≤ r and let M be a finite dimensional kG-module. Then we denote by

(4.6.1)  \xymatrix{Θ̃_{G} : M ≡ M ⊗ O_{ℙ(G)} \ar[r] & M(p^{r−1}) ≡ M ⊗ O_{ℙ(G)}(p^{r−1})}

the associated homomorphism of (locally free) coherent O_{ℙ(G)}-modules determined by the action of Θ_{G,r} ∈ kG ⊗ k[V_{r}(G)].

We denote by

(4.6.2)  \xymatrix{Θ̃_{G}(n) : M(n) \ar[r] & M(p^{r−1} + n)}

the map obtained by tensoring (4.6.1) with O_{ℙ(G)}(n).

For any point x ∈ ℙ(G), we use the notation

M_{k(x)} = M ⊗_{O_{ℙ(G)}} k(x)

for the fiber of the coherent sheaf M at x. Here, we have identified k(x) with the residue field of the stalk O_{ℙ(G),x}.

Proposition 4.7. Let G be an infinitesimal group scheme of height ≤ r, and let M be a finite dimensional kG-module. For any v, v′ ∈ V (G) projecting to the same x ∈ ℙ(G), we have

Im{θ_{v} : M_{k(v)} → M_{k(v)}} ≃ Im{θ_{v′} : M_{k(v′)} → M_{k(v′)}}

and similarly for kernels.

Proof. This is essentially proved in [26, 6.1]. □

In the next section, we shall be particularly interested in kernels and images of Θ̃_{G}. The following proposition relates the fibers of the kernel and image of the global p-nilpotent operator Θ̃_{G} at a point x ∈ ℙ(G) with the kernel and image of the local p-nilpotent operator θ_{v} on M ⊗ k(v) for v representing x.

Proposition 4.8. Let G be an infinitesimal group scheme of height ≤ r, let M be a finite dimensional kG-module, let M = M⊗O_{ℙ(G)}, and let s ∈ Γ(ℙ(G), O_{ℙ(G)}(p^{r−1})) be a non-zero global section with zero locus Z(s) ⊂ ℙ(G). Set U = ℙ(G) \Z(s). Then there is a well defined endomorphism (depending upon s)

Θ̃_{G}/s : M_{|U} → M_{|U}.

Moreover, the image and kernel of the induced map θ_{x}/s : M_{k(x)} → M_{k(x)} on fibers at x ∈ U ⊂ ℙ(G) is independent of s and satisfies

(4.8.1)  Im{θ_{x}/s : M_{k(x)} → M_{k(x)}} ≃ Im{θ_{v} : M_{k(v)} → M_{k(v)}}

and

(4.8.2)  Ker{θ_{x}/s : M_{k(x)} → M_{k(x)}} ≃ Ker{θ_{v} : M_{k(v)} → M_{k(v)}}

for any v ∈ V (G)\{0} that projects onto x.

Proof. Let X = ℙ(G). Let \frac{1}{s} ∈ O_{X}(−p^{r−1})(U) satisfy

s ⊗ \frac{1}{s} = 1 ∈ O_{X}(p^{r−1})(U) ⊗_{O_{X}(U)} O_{X}(−p^{r−1})(U) ≃ O_{X}(U).

Then we define

Θ̃_{G}/s ≡ (Θ̃_{G})_{|U} ⊗ \frac{1}{s} : M_{|U} → M(p^{r−1})_{|U} ⊗ O_{X}(−p^{r−1}))_{|U} ≃ M_{|U}.

Let A = k[V (G)]. Then U is an affine subscheme of ℙ(G) determined by the 0-degree elements of A [\frac{1}{s}], U ≃ Spec A [\frac{1}{s}]_{0}. Via this identification, M_{|U} corresponds to the free A [\frac{1}{s}]_{0}-module M ⊗ A [\frac{1}{s}]_{0}. The fiber M_{k(x)} of M_{|U} at the point x is naturally identified with the fiber of M ⊗ A [\frac{1}{s}]_{0} at x. Since Θ_{G} : M ⊗k[V (G)] → M ⊗k[V (G)] is homogeneous of degree p^{r−1} and s ∈ k[V (G)] is homogeneous of the same degree, the operator Θ_{G} ⊗ \frac{1}{s} is well-defined on M ⊗A [\frac{1}{s}]_{0} and corresponds to the operator Θ̃_{G}/s on M_{|U}. Hence, θ_{x}/s : M_{k(x)} → M_{k(x)} is identified with

(Θ̃_{G} ⊗ \frac{1}{s}) ⊗ k(x) = \frac{Θ_{G}}{s} (x) : M_{k(x)} → M_{k(x)}.

Let v ∈ V (G) be any point projecting onto x. We have k(x) = k(v). By Def. 3.1, the map θ_{v} : M_{k(v)} → M_{k(v)} is given by

Θ_{G}(v) : M_{k(v)} → M_{k(v)}.

We observe that \frac{Θ_{G}}{s} (x) = \frac{Θ_{G}(v)}{s(v)} for any v projecting onto x (and, in particular, is independent of the choice of v). Therefore,

(4.8.3)  θ_{x}/s = \frac{θ_{v}}{s(v)}.

The equalities(4.8.2) and (4.8.1) now follow. □

Remark 4.9. For a finite group G, there is no natural choice of π-point representing a typical equivalence class x ∈ Π(G) ≃ Proj H^{•}(G, k) of π-points. As seen in elementary examples [18, 2.3], the Jordan type of a kG-module M typically can be different for two equivalent π-points representing the same point x ∈ Π(G).

Remark 4.10. Proposition 4.8 immediately generalizes to Θ̃_{G}^{j} for any 1 ≤ j ≤ p−1. Thus, we have the following isomorphisms for any x ∈ X = ℙ(G), v ∈ V (G) projecting onto x, and a global section s of O_{X}(jp^{r−1}) such that s(x) ≠ 0:

Im{(θ_{x}/s)^{j} : M_{k(x)} → M_{k(x)}} ≃ Im{θ_{v}^{j} : M_{k(v)} → M_{k(v)}} ≃
Im{Θ̃_{G}^{j} ⊗_{O_{X}} k(x) : M ⊗_{O_{X}} k(x) → M(jp^{r−1}) ⊗_{O_{X}} k(x)},

and similarly for kernels.

In what follows, we shall use the following abbreviations:

Im{Θ̃_{G}^{j}, M} ≡ Im{Θ̃_{G}^{j}(−jp^{r−1}) : M(−jp^{r−1}) → M},
(4.10.1)  Im{θ_{x}^{j}, M_{k(x)}} ≡ Im{(θ_{x}/s)^{j} : M_{k(x)} → M_{k(x)}},
Ker{Θ̃_{G}^{j}, M} ≡ Ker{Θ̃_{G}^{j} : M → M(jp^{r−1})},
Ker{θ_{x}^{j}, M_{k(x)}} ≡ Ker{(θ_{x}/s)^{j} : M_{k(x)} → M_{k(x)}}.
Coker{Θ̃_{G}^{j}, M} ≡ M/ Im{Θ̃_{G}^{j}, M},
Coker{θ_{x}^{j}, M_{k(x)}} ≡ Coker{(θ_{x}/s)^{j} : M_{k(x)} → M_{k(x)}}.

Note that both Ker and Im are subsheaves of the free sheaf M, and Coker is a quotient sheaf of M.

We shall verify in Theorem 4.13 that a necessary and sufficient condition on a finite dimensional kG-module M for Im{Θ̃_{G}^{j}, M} (and thus Ker{Θ̃_{G}^{j}, M}) to be an algebraic vector bundle on X is that M be a module of constant j-type.

The following proposition is given in [20, 5 ex.5.8] without proof.

Proposition 4.11. Let X be a reduced scheme and M̃ a coherent O_{X}-module. Then M̃ is locally free if and only if dim_{k(x)}(M̃ ⊗_{O_{X}} k(x)) depends only upon the connected component of x in π_{0}(X).

Proof. Assume that the function x ↦ dim_{k(x)}(M̃ ⊗_{O_{X}} k(x)) is constant on each connected component of X. To prove that M̃ is locally free it suffices to assume that X is local so that X = Spec R for some reduced local commutative ring, and that M is a finite R-module (corresponding to the coherent sheaf M̃) with the property that dim_{k(p)}(M ⊗_{R} k(p)) is independent of the prime p ⊂ R. To prove that M is free, we choose some surjective R-module homomorphism g : Q → M from a free R-module Q ≃ R^{n} with the property that \overline{g} : Q ⊗_{R} R/𝔪 → M ⊗_{R} R/𝔪 is an isomorphism where 𝔪 ⊂ R is the maximal ideal. Then g is surjective by Nakayama’s lemma. By assumption, g induces an isomorphism after specialization to any prime 𝔭 ⊂ R: Q ⊗_{R_{𝔭}} k(𝔭) ≃ M ⊗_{R_{𝔭}} k(𝔭). Hence, Q_{𝔭}/𝔭Q_{𝔭} ≃ M_{𝔭}/𝔭M_{𝔭}. We conclude that if a ∈ ker g, then a ∈ 𝔭Q_{𝔭} ∩ Q. Since this happens for any prime ideal, we further conclude that ker g ⊂ ( ⋂_{𝔭∈Spec R} 𝔭Q_{𝔭}) ∩ Q. Recall that Q is a free module so that

Q ≃ R^{n}. We get (⋂ 𝔭Q_{𝔭}) ∩ Q = (⋂ 𝔭R_{𝔭}^{n}) ∩ R^{n} = ((⋂ 𝔭R_{𝔭}) ∩ R)^{n} = (⋂ 𝔭R)^{n} = 0 since R is reduced. □

We shall find it convenient to “localize” the notion of a kG-module of constant j-rank given in Definition 3.12 as follows.

Definition 4.12. Let G be an infinitesimal group scheme, and let M be a finite dimensional kG-module. For any open subset U ⊂ ℙ(G), M is said to be of constant j-rank when restricted to U if rk_{k(x)}((θ_{x}/s)^{j} : M_{k(x)} → M_{k(x)}) is independent of x ∈ U.

Our next theorem emphasizes the local nature of the concept of constant j-rank.

Theorem 4.13. Let G be an infinitesimal group scheme, let M be a finite dimensional kG-module, and let X = ℙ(G). Let U ⊂ X be a connected open subset, and Θ̃_{U}^{j} : M_{|U} → M(jp^{r−1})_{|U} be the restriction to U of the j^{th} iterate of Θ̃_{G} on M = M ⊗ O_{X} as given in (4.6.1). Then the following are equivalent for some fixed j, 1 ≤ j < p:

(1) Im{Θ̃_{U}^{j}, M_{|U}} is a locally free, coherent O_{U}-module.

(2) Im{Θ̃_{G}^{j}, M} ⊗_{O_{X}} k(x) has dimension independent of x ∈ U.

(3) Im{θ_{x}^{j}, M_{k(x)}} ≃ Im{Θ̃_{G}^{j}, M} ⊗_{O_{X}} k(x), ∀ x ∈ U

(4) M has constant j-rank when restricted to U. Moreover, each of these conditions implies that

(5) Coker{Θ̃_{U}^{j}, M_{U}} is a locally free, coherent O_{U}-module.
(6) Coker{θ_{x}^{j}, M_{k(x)}} ≃ Coker{Θ̃_{G}^{j}, M} ⊗_{O_{X}} k(x), ∀ x ∈ U.
(7) Ker{Θ̃_{U}^{j}, M_{U}} is a locally free, coherent O_{U}-module.
(8) Ker{θ_{x}^{j}, M_{k(x)}} ≃ Ker{Θ̃_{G}^{j}, M} ⊗_{O_{X}} k(x), ∀ x ∈ U.

Proof. Clearly, (1) implies (2), whereas Proposition 4.11 implies that (2) implies (1).

If we assume (1), we obtain a locally split short exact sequence of coherent O_{U}-modules

(4.13.1)  0 → Ker{Θ̃_{U}^{j}, M_{|U}} → M_{|U} → Im{Θ̃_{U}^{j}, M_{|U}} → 0.

In particular, Ker{Θ̃_{U}^{j}, M_{|U}} is a locally free, coherent O_{U}-module. Locally on U, Θ̃_{U}^{j} on M_{|U} is isomorphic to the projection

pr_{2} : Ker{Θ̃_{U}^{j}, M_{|U}} ⊕ Im{Θ̃_{U}^{j}, , M_{|U}} → Im{Θ̃_{U}^{j}, , M_{|U}}.

Since θ_{x}^{j} is the base change via O_{U} → k(x) of Θ̃_{U}^{j}, θ_{x}^{j} can be identified with the base change of this projection and thus we may conclude (3).

Let us now assume (3). A simple argument using Nakayama’s Lemma as in the proof of Proposition 4.11 implies that the function x ↦ Im{θ_{x}^{j}, M_{k(x)}} is lower semi-continuous on U whereas the function x ↦ Im{Θ̃_{G}^{j}, M} ⊗_{O_{X}} k(x) is upper semi-continuous on U. Thus, we conclude that each of these functions is constant (since U is connected), thereby concluding (2).

Since rk{(θ_{x}/s)^{j}} = dim_{k(x)}(Im{θ_{x}^{j}, M_{k(x)}}), (2) and (3) imply (4).

Observe that if f : V → V is an endomorphism of a finite dimensional vector space then dim{Coker f} = dim{Ker f}. The assumption that the kG-module M has constant rank (i.e., (4)) implies that

dim_{k(x)}(Coker{θ_{x}^{j}, M_{k(x)}}) = dim_{k(x)}(Ker{θ_{x}^{j}, M_{k(x)}})

is independent of x ∈ U. Hence, Proposition 4.11 implies (5). The right exactness of (−) ⊗_{O_{X}} k(x) applied to

\xymatrix{M(−jp^{r−1}) \ar[r]^{Θ̃_{G}^{j}(−jp^{r−1})} & M \ar[r] & Coker\{Θ̃_{G}^{j}, M\} \ar[r] & 0}

implies (6).

Under the assumption of (4), we obtain a locally split short exact sequence of coherent O_{U}-modules

0 → Im{Θ̃_{U}^{j}, M_{|U}} → M_{|U} → Coker{Θ̃_{U}^{j}, M_{|U}} → 0,

so that Im{Θ̃_{U}^{j}, M_{|U}} is a locally free, coherent O_{U}-module. Now, using the short exact sequence of coherent O_{U}-modules

0 → Ker{Θ̃_{U}^{j}, M_{|U}} → M_{|U} → Im{Θ̃_{U}^{j}, M_{|U}} → 0,

we conclude that (4) implies (7) (i.e., that Ker{Θ̃_{U}^{j}, M_{|U}} is locally free). Since the short exact sequence (4.13.1) is locally split, applying (−) ⊗_{O_{X}} k(x) to (4.13.1) for any x ∈ U yields a short exact sequence, thereby implying (8). □

5. Vector bundles for modules of constant j-rank

In this section, we initiate the study of algebraic vector bundles associated to kG-modules of constant j-rank as defined in 3.12. Our constructions have two immediate consequences. The first is that certain kG-modules with the same “local Jordan type” have non-isomorphic associated vector bundles, so that the isomorphism classes of these vector bundles serve as a new invariant. The second is that our construction yields vector bundles on the highly non-trivial projective schemes ℙ(G).

The reader will find formulas for the ranks of bundles considered, criteria for non-triviality of bundles, a criterion for producing line bundles, a relationship to duality, and another test for the projectivity of kG-modules. We also investigate the dimension of global sections of various bundles.

As in §4, we assume that dim V (G) ≥ 1 throughout this section.

The special case in which U = ℙ(G) of Theorem 4.13 is the following assertion that kG-modules of constant j-rank determine algebraic vector bundles over ℙ(G).

Theorem 5.1. Let G be an infinitesimal group scheme, let M be a finite dimensional kG-module, and let M = M ⊗ O_{ℙ(G)} be a free coherent sheaf on ℙ(G). Then M has constant j-rank if and only if Im{Θ̃_{G}^{j}, M} is an algebraic vector bundle on ℙ(G).

Consequently, if M has constant j-rank, then Ker{Θ̃_{G}^{j}, M} also is an algebraic vector bundle on ℙ(G).

Remark 5.2. Unless M is trivial as a kG-module, Ker{Θ_{G} : M ⊗ k[V (G)] → M ⊗ k[V (G)]} is not projective as a k[V (G)]-module, since the local p-nilpotent operator θ_{0} at 0 ∈ V (G) is the 0-map.

We observe the following elementary functoriality of this construction.

Proposition 5.3. Let i : H → G be an embedding of infinitesimal group schemes, let M be a finite dimensional kG-module, and let N be the restriction of M to kH.

Let M = M ⊗ O_{ℙ(G)}, and N = N ⊗ O_{ℙ(H)}. Then for any j, 1 ≤ j < p, there are natural isomorphisms of coherent sheaves on ℙ(H), where f : ℙ(H) → ℙ(G) is induced by i:

f^{∗} Im{Θ̃_{G}^{j}, M} ≃ Im{Θ̃_{H}^{j}, N }
f^{∗} Ker{Θ̃_{G}^{j}, M} ≃ Ker{Θ̃_{H}^{j}, N }.

Proof. The statement follows immediately from the commutativity of the diagram

(5.3.1)  \xymatrix{M ⊗ O_{ℙ(G)} \ar[r]^{Θ̃_{G}^{j}} \ar[d]_{f^{∗}} & M ⊗ O_{ℙ(G)}(jp^{r−1}) \ar[d]_{f^{∗}} \\ N ⊗ O_{ℙ(H)} \ar[r]^{Θ̃_{H}^{j}} & N ⊗ O_{ℙ(H)}(jp^{r−1}).}

The diagram is commutative by Proposition 2.9.3.

□

The following Corollary will be used later in §6.

Corollary 5.4. Let G_{1}, G_{2} be infinitesimal group schemes, let G = G_{1} × G_{2}, and let f : ℙ(G_{1}) → ℙ(G) be the natural embedding of varieties induced by the embedding of group schemes i : G_{1} ↪ G. Let M_{1}, M_{2} be kG_{1}, kG_{2} modules of dimensions m_{1}, m_{2} respectively. Then for any j, 1 ≤ j ≤ p,

(5.4.1)  f^{∗}(Ker{Θ̃_{G}^{j}, M_{1} ⊠ M_{2}}) ≃ Ker{Θ̃_{G_{1}}^{j}, M_{1}}^{⊕m_{2}}

Here, M_{1} = M_{1}⊗O_{ℙ(G_{1})}, M_{2} = M_{2}⊗O_{ℙ(G_{2})}, and M_{1}⊠M_{2} ≃ (M_{1}⊗M_{2})⊗O_{ℙ(G)}.

Proof. By Proposition 5.3, it suffices to observe that (M_{1} ⊗ M_{2}) ↓_{G_{1}}≃ M_{1}^{⊕m_{2}}, and that f^{∗} and Θ̃^{j} commute with direct sums. □

We have a duality for kernel and cokernel bundles. For a vector bundle E on a projective variety X, we denote by E^{∨} = Hom_{O_{X}} (E, O_{X}) the dual bundle.

Proposition 5.5. Let M be a finite dimensional kG-module of constant j-rank. Let N = M^{#} ⊗ O_{ℙ(G)}, and M = M ⊗ O_{ℙ(G)}. Then

Ker{Θ̃_{G}^{j}, M}^{∨} ≃ Coker{Θ̃_{G}^{j}, N }

as vector bundles on O_{ℙ(G)}.

Proof. Choosing dual bases for M and M^{#}, we get an isomorphism of trivial bundles M and N. Hence, we may identify the dual bundle M(jp^{r−1})^{∨} with N (−jp^{r−1}). Under this identification, the O_{ℙ(G)}-dual of the map

Θ̃_{G}^{j} : M → M(jp^{r−1})

is identified with

Θ̃_{G}^{j}(−jp^{r−1}) : N (−jp^{r−1}) → N .

Since Ext_{O_{ℙ(G)}}^{1}(−, O_{ℙ(G)}) vanishes on locally free sheaves, taking the O_{ℙ(G)}-dual of a short exact sequence of vector bundles

\xymatrix{0 \ar[r] & Ker\{Θ̃_{G}^{j}, M\} \ar[r] & M \ar[r]^{Θ̃_{G}^{j}} & M(jp^{r−1}) \ar[r] & Coker\{Θ̃^{j}, M\}(jp^{r−1}) \ar[r] & 0,}

we get an exact sequence

[figure: 0; Ker{Θ̃_{G}^{j}, M}^{∨}; N; N (−jp^{r−1}); Θ_{G}^{j}(−jp^{r−1})]
\xymatrix{Coker\{Θ̃^{j}, M\}^{∨}(−jp^{r−1}) & 0 \ar[l]}

Therefore, Ker{Θ̃_{G}^{j}, M}^{∨} ≃ Coker{Θ̃_{G}^{j}, N }.

□

Example 5.6. For each of our four examples of infinitesimal group schemes (initially investigated in Example 1.5), we give examples of kG-modules of constant Jordan type taken from [8].

(1) Let 𝔤 be a finite dimensional p-restricted Lie algebra of dimension at least 2. For any Tate cohomology class of negative dimension, ζ ∈ Ĥ^{n}(𝔲(𝔤), k) ≃ Ext_{𝔲(𝔤)}^{1}(Ω^{n−1}(k), k), we consider the extension of 𝔲(𝔤)-modules

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

determined by ζ. By [8, 6.3], M is a 𝔲(𝔤)-module of constant Jordan type. We verify by inspection that the Jordan type of M is (a, 0, …, 0, 2) for some a > 0 if n is odd, and (b, 1, 0, …, 0, 1) for some b > 0 if n is even (see (3.1.2) for notation).

(2) Let G = 𝔾_{a(r)}, and set I equal to the augmentation ideal of kG ≃ k[u_{0}, …, u_{p−1}]/(u_{0}^{p}, …, u_{p−1}^{p}). As observed in [8], I^{i}/I^{t} is a module of constant Jordan type for any t > i. As proven in [9], the only ideals of k𝔾_{a(2)} which are of constant Jordan type are of the form I^{i}.

(3) As observed in [8], the n^{th} syzygy module Ω^{n}(k), n ∈ ℤ, is a module of constant Jordan type for any infinitesimal group scheme G. For n even, Ω^{n}(k) has constant Jordan type (a, 0, …, 0, 1) for some a > 0; whereas for n odd, Ω^{n}(k) has constant Jordan type (b, p − 1, 0, …, 0) for some b > 0.

(4) For G = SL_{2(2)}, we recall that the cohomology algebra H^{•}(G, k) is generated modulo nilpotents by classes ζ_{1}, ζ_{2}, ζ_{3} ∈ H^{2}(G, k) and classes ξ_{1}, ξ_{2}, ξ_{3} ∈ H^{2p}(G, k) ([19]). As in [8, 6.8], the kG-module

M ≡ Ker{∑ ζ_{i} + ∑ ξ_{j} : (Ω^{2}(k))^{⊕3} ⊕ (Ω^{2p}(k))^{⊕3} → k}

is a kG-module of constant Jordan type (a, 0, …, 0, 1) for some a > 0.

We elaborate on the Example 5.6(2), constructing 𝔾_{a(r)}-modules of constant j-rank for but not of constant Jordan type.

Example 5.7. We start with the following simple observation. Let M_{1} ⊂ M_{2} ⊂ M be a chain of k-vector spaces, and let ϕ be an endomorphism of M such that ϕ(M_{1}) ⊂ M_{1} and ϕ(M_{2}) ⊂ M_{2}. If dim(Ker ϕ_{|_{M_{1}}} ) = dim(Ker ϕ), then dim(Ker ϕ_{|_{M_{1}}} ) = dim(Ker ϕ_{|_{M_{2}}} ) = dim(Ker ϕ).

Let G = 𝔾_{a(r)}, and set I equal to the augmentation ideal of kG ≃ k[u_{0}, …, u_{p−1}]/(u_{0}^{p}, …, u_{p−1}^{p}). Consider any ideal J of kG with the property that I^{i} ⊂ J for some i, i ≤ p − 1. Note that for any \underline{a} ∈ 𝔸^{r}, and any j ≤ p − i,

(5.7.1)  dim(Ker\{θj ; \underline{a} : I^{i} → I^{i}\}) = pj = dim(Ker\{θj ; \underline{a} : kG → kG\}).

Indeed, since I^{i} is a module of constant Jordan type, it suffices to check the statement for θ\underline{a} = u_{0} for which it is straightforward. The observation in the previous paragraph together with (5.7.1) and the inclusions I^{i} ⊂ J ⊂ kG imply

dim(Ker\{θj ; \underline{a} : J → J\}) = pj

for any j ≤ p − i and any \underline{a} ∈ 𝔸^{r}. Hence, J has constant j-rank for 1 ≤ j ≤ p − i.

In the following example, we offer a method applicable to almost all infinitesimal group schemes G of constructing kG-modules which are of constant rank but not constant Jordan type.

Example 5.8. Let G be an infinitesimal group scheme with the property that V (G) has dimension at least 2. Assume that p is odd, and let n > 0 be an odd positive integer. Let ζ ∈ H^{n}(G, k) be a non-zero cohomology class and let M denote the kernel of ζ : Ω^{n}(k) → k. Then M has constant rank but not constant Jordan type. Namely, the local Jordan type of M at 0 ≠ v ∈ V (G) is (a, 0, 1, 0, …, 0) if ζ(v) ≠ 0, and is (a − 1, 2, 0, …, 0) if ζ(v) = 0. These Jordan types have the same rank.

For G = SL_{2(1)}, the restriction of any rational SL_{2}–module is a module of constant Jordan type (see [8]). Irreducible SL_{2}-modules S_{λ} are parameterized by their highest weight, a non-negative integer λ. Irreducible SL_{2(1)} modules are the restrictions of S_{λ} to SL_{2(1)} for 0 ≤ λ ≤ p − 1.

Another important family of SL_{2}-modules are the V_{λ} (also denoted H^{0}(λ)) defined as the subspace of k[s, t] (i.e., the symmetric algebra on the natural 2dimensional representation for SL_{2}) consisting of homogeneous vectors of degree λ. For 0 ≤ λ ≤ p − 1, we have an isomorphism of SL_{2(1)}–modules: S_{λ} ≃ V_{λ}.

Recall that V (G) is the nullcone in sl_{2}, and, hence, A = k[V (G)] ≃ k[x, y, z]/(xy+ z^{2}). Let

(5.8.1)  i : ℙ^{1} → ℙ(G)

be the isomorphism given on homogeneous coordinates by

\frac{k[x, y, z]}{(xy + z^{2})} → k[s, t] (x, y, z) ↦ (s^{2}, −t^{2}, st).

In the next proposition, we compute explicitly the kernel bundles associated to the irreducible SL_{2(1)}-modules, and for the induced modules V_{λ} for p ≤ λ ≤ 2p−2. For convenience, we give the answer in terms of pull-backs to ℙ^{1} via the isomorphism i.

Proposition 5.9. Let G = SL_{2(1)}, and let i : ℙ^{1} →^{∼} ℙ(G) be the isomorphism defined in (5.8.1).

(1) For 0 ≤ λ ≤ p − 1,

i^{∗}(Ker{Θ̃_{G}, S_{λ} ⊗ O_{ℙ(G)}}) ≃ O_{ℙ^{1}}(−λ).
(2) For p ≤ λ ≤ 2p − 2,
i^{∗}(Ker{Θ̃_{G}, V_{λ} ⊗ O_{ℙ(G)}}) ≃ O_{ℙ^{1}}(−λ) ⊕ O_{ℙ^{1}}(λ − 2(p − 1)).

Proof. We adopt the conventions of [2, §1]; in particular, we replace λ by m. Let v_{0}, v_{1}, …, v_{m} be a basis for V_{m} such that the generators e, f and h of sl_{2} act as follows: hv_{i} = (2i − m)v_{i}, fv_{i} = (m − i + 1)v_{i−1} for i > 0, fv_{0} = 0, ev_{i} = (i + 1)v_{i+1} for i < m, ev_{m} = 0 (see [21, 7.2] or [2, §1]). Recall that

Θ_{G} = xe + yf + zh

(see Example 2.8(1)). Hence, the operator

\xymatrix{Θ_{G} : V_{m} ⊗ A ≃ A^{m+1} \ar[r] & V_{m} ⊗ A ≃ A^{m+1}}

is represented by the matrix

(5.9.1)  (\begin{matrix}−mz & my & 0 & … & … \\ x & −(m − 2)z & (m − 1)y & 0 & … \\ 0 & 2x & −(m − 4)z & (m − 2)y & 0 & … \\ 0 & 0 & 3x & −(m − 6)z & (m − 3)y & … \\ … & … & … & … & … \\ … & … & … & 0 & mx & mz \end{matrix})_{.}

Substituting (s^{2}, −t^{2}, st) for (x, y, z), we get a degree two operator on k[s, t]^{m+1} given by (5.9.2)

B_{m}(s, t) = (\begin{matrix}−mst & −mt^{2} & 0 & … & … \\ s^{2} & (2 − m)st & (1 − m)t^{2} & 0 & … \\ 0 & 2s^{2} & (4 − m)st & (2 − m)t^{2} & 0 & … \\ 0 & 0 & 3s^{2} & (6 − m)st & (3 − m)t^{2} & … \\ … & … & … & … & … \\ … & … & … & 0 & ms^{2} & mst \end{matrix})_{.}

One verifies easily that the vector

w_{m} = [t^{m}, −st^{m−1}, s^{2}t^{m−2}, …, ±s^{m}]

is annihilated by B_{m}(s, t). For 0 ≤ m ≤ p − 1, the module V_{m} is irreducible and the kernel bundle has rank 1 because the (constant) Jordan type of V_{m} has a single block. The vector w_{m} generates the kernel as a graded k[s, t]-module (no element of smaller degree lies in the kernel). Because w_{m} is homogeneous of degree m in k[s, t]^{m+1}, we conclude for 0 ≤ m ≤ p − 1 that

i^{∗}(Ker{Θ̃_{G}, V_{m} ⊗ O_{ℙ(G)}}) ≃ O_{ℙ^{1}}(−m), 0 ≤ m ≤ p − 1.

For p ≤ m ≤ 2p − 2, V_{m} has a decomposition series which can be represented as follows:

\xymatrix{S_{μ} \ar[dr] &  & S_{μ} \ar[dl] \\  & S_{λ}}

where λ = 2(p − 1) − m, μ = m − p = p − 2 − λ, and S_{λ}, the irreducible module of highest weight λ, is the socle of V_{m} ([2, §1]). By [8], V_{m} has constant Jordan type.

Plugging x = 1, y = z = 0 in (5.9.1), we get that the Jordan type is [p] + [μ + 1]. In particular, the rank of the kernel bundle is 2.

Using the relations μ ≡ m (mod p) and −λ ≡ μ + 2 (mod p), we obtain that the matrix B_{m}(s, t) has the following form: (5.9.3)

B_{m}(s, t) = [  \begin{matrix}& | & & & & | \\ B_{μ}(s, t) & | & & & & | \\ & | & 0 & & & | \\ & _{(μ+1)s^{2}}| & & & & | \\ & | & & B_{λ}(t, s)^{T} & & | \\ & | & & & & |_{(λ+1)t^{2}} \\ & | & & & 0 & | \\ & | & & & & | & B_{μ}(s, t) \\ & | & & & & | \end{matrix}  ]

Here, the top left and bottom right corners are of size μ + 1 × μ + 1, whereas the matrix in the center is of size λ + 1 × λ + 1. The only non-zero entries outside of these three square diagonal blocks are at (μ + 2, μ + 1) and (μ + λ + 2, μ + λ + 3), equaled to (μ + 1)s^{2} and (λ + 1)t^{2} respectively.

In particular, the μ + 1 × λ + 1 blocks above and below the matrix B_{λ}(t, s)^{T} in the middle are zero. This implies that the kernel of this operator contains a copy of the kernel of B_{λ}(s, t) (since it coincides with the kernel of B_{λ}(t, s)^{T}). We therefore obtain the vector

w′_{m} = [0, …, 0, t^{λ}, −st^{λ−1}, …, ∓s^{λ}, 0, …, 0]

in the kernel, where the non-zero entries are at the positions (μ + 2, …, μ + λ + 2).

One verifies that {w_{m}, w′_{m}} generate the kernel of B_{m}(s, t) as a graded k[s, t]- module. For example, one can check this by restricting to the affine pieces U(s ≠ 0) and U(t ≠ 0). Hence,

i^{∗}(Ker{Θ̃_{G}, V_{m} ⊗ O_{ℙ(G)}}) ≃ O_{ℙ^{1}}(−m) ⊕ O_{ℙ^{1}}(m − 2(p − 1)).
□

One may readily determine the rank of various bundles of ℙ(G) associated to modules of constant Jordan type using the next proposition.

Proposition 5.10. Let G be an infinitesimal group scheme, let M a kG-module of constant Jordan type ∑_{i=1}^{p} a_{i}[i], and let M = M⊗O_{ℙ(G)}. Then for any j, 1 ≤ j < p,

(5.10.1)  rk(Im{Θ̃_{G}^{j}, M}) = ∑_{i=j+1}^{p} a_{i}(i − j).

In particular,

Ker{Θ̃_{G}, M} ⊂ Ker{Θ̃_{G}^{2}, M} ⊂ ⋯ ⊂ Ker{Θ̃_{G}^{p−1}, M} ⊂ M

is a chain of O_{ℙ(G)}-submodules with rk(Ker{Θ̃_{G}^{j−1}, M}) < rk(Ker{Θ̃_{G}^{j}, M}) if and only if a_{i} ≠ 0 for some 1 ≤ j ≤ i ≤ p.

Proof. The formula (5.10.1) is the formula for the rank of u^{j} on the k[u]/u^{p}-module ⊕_{i}(k[u]/u^{i})^{⊕a_{i}} of Jordan type ∑_{i=1}^{p} a_{i}[i]. This is therefore the dimension of the image of θ_{v}, 0 ≠ v ∈ V (G) on M_{k(v)}, and thus the rank of the vector bundle Im{Θ̃_{G}^{j}, M} by Theorem 4.13. □

The following class of modules, of interest in its own right, is currently being studied by Jon Carlson and the authors.

Definition 5.11. Let G be an infinitesimal group scheme, M a finite dimensional kG-module, and j < p a positive integer. We say that a kG-module M has the constant j-image property if there exists a subspace I(j) ⊂ M such that for every v ≠ 0 in V (G), the image of θ_{v}^{j} : M_{k(v)} → M_{k(v)} equals I(j)_{k(v)}. Similarly, we say that M has constant j-kernel property if there exists some submodule K(j) ⊂ M such that for every v ≠ 0 in V (G), the kernel of θ_{v}^{j} : M_{k(v)} → M_{k(v)} equals K(j)_{k(v)}.

We see that these modules are precisely those whose associated vector bundles are trivial vector bundles.

Proposition 5.12. Let G be an infinitesimal group scheme, and let M be a kGmodule of constant j-rank. Then the algebraic vector bundle Im{Θ̃_{G}^{j}, M} is trivial (i.e., a free coherent sheaf) on ℙ(G) if and only if M has the constant j-image property. Similarly, Ker{Θ̃_{G}^{j}, M} is trivial if and only if M has the constant j-kernel property.

Proof. If M has a constant j-image property then Im{Θ̃_{G}^{j}, M} is a free O_{X}-module generated by I(j). Conversely, assume that Im{Θ̃_{G}^{j}, M} is a free O_{X}-module. Then there exists a subspace I(j) ⊂ M = Γ(X, M) which maps to and spans each fiber Im{θ_{v}^{j}, M_{k(v)}}, for 0 ≠ v ∈ V (G). The argument for kernels is similar. □

Remark 5.13. We point out the properties of constant j-image and constant jkernel are independent of each other. Consider the module M^{#} of Example 6.1. As shown in that example, Ker{Θ̃_{G}, M^{#}} is locally free of rank 2 but not free, since the global sections have dimension one. On the other hand, Im{Θ̃_{G}, M^{#}} is a free O_{X}-module generated by the global section n_{3}. In particular, M^{#} has constant 1-image property but not constant 1-kernel property.

For the module M of Example 6.1, the sheaf Ker{Θ̃_{G}, M} is free of rank 2 whereas Im{Θ̃_{G}, M} is locally free of rank 1 but not free since it does not have any global sections. Hence, M has a constant 1-kernel property but not constant 1-image property.

We consider an analogue of the sheaf construction of Duflo-Serganova for Lie superalgebras [11]. This construction enables us to produce additional algebraic vector bundles on ℙ(G). We implicitly use the observation Θ̃_{G}^{p} = 0.

Definition 5.14. Let G be an infinitesimal group scheme, and let M be a finite dimensional kG–module. Let M = M ⊗ O_{ℙ(G)}. For any i, 1 ≤ i ≤ p − 1, we define coherent O_{ℙ(G)}-modules, subquotients of M:

M^{[i]} ≡ Ker{Θ̃_{G}^{i}, M}/ Im{Θ̃_{G}^{p−i}, M}.

The following simple lemma helps to motivate these subquotients.

Lemma 5.15. Let V be a finite dimensional k[t]/t^{p}-module, and let JT ype(V, t) = (a_{p}, …, a_{1}) (using the notation introduced in (3.1.2)). Let

V^{[j]} = Ker{t^{j} : V → V }/ Im{t^{p−j} : V → V }

for j ≤ p − 1. Then

dim(V^{[j]}) = ∑_{1≤i≤j} ia_{i} + ∑_{i>j} ja_{i} − ∑_{i+j>p}(i + j − p)a_{i}.

In particular, V is projective as a k[t]/t^{p}-module if and only if V^{[1]} = 0.

Furthermore, for j ≤ p − 1, V^{[j]} ≃ V^{[p−j]} as k[t]/t^{p}-modules.

As seen in the next proposition, these subquotients can provide additional examples of algebraic vector bundles over ℙ(G).

Proposition 5.16. Let G be an infinitesimal group scheme and let M be a finite dimensional kG-module which is of constant j-rank and constant (p − j)-rank for some j, 1 ≤ j < p. Then M^{[j]} is a locally free O_{X}-module and M^{[j]} ⊗_{O_{X}} k(x) → M_{k(x)}^{[j]} is an isomorphism for all x ∈ X ≡ ℙ(G).

Proof. For any x ∈ X, consider the map of exact sequences

\xymatrix{Im\{Θ̃_{G}^{p−i}, M\} ⊗_{O_{X}} k(x) \ar[d] \ar[r] & Ker\{Θ̃_{G}^{i}, M\} ⊗_{O_{X}} k(x) \ar[d] \ar[r] & M^{[j]} ⊗_{O_{X}} k(x) \ar[d] \ar[r] & 0 \\ Im\{θ_{x}^{p−i}, M_{k(x)}\} \ar[r] & Ker\{θ_{x}^{i}, M_{k(x)}\} \ar[r] & M_{k(x)}^{[j]} \ar[r] & 0.}

The left and middle vertical maps are isomorphisms by Theorem 4.13. Thus, the 5-Lemma implies that the right vertical arrow is also an isomorphism. □

We give an application of this (−)^{[1]} construction to endotrivial modules. An interested reader can compare our construction to [1]. Recall that a module M of a finite group scheme G is endotrivial if End_{k}(M) ≃ k + proj. It was shown in [8, §5] that an endotrivial module is a module of constant Jordan type with possible types [1] + proj and [p − 1] + proj.

Proposition 5.17. Let G be an infinitesimal group scheme, and assume that G has a subgroup scheme isomorphic to 𝔾_{a(1)} × 𝔾_{a(1)} or 𝔾_{a(2)}. Let M be a module of constant Jordan type, and set M = M ⊗ O_{ℙ(G)}. Then M^{[1]} is a line bundle (i.e., an algebraic vector bundle of rank one) if and only if M is endotrivial.

Proof. The sheaf M^{[1]} is locally free by Lemma 5.15. Let ∑_{i=1}^{p} a_{i}[i] be the Jordan

type of M. Proposition 5.16 implies that the rank of the vector bundle M^{[1]} equals

∑_{i=0}^{p−1} a_{i}. Hence, M^{[1]} is a line bundle if and only if the Jordan type of M has

only one non-projective block. A theorem of D. Benson [4] states that modules of constant Jordan type with unique non-projective block must be of type [1] + proj or [p − 1] + proj. By [8, §5], this happens if and only if M is endotrivial. □

We next give a global version of the observation in Lemma 5.15 that V^{[j]} ≃ V^{[p−j]} for j ≤ p − 1. Recall that for a variety X, and a coherent sheaf E, we denote by E^{∨} = Hom_{O_{X}} (E, O_{X}) the dual sheaf.

Proposition 5.18. Let G be an infinitesimal group scheme, and let M be a kGmodule which is of constant j-rank and of constant (p − j)-rank for some j, 1 ≤ j < p. Let M = M ⊗ O_{ℙ(G)}, N = M^{#} ⊗ O_{ℙ(G)}. Then

N^{[p−j]} ≃ (M^{[j]})^{∨}

as O_{ℙ(G)}–modules.

Proof. Let X = ℙ(G). As discussed in the proof of Proposition 5.5, the O_{X}-linear dual of the complex of O_{X}-modules

\xymatrix{M(−(p − j)p^{r−1}) \ar[r]^{Θ̃_{G}^{p−j}(−(p−j)p^{r−1})} & M \ar[r]^{Θ̃_{G}^{j}} & M(jp^{r−1})}

is the complex

\xymatrix{N ((p − j)p^{r−1}) & N \ar[l]_{Θ̃_{G}^{p−j}} & N (−jp^{r−1}) \ar[l]_{Θ̃_{G}^{j}(−jp^{r−1})}} .

A similar statement applies with θ_{v} in place of Θ̃_{G}.

For any scheme Y and any complex of O_{Y} -modules

\xymatrix{S_{1} \ar[r]^{f} & S_{2} \ar[r]^{g} & S_{3}}

with O_{Y} -linear dual

\xymatrix{S_{1}^{∨} & S_{2}^{∨} \ar[l]_{f^{∨}} & S_{3}^{∨} \ar[l]_{g^{∨}}} ,

there is a natural pairing

(5.18.1)  \xymatrix{(Ker\{g\}/ Im\{f\}) ⊗ (Ker\{f^{∨}\}/ Im\{g^{∨}\}) \ar[r] & O_{Y} ,}

induced by the evident pairing

S_{2} ⊗ S_{2}^{∨} → O_{Y} .

In particular, we have a pairing

M^{[j]} ⊗ N^{[p−j]} → O_{X},

and, hence, a map

f : N^{[p−j]} → (M^{[j]})^{∨}.

By Proposition 5.1,

M^{[j]} ⊗_{O_{X}} k(x) ≃ M_{k(x)}^{[j]}, N^{[p−j]} ⊗_{O_{X}} k(x) ≃ N_{k(x)}^{[p−j]}

for any x ∈ X. By naturality, the specialization of f at a point x corresponds to the map f_{x} : N_{k(x)}^{[p−j]} → (M_{k(x)}^{[j]})^{#} induced by the paring (5.18.1) for Y = Spec k(x). One readily verifies that this is a perfect pairing if O_{Y} is a field. Hence,

f ⊗_{O_{X}} k(x) : N^{[p−j]} ⊗_{O_{X}} k(x) → (M^{[j]})^{∨} ⊗_{O_{X}} k(x)

is an isomorphism for any x ∈ X. Therefore, N^{[p−j]} ≃ (M^{[j]})^{∨}. □

Consideration of M^{[1]} leads to another characterization of projective kG-modules.

Proposition 5.19. Let G be an infinitesimal group scheme and let M be a finite dimensional kG-module. Then M is projective if and only if M has constant rank, has constant (p − 1)–rank, and satisfies M^{[1]} = 0.

Proof. Assume that M is a projective kG-module. Then M has constant Jordan type (which is some multiple of [p]), and hence has constant rank and constant (p− 1)-rank. For any x ∈ ℙ(G) = X, θ_{x}^{∗}(M_{k(x)}) is a free k(x)[t]/t^{p}-module of rank equal to \frac{dim(M)}{p}. If we lift a basis of this free module to M_{(x)} ≡ M⊗_{O_{X}} O_{X,x}, then an application of Nakayama’s Lemma tells us that M_{(x)} is free as an O_{X,x}[t]/t^{p}-module.

This readily implies that (M_{(x)})^{[p−1]} ≡ Ker{Θ̃_{G,(x)}^{p−1}, M_{(x)}}/ Im{Θ̃_{G,(x)}, M_{(x)}} vanishes. Using the exactness of localization, we conclude that (M^{[p−1]})_{(x)} = (M_{(x)})^{[p−1]}. Consequently, M^{[p−1]} = 0. By Proposition 5.18, we conclude that M^{[1]} = 0.

Conversely, if M has constant rank and constant (p − 1)-rank and if M^{[1]} = 0, then Proposition 5.16 tells us that M_{k(x)}^{[1]} ≡ Ker θ_{x}/ Im θ_{x}^{p−1} equals 0 for all x ∈ X. Lemma 5.15 thus implies that each M_{k(x)} is projective, so that the local criterion for projectivity [25] implies that M is projective. □

One very simple invariant of the algebraic vector bundle Ker{Θ̃_{G}^{j}, M} is the dimension of its vector space of global sections. The following proposition gives some understanding of Γ(ℙ(G), Ker{Θ̃_{G}^{j}, M}) ⊂ Γ(ℙ(G), M).

Proposition 5.20. Let G be an infinitesimal group scheme, and assume that V (G) is reduced. Let M be a kG-module and let M = M ⊗ O_{ℙ(G)}. Then

Γ(ℙ(G), Ker{Θ̃_{G}^{j}, M}) ⊂ M

consists of those m ∈ M such that θ_{x}^{j}(m) = 0 for all x ∈ ℙ(G).

Proof. Recall that ℙ(G) is connected by [8, 3.4] and thus Γ(ℙ(G), M) = M. Under this identification, the global sections of Ker{Θ̃_{G}^{j}, M} coincide with the subset

{m ∈ M | Θ_{G}^{j}(m ⊗ 1) = 0}.

Since V (G) is reduced, we have Θ_{G}^{j}(m ⊗ 1) = 0 if and only if θ_{v}^{j}(m ⊗ 1) = Θ_{G}^{j}(m ⊗ 1) ⊗_{k[V (G)]} k(v) = 0 for every v ∈ V (G). Hence, m ∈ Γ(ℙ(G), Ker{Θ̃_{G}^{j}, M}) if and only if m ∈ Ker{θ_{v}^{j}, M_{k(v)}} for every v ∈ V (G) if and only if θ_{x}^{j}(m) = 0 for all x ∈ ℙ(G). □

We make Proposition 5.20 more explicit in the case of a classical Lie algebra.

Proposition 5.21. Let 𝔊 be a (reduced, irreducible) algebraic group over k, let G = 𝔊_{(1)}, and let 𝔤 = Lie(𝔊). If 𝔤 is generated by p-nilpotent elements and if M is a rational 𝔊-module, then Γ(ℙ(G), Ker{Θ̃_{G}^{j}, M}) is a rational 𝔊-submodule of M.

Furthermore,

Γ(ℙ(G), Ker{Θ̃_{G}, M}) = H^{0}(G, M).

Proof. To prove that Γ(ℙ(G), Ker{Θ̃_{G}^{j}, M}) is a 𝔊-submodule of M, we may base change to the algebraic closure of k, and thus we may assume k is algebraically closed. Let g ∈ 𝔊 be a k-rational point. Then

(5.21.1)  θ_{v}^{j}(gm ⊗ 1) = gθ_{v^{g^{−1}}}^{j} (m ⊗ 1), .

where the action of 𝔊 on V (G) = N_{p}(𝔤) (the p-nilpotent cone of 𝔤) is via the adjoint action of 𝔊.

Hence, we have the following equalities:

{m ∈ M | Θ_{G}^{j}(m ⊗ 1) = 0} = ⋂_{0≠v∈V (G)}{m ∈ M | θ_{v}^{j}(m ⊗ 1) = 0} =

⋂_{0≠Ad(g^{−1})v∈V (G)}{m ∈ M | gθ_{Ad(g^{−1})v}^{j}(m⊗1) = 0} = ⋂_{0≠v∈V (G)}{m ∈ M | θ_{v}^{j}(gm⊗1) = 0}

= {m ∈ M | Θ_{G}^{j}(gm ⊗ 1) = 0},

where the first and the last equality follow from Proposition 5.20, the second equality follows from the fact that Ad(g^{−1}) : V (G) → V (G) is a bijection, and the third equality from (5.21.1). We conclude that {m ∈ M | Θ_{G}^{j}(m ⊗ 1) = 0} is a 𝔊-stable subspace of M.

The second assertion follows immediately from Proposition 5.20 and the facts that v ∈ V (G) corresponds to a p-nilpotent element X_{v} of 𝔤 and that the action of θ_{v} is the action of X_{v}.

□

Combining Proposition 5.10 and Corollary 5.21 in the special case j = 1 yields the following criterion for the non-triviality of Ker{Θ̃_{G}, M}.

Corollary 5.22. Let G be an infinitesimal group scheme such that V (G) is reduced and positive dimensional. Assume that for any field extension K/k, KG is generated by θ_{v} ∈ k(v)G, for all v ∈ V (G) such that k(v) ⊂ K. Let M a finite dimensional kG-module of constant Jordan type ∑_{i} a_{i}[i]. If

dim H^{0}(G, M) < ∑_{i=1}^{p} a_{i},

then Ker{Θ̃_{G}, M} is a non-trivial algebraic vector bundle over ℙ(G)

Proof. By Proposition 5.10, the dimension of the fibers of Ker{Θ̃_{G}, M} is dim M − ∑_{i=2}^{p} a_{i}(i − 1) = ∑_{i=1}^{p} a_{i}. By Proposition 5.20, the global sections of Ker{Θ̃_{G}, M} equal H^{0}(G, M). Hence, the inequality dim H^{0}(G, M) < ∑_{i=1}^{p} a_{i} implies that the dimension of the global sections is less than the dimension of the fibers. Therefore, the sheaf is not free.

□

The following two lemmas will be applied to prove Proposition 5.25.

Lemma 5.23. Let R be a local commutative ring with residue field k and let M be a finite R[t]/t^{p}-module which is free as an R-module. If M ⊗_{R} k is a free k[t]/t^{p}- module, then M is free as an R[t]/t^{p}-module.

Proof. Let m_{1}, …, m_{s} ∈ M be such that \overline{m}_{1}, …, \overline{m}_{s} form a basis for M ⊗_{R} k as a k[t]/t^{p}-module. Let Q be a free R]t]/t^{p}-module of rank s with basis q_{1}, …, q_{s} and consider the R[t]/t^{p}-module homomorphism f : Q → M sending q_{i} to m_{i}.

By Nakayama’s Lemma, f : Q → M is surjective. Because M is free as an Rmodule, applying − ⊗_{R} k to the short exact sequence 0 → Ker{f} → Q → M → 0 determines the short exact sequence

0 → Ker{f} ⊗_{R} k → Q ⊗_{R} k → M ⊗_{R} k → 0.

Consequently, Ker{f}⊗_{R} k = 0, so that another application of Nakayama’s Lemma implies that Ker{f} = 0. Hence, f is an isomorphism, and thus M is free as an R[t]/t^{p}-module. □

Lemma 5.24. Let G be an infinitesimal group scheme and M be a finite dimensional kG-module. Set A = k[V (G)]; for any f ∈ A, set A_{f} = A[1/f]. Assume that Spec A_{f} ⊂ V (G) has empty intersection with V (G)_{M}. Then (U_{G} ∘ ϵ)^{∗}(M ⊗ A_{f}) is a projective A_{f}[t]/t^{p} -module.

Proof. By definition, V (G)_{M} consists of those points v ∈ V (G) such that θ_{v}^{∗}(M_{(k(v)}) is not free as a k(v)[t]/t^{p}-module. By the universal property of U_{G} ∘ ϵ, the assumption that Spec A_{f} ∩ V (G)_{M} = ∅ implies for every point v ∈ Spec A_{f} that θ_{v}^{∗}(M_{k(v)}) = (U_{G} ∘ ϵ)^{∗}(M ⊗ A_{f}) ⊗_{A_{f}} k(v) is free as a k(v)[t]/t^{p}-module. Let A_{(v)} denote the localization of A at v. Then Lemma 5.23 implies for every point v ∈ Spec A_{f} that the localization (U_{G} ∘ ϵ)^{∗}(M ⊗ A_{f}) ⊗_{A_{f}} A_{(v)} is free as a A_{(v)}[t]/t^{p}- module. This implies that M ⊗ A_{f} is projective (since projectivity of a finitely generated module over a A is determined locally). □

We conclude with a property of the (projectivized) rank variety ℙ(G)_{M} of a kG-module M.

Proposition 5.25. Let G be an infinitesimal group scheme, M be a finite dimensional kG-module, and set M = M ⊗ O_{ℙ(G)}. Then

Supp_{O_{ℙ(G)}}(M^{[1]}) ⊂ ℙ(G)_{M},

where Supp_{O_{ℙ(G)}}(M^{[1]}) is the support of the coherent sheaf M^{[1]} (the closed subset

of points x ∈ ℙ(G) at which M_{(x)}^{[1]} ≠ 0).

Proof. Let A denote k[V (G)] and let X denote ℙ(G). Consider some x ∉ X_{M} and choose some homogeneous polynomial F ∈ A vanishing on X_{M} such that F (x) ≠ 0. Thus, x ∈ Spec(A_{F})_{0} ⊂ X and Spec(A_{F})_{0} ∩ X_{M} = ∅, where (A_{F})_{0} denote the elements of degree 0 in the localization A_{F} = A[1/F]. It suffices to prove that x ∉ Supp_{O_{X}} (M^{[1]}) ∩ Spec(A_{F})_{0}. Since localization is exact, this is equivalent to proving that v ∉ Supp_{A_{F}} ((M ⊗ A_{F})^{[1]}) for some v ∈ Spec A_{F} mapping to x.

The condition Spec(A_{F})_{0} ∩X_{M} = ∅ implies that Spec(A_{F})∩V (G)_{M} = ∅. Hence, by Lemma 5.24, (U_{G} ∘ ϵ)^{∗}(M ⊗ A_{F}) is a projective A_{F} [t]/t^{p} -module. This implies that (U_{G} ∘ ϵ)^{∗}((M ⊗ A_{F})^{[1]}) = 0, and thus that v ∉ Supp_{A_{F}} ((M ⊗ A_{F})^{[1]}). □

Remark 5.26. The reverse inclusion ℙ(G)_{M} ⊂ Supp_{O_{ℙ(G)}}(M^{[1]}) seems closely

related to the condition that Ker{Θ̃_{G}, M} ⊗_{O_{ℙ(G)}} k(x) → Ker{θ_{x}, M_{k(x)}} be surjective.

6. Examples and calculations with bundles

In this final section, we investigate numerous specific examples. The case in which G equals either 𝔾_{a(1)}^{×2} or \underline{s}l_{2} (the infinitesimal group scheme associated to the restricted Lie algebra sl_{2}) is particularly amenable to computation for ℙ(G) is isomorphic to ℙ^{1}. Specifically, we do calculations for projective kG-modules, examples of modules of constant Jordan type which are not distinguished by support varieties. We also compute bundles obtained from“zig-zag modules” and syzygies.

As we see in the following simple example, the isomorphism type of the vector bundles discussed in Theorem 5.1 can be used to distinguish certain kG-modules which have the same local Jordan type. We remind the reader that the local Jordan type of a finite dimensional kG-module M of constant Jordan type is the same as that of its linear dual M^{#}.

Example 6.1. Let G = 𝔾_{a(2)}, so that k𝔾_{a(2)} = k[u_{0}, u_{1}]/(u_{0}^{p}, u_{1}^{p}), V (𝔾_{a(2)}) = 𝔸^{2}, A = k[V (𝔾_{a(2)})] = k[x_{0}, x_{1}] graded so that x_{0} is given degree 1 and x_{1} is given degree p. Then

Θ_{G} = x_{1}u_{0} + x_{0}^{p}u_{1} ∈ A[u_{0}, u_{1}]/(u_{0}^{p}, u_{1}^{p}).

(see (2.6(2)). We consider the 3-dimensional kG-module M of constant Jordan type [2] + [1] and its linear dual M^{#}, which we represent diagrammatically as follows:

\xymatrix{ & m_{1} \ar[dl]_{u_{2}} \ar[dr]^{u_{1}} &  & n_{1} \ar[dr]^{u_{1}} &  & n_{2} \ar[dl]_{u_{2}} \\ m_{2} &  & m_{3} &  & n_{3}}

The k 𝔾_{a(2)}-invariant subspace of M is two dimensional, and, hence, the global sections of Ker{Θ̃_{G}, M ⊗ O_{ℙ(G)}} have dimension two by Prop. 5.20 (in fact, an explicit calculation shows that this is a trivial bundle of rank 2). On the other hand, M^{#} has only one-dimensional invariant subspace, and, hence, the global sections of Ker{Θ̃_{G}, M^{#} ⊗ O_{ℙ(G)}})) have dimension one. Thus,

Ker{Θ̃_{G}, M ⊗ O_{ℙ(G)}} ≇ Ker{Θ̃_{G}, M^{#} ⊗ O_{ℙ(G)}}.

We next give a somewhat more interesting example of pairs of modules of the same constant Jordan type with different associated bundles.

Example 6.2. As in Proposition 5.9, let S_{λ} be the irreducible SL_{2}-module of highest weight λ, 0 ≤ λ ≤ p − 1, and consider S^{p}(S_{λ}), the p^{th} symmetric power of S_{λ}. Since S_{λ} is self-dual, the dual of S^{p}(S_{λ}) is Γ^{p}(S_{λ}), the p^{th} divided power of S_{λ}. We have a short exact sequence of rational SL_{2}-modules

\xymatrix{0 \ar[r] & S_{λ}^{(1)} \ar[r] & S^{p}(S_{λ}) \ar[r] & Γ^{p}(S_{λ}) \ar[r] & S_{λ}^{(1)} \ar[r] & 0.}

Here, S_{λ}^{(1)} is the first Frobenius twist of S_{λ}, thus trivial as a 𝔲(sl_{2})-module.

Let X = Proj k[N(sl_{2})]. By Prop. 5.20, the space of global sections of Ker\{Θ̃_{\underline{s}l_{2}}, S^{p}(S_{λ}) ⊗ O_{X}\} equals the sl_{2} invariants of S^{p}(S_{λ}). Hence,

Γ(X, Ker\{Θ̃_{\underline{s}l_{2}}, S^{p}(S_{λ}) ⊗ O_{X}\}) = S_{λ}^{(1)}.

On the other hand, Γ^{p}(S_{λ}) does not have any sl_{2}-invariants, and, hence, Ker\{Θ̃_{\underline{s}l_{2}}, Γ^{p}(S_{λ}) ⊗ O_{X}\} does not have any global sections. We conclude that the kernel bundles associated to the dual modules S^{p}(S_{λ}) and Γ^{p}(S_{λ}) are nonisomorphic.

We continue our consideration of SL_{2(1)} ≡ \underline{s}l_{2} in the following proposition.

Proposition 6.3. Let G = \underline{s}l_{2}, let S_{λ} be the irreducible kG-module of highest weight λ, 0 ≤ λ ≤ p − 1, and let P_{λ} → S_{λ} be the projective cover of S_{λ}. Then

i^{∗}(Ker\{Θ̃_{G}, P_{λ} ⊗ O_{ℙ(G)}\}) ≃ \begin{cases}O_{ℙ^{1}}(1 − p) if λ = p − 1 \\ O_{ℙ^{1}}(λ − 2(p − 1)) ⊕ O_{ℙ^{1}}(−λ) if 0 ≤ λ ≤ p − 2 \end{cases}

where i : ℙ^{1} → ℙ(G) is the isomorphism (5.8.1).

Proof. For λ = p − 1, P_{λ} = St is the Steinberg module for sl_{2}, and is irreducible. Hence, in this case the statement follows from Proposition 5.9.

The decomposition series of P_{λ} for 0 ≤ λ < p − 1 is represented by the following diagram (see [14, 2.4]):

(6.3.1)  \xymatrix{ & S_{λ} \ar[dl] \ar[dr] \\ S_{p−2−λ} \ar[dr] &  & S_{p−2−λ} \ar[dl] \\  & S_{λ}}

Thus, we have a short exact sequence of SL_{2}-modules

(6.3.2)  0 → V_{2p−2−λ} → P_{λ} → S_{λ} → 0.

By Proposition 5.9, it suffices to prove that V_{2p−2−λ} ⊂ P_{λ} induces an isomorphism

Ker{Θ̃_{G}, V_{2p−2−λ} ⊗ O_{ℙ(G)}} ≃ Ker{Θ̃_{G}, P_{λ} ⊗ O_{ℙ(G)}}.

By Theorem 4.13(8) and Nakayama’s Lemma, it suffices to prove that

Ker{θ_{x}, V_{2p−2−λ} ⊗ k(x)} → Ker{θ_{x}, P_{λ} ⊗ k(x)}

is an isomorphism for all x ∈ ℙ(G). This last statement follows from the observation that the Jordan decomposition of θ_{x} on both V_{λ} and P_{λ} consists of two blocks: on P_{λ}, because P_{λ} is projective of dimension 2p; and on V_{2p−2−λ} as discussed in Proposition 5.9. □

If X is an algebraic variety over k (for example, X = ℙ(G)), then K_{0}(X) denotes the Grothendieck group of algebraic vector bundles over X (i.e., finitely generated, locally free O_{X}-modules) defined as the free abelian group on the set of isomorphism classes of such vector bundles modulo relations given by short exact sequences. We shall also consider K_{0}^{⊕}(X) defined as the free abelian group on the same set of generators modulo relations given by split short exact sequences. Thus, there is a canonical surjective homomorphism

K_{0}^{⊕}(X) → K_{0}(X).

Definition 6.4. Let G be an infinitesimal group scheme and let K_{0}(kG) denote the Grothendieck group of finitely generated projective kG-modules. For any j, 0 ≤ j ≤ p − 1, the homomorphism

κ_{G,j}^{⊕} : K_{0}(kG) → K_{0}^{⊕}(ℙ(G))

is defined by sending a projective kG-module Q to Ker{Θ̃_{G}^{j}, Q ⊗ O_{ℙ(G)}}. We define \underline{κ}_{G}^{⊕} by

\underline{κ}_{G}^{⊕} = (κ_{G,1}^{⊕}, … κ_{G,p}^{⊕}) : K_{0}(kG) → K_{0}^{⊕}(ℙ(G))^{⊕p}.

Moreover, the homomorphism

κ_{G,j} : K_{0}(kG) → K_{0}(ℙ(G))

is defined to be the composition of κ̃_{G,j}^{⊕} with the canonical projection K_{0}^{⊕}(ℙ(G)) → K_{0}(ℙ(G)), and the homomorphism

\underline{κ}_{G} : K_{0}(kG) → K_{0}(ℙ(G))^{⊕p}

is defined to be the composition of \underline{κ}_{G} with the canonical projection K_{0}^{⊕}(ℙ(G))^{⊕p} → K_{0}(ℙ(G))^{⊕p}.

We shall omit the subscript G is κ_{G} when the group scheme is clear from the context. Note that since Θ̃^{p} = 0, κ_{p} returns the entire class [Q ⊗ O_{ℙ(G)}].

The dimension of the global sections function:

E ↦ dim Γ(ℙ(G), E)

extends to a homomorphism

ρ : K_{0}^{⊕}(ℙ(G)) → ℤ

Observe that since Γ(ℙ(G), −) is not right exact for non-split exact sequences of algebraic vector bundles on ℙ(G), ρ does not factor through K_{0}(ℙ(G)).

Proposition 6.5. Let G = \underline{s}l_{2}. The composition

ρ ∘ \underline{κ}^{⊕} : K_{0}(kG) → K_{0}^{⊕}(ℙ(G))^{⊕p} → ℤ^{⊕p}

is a rational isomorphism.

Proof. Recall that K_{0}(kG) = K_{0}(u(sl_{2})) ≃ ℤ^{⊕p}, spanned by the projective indecomposable u(sl_{2})-modules P_{λ}, 0 ≤ λ ≤ p − 1. By Proposition 5.21, the global sections of Ker{Θ̃_{G}^{j}, P_{λ} ⊗ O_{ℙ(G)}}) is a rational SL_{2}-submodule of P_{λ}. Hence, Γ(ℙ(G), Ker{Θ̃_{G}^{j}, P_{λ} ⊗ O_{ℙ(G)}}) ≠ 0 if and only if S_{λ}, which is the socle of P_{λ}, belongs to the global sections. Hence,

Γ(ℙ(G), Ker{Θ̃_{G}^{j}, P_{λ} ⊗ O_{ℙ(G)}}) ≠ 0 ⇔ Θ̃^{j}(S_{λ} ⊗ O_{ℙ(G)}) = 0 ⇔ θ_{v}^{j}(S_{λ}) = 0

for any v ∈ V (G) ≃ N (sl_{2}). Since S_{λ} is a module of constant Jordan type [λ + 1], θ_{v}^{j} annihilates S_{λ} if and only if j > λ. Thus,

Γ(ℙ(G), Ker{Θ̃_{G}^{j}, P_{λ} ⊗ O_{ℙ(G)}}) ≠ 0 if and only if j > λ.

Moreover, the decomposition series for P_{λ} (6.3.1) implies that the Jordan type of any rational submodule of P_{λ} that is bigger than the socle S_{λ} has a Jordan block of size p. Hence,

Γ(ℙ(G), Ker{Θ̃_{G}^{λ+1}, P_{λ} ⊗ O_{ℙ(G)}}) = S_{λ}.

Note that the last equality holds trivially for λ = p − 1, since in this case the kernel bundle is the entire free sheaf P_{p−1} ⊗ O_{ℙ(G)}, where P_{p−1} = S_{p−1} is the Steinberg module for sl_{2}.

We conclude that the homomorphism

ρ ∘ \underline{κ}^{⊕} : K_{0}(kG) ≃ ℤ^{⊕p} → ℤ^{⊕p}

is given by a non-singular upper-triangular matrix

[ \begin{matrix}1 & ⋯ & & & · \\ 0 & 2 & ⋯ & & · \\ 0 & 0 & 3 & ⋯ & · \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & 0 & p \end{matrix} ]

Hence, ρ ∘ \underline{κ}^{⊕} is a rational isomorphism. □

In contrast with Proposition 6.5, we have the following computations for κ_{1} : K_{0}(kG) → K_{0}(ℙ(G)) and \underline{κ} : K_{0}(kG) → K_{0}(ℙ(G))^{⊕p} for G = \underline{s}l_{2}.

Lemma 6.6. Let G = \underline{s}l_{2}, and denote by St the Steinberg module for sl_{2}. The image of the homomorphism

κ_{1} : K_{0}(kG) → K_{0}(ℙ(G))

is generated by κ_{1}(St). Consequently, rk κ_{1} = 1.

Proof. The isomorphism i : ℙ(G) ≃ ℙ^{1} (5.8.1) induces an isomorphism K_{0}(ℙ(G)) ≃ K_{0}(ℙ^{1}). We denote by the letter κ_{1} the composition K_{0}(kG) → K_{0}(ℙ(G)) →^{∼} K_{0}(ℙ^{1}). Clearly, it suffices to prove the statement of the Lemma for this composition.

Let a_{n} = [O_{ℙ^{1}}(−n)] ∈ K_{0}(ℙ^{1}), n ∈ ℤ. Then a_{0} = [O_{ℙ^{1}}], a_{1} = [O_{ℙ^{1}}(−1)] generate K_{0}(ℙ^{1}) ≃ ℤ^{⊕2}. Using the short exact sequence of O_{ℙ^{1}}-modules

0 → O_{ℙ^{1}}(−(n + 1)) → O_{ℙ^{1}}(−n) ⊕ O_{ℙ^{1}}(−n) → O_{ℙ^{1}}(−(n − 1)) → 0,

we obtain the recurrence relation a_{n+1} = 2a_{n} − a_{n−1}. By induction, a_{n} = na_{1} − (n − 1)a_{0}.

By Proposition 6.3, for 0 ≤ λ < p − 1,

κ_{1}(P_{λ}) = [O_{ℙ^{1}}(−λ)] + [O_{ℙ^{1}}(λ − 2(p − 1)].

Hence, κ_{1}(P_{λ}) = a_{λ} + a_{2(p−1)−λ} = λa_{1} − (λ − 1)a_{0} + (2(p − 1) − λ)a_{1} − (2(p − 1) − λ − 1)a_{0} = 2(p − 1)a_{1} − 2(p − 2)a_{0}. Moreover,

κ_{1}(St) = [O_{ℙ^{1}}(1 − p)] = (p − 1)a_{1} − (p − 2)a_{0}

by Proposition 5.9. Hence,

κ_{1}(P_{λ}) = 2κ_{1}(St)

for 0 ≤ λ ≤ p − 2 which proves the statement. □

The second conclusion of the following Proposition is in sharp contrast with Proposition 6.5.

Proposition 6.7. Let G = \underline{s}l_{2}, and denote by St the Steinberg module for sl_{2}. (1) The image of κ_{j} : K_{0}(kG) → K_{0}(ℙ(G)) is generated by κ_{j}(St).

(2) The image of

\underline{κ} : K_{0}(kG) → K_{0}(ℙ(G))^{⊕p}

is generated by \underline{κ}(St) and, consequently, has rank 1.

Proof. As in the proof of Lemma 6.6, we identify K_{0}(ℙ(G)) with K_{0}(ℙ^{1}). We have a short exact sequence of bundles:

(6.7.1)  [figure: 0; Ker{Θ̃, P_{λ} ⊗ O_{ℙ(G)}}; Ker{Θ̃^{j}, P_{λ} ⊗ O_{ℙ(G)}}; Θ̃]
\xymatrix{Ker\{Θ̃^{j−1}, P_{λ} ⊗ O_{ℙ(G)}\}(2) \ar[r] & 0}

Indeed, the composition is clearly zero and the first map is an embedding. Moreover, by Theorem 4.13, the specialization of this sequence at any point x ∈ ℙ(G) looks as follows:

\xymatrix{0 \ar[r] & Ker\{θ_{x}, P_{λ} ⊗ k(x)\} \ar[r] & Ker\{θ_{x}^{j}, P_{λ} ⊗ k(x)\} \ar[r]^{θ_{x}} & Ker\{θ_{x}^{j−1}, P_{λ} ⊗ k(x)\} \ar[r] & 0}

The projectivity of P_{λ} implies that this sequence is exact. Hence, (6.7.1) is exact by Nakayama’s lemma.

We conclude that in K_{0}(ℙ^{1}),

(6.7.2)  κ_{j}(P_{λ}) = κ_{1}(P_{λ}) + κ_{j−1}(P_{λ})(2).

By Lemma 6.6, κ_{1}(P_{λ}) = 2κ_{1}(St) for 0 ≤ λ ≤ p − 2. Applying induction, we get κ_{j}(P_{λ}) = 2κ_{1}(St)+2κ_{j−1}(St)(2). Now applying the formula (6.7.2) to the Steinberg module, we get 2κ_{1}(St) + 2κ_{j−1}(St)(2) = 2κ_{j}(St). To summarize,

κ_{j}(P_{λ}) = \begin{cases}κ_{j}(St) λ = p − 1 \\ 2κ_{j}(St) 0 ≤ λ ≤ p − 2 \end{cases}

which implies (1). Moreover, we conclude that \underline{κ} : K_{0}(kG) → K_{0}(ℙ^{1})^{⊕p} is given by the formula

\underline{κ}(P_{λ}) = \begin{cases}(κ_{1}(St), κ_{2}(St), …, κ_{p}(St)) λ = p − 1 \\ (2κ_{1}(St), 2κ_{1}(St), …, 2κ_{p}(St)) = 2\underline{κ}(St) 0 ≤ λ ≤ p − 2 \end{cases}

which proves (2). □

The Proposition 6.9 gives us some information about the behavior of κ and κ^{⊕} with respect to products. First, we need a trivial linear algebra lemma.

Lemma 6.8. Let V, W be vector spaces over a field k. Let {v_{1}, …, v_{r}} be a basis of V, and {w_{1}, …, w_{s}} be a basis of W. Let v ∈ V, w ∈ W, and let X ⊂ V ⊕ W be the span of {(v_{1}, w), …, (v_{r}, w), (v, w_{1}), …, (v, w_{s})}. Then the dimension of X is at least r + s − 1.

Proof. If v = 0 and w = 0 then we obviously have dim X = r + s. We assume v ≠ 0. Observe that

(6.8.1)  \xymatrix{X \ar@{->>}[r] & (V ⊕ W)/⟨(v, 0) + (0, w)⟩}

is surjective (thus has image of dimension r + s − 2). Let v = a_{1}v_{1} + … + a_{r}v_{r}. Then the vector a_{1}(v_{1}, w) + … + a_{r}(v_{r}, w) = (v, w) is non-trivial since v ≠ 0 and belongs to the kernel of the projection (6.8.1). Hence, dim X ≥ r + s − 1. □

Proposition 6.9. Let G_{1} and G_{2} be infinitesimal group schemes, and let G = G_{1} × G_{2}. Then

rk κ_{G}^{⊕} ≥ rk κ_{G_{1}}^{⊕} + rk κ_{G_{2}}^{⊕} − 1,

where κ^{⊕} = κ_{j}^{⊕} for any j, 1 ≤ j ≤ p − 1.

Proof. Let i_{ℓ}^{∗} : K_{0}^{⊕}(ℙ(G)) → K_{0}^{⊕}(ℙ(G_{ℓ})) be the map induced by the pull-back of vector bundles along the embedding ℙ(G_{ℓ}) ↪ ℙ(G), for ℓ = 1, 2. We consider the composition

\xymatrix{K_{0}(kG) \ar[r]^{κ^{⊕}} & K_{0}^{⊕}(ℙ(G)) \ar[r]^{i_{1}^{∗}+i_{2}^{∗}} & K_{0}^{⊕}(ℙ(G_{1})) ⊕ K_{0}^{⊕}(ℙ(G_{2}))} .

Let P be a projective kG_{1}-module of dimension m, and Q be a projective kG_{2}- module of dimension n. Then P ⊗ Q is a projective kG-module. By Corollary 5.4,

(i_{1}^{∗} + i_{2}^{∗}) ∘ κ_{G}^{⊕}(P ⊗ Q) = (i_{1}^{∗} ∘ κ_{G}^{⊕}(P ⊗ Q), i_{2}^{∗} ∘ κ_{G}^{⊕}(P ⊗ Q)) =
(6.9.1)  (n κ_{G_{1}}^{⊕}(P), m κ_{G_{2}}^{⊕}(Q)).

Let {P_{1}, …, P_{r}} be projective kG_{1}-modules of dimensions {p_{1}, …, p_{r}} such that

{κ_{G_{1}}^{⊕}(P_{1}), …, κ_{G_{1}}^{⊕}(P_{r})}

are linearly independent generators of Im κ_{G_{1}}^{⊕} ⊂ K_{0}^{⊕}(ℙ(G_{1})), so that rk κ_{G_{1}}^{⊕} = r. Similarly, let {Q_{1}, …, Q_{r}} be projective kG_{2}-modules such that

{κ_{G_{2}}^{⊕}(Q_{1}), …, κ_{G_{2}}^{⊕}(Q_{s})}

are linearly independent generators of Im κ_{G_{2}}^{⊕} ⊂ K_{0}^{⊕}(ℙ(G_{2})), so that rk κ_{G_{2}}^{⊕} = s. Finally, let P be any projective kG_{1}-module, m = dim_{k} P, and Q be any projective kG_{2}-module, n = dim_{k} Q. Consider

S = Span{P_{1} ⊗ Q, P_{2} ⊗ Q, …, P_{r} ⊗ Q, P ⊗ Q_{1}, P ⊗ Q_{2}, …, P ⊗ Q_{s}} ⊂ K_{0}(kG)

Then the image of (i_{1}^{∗} + i_{2}^{∗}) ∘ κ_{G} contains

{(n κ_{G_{1}}^{⊕}(P_{1}), p_{1}κ_{G_{2}}^{⊕}(Q)), (n κ_{G_{1}}^{⊕}(P_{2}), p_{2}κ_{G_{2}}^{⊕}(Q)), …, (n κ_{G_{1}}^{⊕}(P_{r}), p_{r}κ_{G_{2}}^{⊕}(Q))}

and

{(q_{1}κ_{G_{1}}^{⊕}(P), m κ_{G_{2}}^{⊕}(Q_{1})), (q_{2}κ_{G_{1}}^{⊕}(P), m κ_{G_{2}}^{⊕}(Q_{2})), …, (q_{s}κ_{G_{1}}^{⊕}(P), m κ_{G_{2}}^{⊕}(Q_{s}))}

Since all the coefficients m, n, p_{i}, q_{j} are positive, Lemma 6.8 implies that the dimension of the image of (i_{1}^{∗} + i_{2}^{∗}) ∘ κ_{G}^{⊕} is at least r + s − 1.

□

Remark 6.10. As the reader can easily check, Proposition 6.9 and its proof hold for κ : K_{0}(kG) → K_{0}(ℙ(G)) in place of κ^{⊕} : K_{0}(kG) → K_{0}^{⊕}(ℙ(G)).

Observe that ℙ(\underline{sl_{2}^{⊕r}}) ≃ ℙ^{2r−1}, the join of r copies of ℙ(\underline{s}l_{2}) ≃ ℙ^{1}.

Corollary 6.11. Let G = \underline{sl}_{2}^{×r} be the infinitesimal group scheme corresponding to the restricted Lie algebra sl_{2}^{⊕r}. Then

κ_{1}^{⊕} : K_{0}(kG) → K_{0}^{⊕}(ℙ^{2r−1})

has rank at least r(p − 1) + 1.

Proof. For r = 1, κ_{\underline{sl}_{2}}^{⊕} : K_{0}(u(sl_{2})) → K_{0}^{⊕}(ℙ^{1}) is injective by Proposition 6.3 and, hence, has rank p. The statement now follows by induction and Proposition 6.9. □

Recall that if H ↪ G is a subgroup scheme of a finite group scheme G, then kG is free as a kH–module [24, 2.4]. Hence, the restriction functor res : (kG − mod) → (kH − mod) induces a well-defined map on K-groups: res^{∗} : K_{0}(kG) → K_{0}(kH). The commutativity of the diagram (5.3.1) implies that restriction commutes with κ_{j}. That is, we have a commutative diagram:

(6.11.1)  [figure: K_{0}(kG); _{res_{∗}}; κ_{G,j}; K_{0}(ℙ(G)); i^{∗}; K_{0}(kH); κ_{H,j}; K_{0}(ℙ(H)),]

where i : ℙ(H) ↪ ℙ(G) is the closed embedding of projective varieties induced by the embedding of group schemes and i^{∗} : K_{0}(ℙ(G)) → K_{0}(ℙ(H)) is the pull-back via i of locally free O_{P (G)}-modules. An analogous commutative diagram holds for κ^{⊕}.

Proposition 6.12. Let G be an infinitesimal group scheme, and let H ≃ \underline{s}l_{2}^{×r} ⊂ G be a closed subgroup scheme with the property that res^{∗} : K_{0}(kG) → K_{0}(kH) is rationally surjective. Then the composition

\xymatrix{K_{0}(kG) \ar[r]^{κ_{1}^{⊕}} & K_{0}^{⊕}(ℙ(G)) \ar[r]^{i^{∗}} & K_{0}^{⊕}(ℙ(H)) ≃ K_{0}^{⊕}(ℙ^{2r−1})}

has rank at least r(p − 1) + 1.

Proof. We apply the diagram (6.11.1) for κ_{1}^{⊕}:

[figure: K_{0}(kG); _{res_{∗}}; κ_{G,1}^{⊕}; K_{0}^{⊕}(ℙ(G)); i^{∗}; K_{0}(kH); κ_{H,1}^{⊕}; K_{0}^{⊕}(ℙ(H)),]

Since res^{∗} is assumed to be rationally surjective, rk (i^{∗} ∘ κ_{G,1}^{⊕}) = rk κ_{H,1}^{⊕}. Since

H = \underline{sl}_{2}^{×r}, rk κ_{H,1}^{⊕} ≥ r(p − 1) + 1 by Corollary 6.11. This proves the statement. □

For the rest of this computational section, we calculate some examples of bundles for E = 𝔾_{a(1)} × 𝔾_{a(1)}. We have kE ≃ k[x, y]/(x^{p}, y^{p}). Let X_{n} be (2n + 1)– dimensional “zig-zag” module. Pictorially, we represent X_{n} by the following diagram:

[figure: •^{⟨v_{0}⟩}; y; •^{⟨v_{1}⟩}; x; y; …; •^{⟨v_{n}⟩}; x; •; •; …; •]

It is straightforward to check that X_{n} has constant Jordan type n[2] + [1] (see [8, §2]). We proceed to prove that for any integer m we can obtain the line bundle O_{ℙ^{1}}(m) on ℙ(E) = ℙ^{1} by applying our constructions to some X_{n} or its linear dual X_{n}^{#}.

Note that for X_{n}, the map

\xymatrix{Θ̃_{E} : X_{n} ⊗ O_{ℙ^{1}} \ar[r] & X_{n} ⊗ O_{ℙ^{1}}(1)}

as defined in 4.6.1 has nilpotentcy degree 2. Hence, there is an inclusion Im{Θ̃_{E}, X_{n}⊗ O_{ℙ^{1}}} ⊂ Ker{Θ̃_{E}, X_{n} ⊗ O_{ℙ^{1}}(1)}. We, therefore, may define a subquotient sheaf of the free sheaf X_{n} ⊗ O_{ℙ^{1}} as

X_{n} := Ker{Θ̃_{E}, X_{n} ⊗ O_{ℙ^{1}}}/ Im{Θ̃_{E}(−1), X_{n} ⊗ O_{ℙ^{1}}(−1)}.

Arguing as in the proof of Proposition 5.16, one verifies that X_{n} is locally free with the fiber at a point t ∈ ℙ^{1} isomorphic to the 1-dimensional vector space \frac{Ker\{θ_{t}:X_{n,k(t)}→X_{n,k(t)}\}}{Im\{θ_{t}:X_{n,k(t)}→X_{n,k(t)}\}}. Hence, X_{n} is a line bundle. The linear dual X_{n}^{#} of X_{n} is represented by the diagram:

[figure: •; x; y; •; x; …; •; y; ⟨w_{0}⟩^{•}; ⟨w_{1}⟩^{•}; …; ⟨w_{n}⟩^{•}]

Define a subquotient sheaf of X_{n}^{#} ⊗ O_{ℙ^{1}} as

Y_{n} := Ker{Θ̃_{E}, X_{n}^{#} ⊗ O_{ℙ^{1}}}/ Im{Θ̃_{E}(−1), X_{n}^{#} ⊗ O_{ℙ^{1}}(−1)}.

Proposition 6.13. X_{n} ≃ O_{ℙ^{1}}(−n), Y_{n} ≃ O_{ℙ^{1}}(n).

Proof. Let k[s, t] = k[𝔸^{2}] ≃ k[V (E)]. The universal p-nilpotent operator Θ_{E} ∈ k[x, y]/(x^{p}, y^{p}) ⊗ k[s, t] is given by

Θ_{E} = xs + yt,

(see, for example, Ex. 2.7). We identify the graded k[s, t]-module Ker{Θ_{E}, X_{n} ⊗ k[s, t]}/ Im{Θ_{E}, X_{n} ⊗ k[s, t]}, thereby determining the vector bundle X_{n}. It is easy to see that Im{Θ_{E}, X_{n} ⊗ k[s, t]} is generated by the bottom row of the diagram representing X_{n} as a k[s, t]–module and Ker{Θ_{E}, X_{n} ⊗ k[s, t]} is generated by the same bottom row and the vector s^{n}v_{0} + s^{n−1}tv_{1} + ⋯ t^{n}v_{n}. Hence, Ker{Θ_{E}, X_{n} ⊗ k[s, t]}/ Im{Θ_{E}, X_{n} ⊗ k[s, t]} is generated by s^{n}v_{0} + s^{n−1}tv_{1} + ⋯ t^{n}v_{n} as a k[s, t]- module. Since the generator is in degree n, we conclude that the corresponding locally free sheaf of rank 1 is O_{ℙ^{1}}(−n).

We now compute Y_{n}. The graded k[s, t]-module Ker{Θ_{E}, X_{n}^{#} ⊗ k[s, t]} is generated by ⟨w_{0}, …, w_{n}⟩ in degree 0, and Im{Θ_{E}, X_{n}^{#} ⊗ k[s, t](−1)} is generated by ⟨sw_{0}+tw_{1}, sw_{1}+tw_{2}, …, sw_{n−1}+tw_{n}⟩, also in degree 0. Hence, on U_{0} = ℙ^{1}−Z(s), the restriction of Y_{n} is generated by w_{n}, with w_{0} = (−\frac{t}{s})^{n}w_{n}. We map Y_{n}(U_{0}) to K = k(t/s), the residue field at the generic point of ℙ^{1}, by sending w_{n} to 1. The image of w_{0} is (−\frac{t}{s})^{n}. On the other affine piece, U_{1} = ℙ^{1} − Z(t), the restriction of Y_{n} is generated by w_{0}, with the relation w_{n} = (−\frac{s}{t})^{n}w_{0}. We map this to K = k(s/t) by sending w_{0} to (−\frac{t}{s})^{n}. Hence, the vector bundle is given by the Cartier divisor (U_{0}, 1), (U_{1}, (−\frac{t}{s})^{n}). This divisor is equivalent to the Cartier divisor (U_{0}, 1), (U_{1}, (\frac{t}{s})^{n}) which correspond to the line bundle O_{ℙ^{1}}(n). Hence, Y_{n} ≃ O_{ℙ^{1}}(n). □

In the next Proposition we calculate explicitly the line bundles corresponding to the syzygies of the trivial modules, Ω^{n}k. For convenience, we use the notation H^{[1]}(M) for the bundle M^{[1]} associated to M as defined in (5.14).

Proposition 6.14. Let E = 𝔾_{a(1)}^{×r}. Then

H^{[1]}(Ω^{n}k) ≃ \begin{cases}O_{ℙ^{r−1}}(−\frac{np}{2}) & if n is even, \\ O_{ℙ^{r−1}}(−\frac{n+1}{2}p + 1) & if n is odd \end{cases}

for p odd and

H^{[1]}(Ω^{n}k) ≃ O_{ℙ^{r−1}}(−n)

for p = 2.

Proof. Let r = 2, and assume n ≥ 0. As in the proof of Prop. 6.13, the universal operator Θ_{E} = sx + ty where k[V (E)] = k[s, t]. The structure of a minimal kE ≃ k[x, y]/(x^{p}, y^{p})-projective resolution P_{•} → k is well known [10], with P_{n−1} = kG^{×n}. A set of generators a_{1}, …, a_{n} for P_{n−1} can be chosen so that Ω^{n}(k) is the submodule generated by the elements

x^{p−1}a_{1}, ya_{1} − xa_{2}, y^{p−1}a_{2} − x^{p−1}a_{3}, ya_{3} − xa_{4}, …, ya_{n−1} − xa_{n}, y^{p−1}a_{n}

for n even, and

xa_{1}, ya_{1} − x^{p−1}a_{2}, y^{p−1}a_{2} − xa_{3}, ya_{3} − x^{p−1}a_{4}, …, y^{p−1}a_{n−1} − xa_{n}, ya_{n}

for n odd.

Let n be even. For illustrational purposes, we include a picture of Ω^{4}k for p = 3,

\xymatrix{ &  &  & • \ar[dr] \ar[dl] &  &  &  &  &  & • \ar[dr] \ar[dl] \\ •^{v_{1}} \ar[dr] &  & •^{v_{2}} \ar[dl] &  & •^{v_{3}} \ar[dr] &  & •^{v_{4}} \ar[dl] \ar[dr] &  & •^{v_{5}} \ar[dl] &  & •^{v_{6}} \ar[dr] &  & •^{v_{7}} \ar[dl] \\  & • \ar[dr] &  & • \ar[dr] \ar[dl] &  & • \ar[dl] &  & • \ar[dr] &  & • \ar[dr] \ar[dl] &  & • \ar[dl] \\  &  & • &  & • &  &  &  & • &  & •}

The kernel of Θ_{E} = sx + ty on Ω^{n}k ⊗ k[s, t] is a submodule of a free k[s, t]- module generated by the “middle layer” of Ω^{n}k, that is, by v_{1} = x^{p−1}a_{1}, v_{2} = x^{p−2}(ya_{1} −xa_{2}), v_{3} = x^{p−3}y(ya_{1} −xa_{2}), …, v_{p} = y^{p−2}(ya_{1} −xa_{2}), v_{p+1} = y^{p−1}a_{2} − x^{p−1}a_{3}, …, v_{\frac{np}{2} +1} = y^{p−1}a_{n}. Everything below the middle layer which is in Ker Θ_{E} also lies in Im Θ_{E}^{p−1}. One verifies that the quotient

Ker{Θ_{E} : Ω^{n}k ⊗ k[s, t] → Ω^{n}k ⊗ k[s, t]}/ Im{Θ_{E}^{p−1} : Ω^{n}k ⊗ k[s, t] → Ω^{n}k ⊗ k[s, t]}

is generated by

s^{\frac{np}{2}} v_{1} − s^{\frac{np}{2} −1}tv_{2} + … ± t^{\frac{np}{2}} v_{\frac{np}{2} +1}.

Arguing as in the proof of Prop. 6.13, we conclude that the corresponding locally free sheaf of rank 1 is O_{P^{1}}(−\frac{np}{2}). The calculation for odd positive n is similar. For negative values of n, one verifies the formula by doing again a similar calculation with dual modules.

Now let r > 2, let i : F ⊂ E be a subgroup scheme isomorphic to 𝔾_{a(1)}^{×2}, and let f : ℙ(E) → ℙ(F) be the map induced by the embedding i. Since (Ω_{E}^{n}k) ↓_{F} ≃ Ω_{F}^{n} k ⊕proj, we conclude that H^{[1]}((Ω_{E}^{n}k) ↓_{F}) ≃ H^{[1]}(Ω_{F}^{n} k). Proposition 5.3 implies an isomorphism f^{∗}(H^{[1]}(Ω_{E}^{n}k)) ≃ H^{[1]}((Ω_{E}^{n}k) ↓_{F}). Hence,

f^{∗}(H^{[1]}(Ω_{E}^{n}k)) ≃ H^{[1]}(Ω_{F}^{n} k).

The proposition now follows from the observation that

\xymatrix{f : ℙ(F) ≃ ℙ^{1} \ar[r] & ℙ(E) ≃ ℙ^{r−1}}

induces an isomorphism on Picard groups via f^{∗}. □

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Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532

E-mail address: eric@math.northwestern.edu,  efriedla@usc.edu

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

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