Constructions for infinitesimal group schemes

Let G be an infinitesimal group scheme over a field k of positive characteristic p. We introduce the global p-nilpotent operator $\Theta_G: k[G] \to k[V(G)]$, where V(G) is the scheme which represents 1-parameter subgroups of G. This operator applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain G-modules (including those of constant Jordan type), we employ the global p-nilpotent operator to associate various algebraic vector bundles on the projective scheme $\bP(G)$, the projectivization of the scheme of one-parameter subgroups of 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 $\bP(G)$.


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.
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.
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 , a p-nilpotent operator on the free, coherent sheaf M ⊗ O P(G) on P(G). We verify in Proposition 4.8 that Θ 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 Θ j G be locally free. In the fifth section, we initiate an investigation of the algebraic vector bundles Ker{ Θ j G , M}, Im{ Θ j G , 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.

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, (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). 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]): 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 g ( [22, I.8.5]). The group algebra of 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 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 , 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 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 is defined by raising each matrix entry to the p th power. The coordinate algebra of GL n(r) is given by whereas the group algebra of GL n(r) is given as the k-space of linear functionals k[GL n(r) ] to k. The coproduct 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 The coordinate algebra of SL 2(2) is given by whereas the group algebra of SL 2(2) is given as 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 p 1,2 , X p 2,1 , (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 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 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 is representable by an affine scheme V r (G) = Spec k[V r (G)]. Namely, this functor is naturally isomorphic to the functor 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 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 Conversely, any one-parameter subgroup G a(s ′ ),R → G R can be decomposed as 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), 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 α : G a(r),R → GL n,R of R-group schemes corresponding to an r-tuple 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 where for any p-nilpotent matrix A ∈ GL n (S) we set 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 [r] p (gl n ); the identification is given by sending α = (α 0 , . . . , α r−1 ) ∈ N (1) V (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 → g R is the associated 1-parameter subgroup of g.
(2) V (G a(r) ) ≃ A r . The r-tuple a = (a 0 , . . . , a r−1 ) ∈ R ×r = A r (R) corresponds to the 1-parameter subgroup µ 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 (gl n ), the variety of r-tuples of pairwise commuting, p-nilpotent n × n matrices. The embedding i r,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 ). r) ] is given by sending X i,j for some 1 ≤ i, j ≤ n to the (i, j)-entry of the polynomial p 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 α (t) multiplying s p ℓ is α ℓ for 0 ≤ ℓ < r, whereas coefficients of p 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 * 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 α = (α 0 , α 1 ). This variety is given explicitly as the affine scheme with coordinate algebra . We give an explicit description of the map on coordinate algebras (2) ] ≃ R[T ]/T p 2 induced by the one-parameter subgroup exp α : G a(2),R → SL 2(2),R . This description follows immediately from the general discussion in the previous example. Let α = . Then exp * α is determined by the formulae . 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): 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 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 , 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), 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: 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.
]) Let G be an infinitesimal group scheme of height ≤ r. There is a natural homomorphism of k-algebras with nilpotent kernel whose image contains the p r -th power of each element of k[V (G)]. Hence, the associated morphism of schemes If M is a finite dimensional kG-module, then Ψ restricts to a homeomorphism In the special case of of G = GL n(r) the isogeny Ψ has an explicitly constructed inverse. 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. (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 (g) M ⊂ V (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 u 0 , . . . , u 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 a : G a(r),K → G a(r),K for some r-tuple a = (a 0 , . . . , a r−1 ) of K-rational points as in Example 1.12 (2). The condition that µ a be a point of V (G a(r) ) M is the condition that (µ a • ǫ) * (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[ u]/ u p -module where u = a r−1 u 0 + a p r−2 u 1 · · · + a p r−1 0 u 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 α : G a(r),K → GL n(r),K indexed by rtuples α = (α 0 , . . . , α r−1 ) ∈ M n (K) of p-nilpotent, pairwise commuting matrices such that (exp α, * • ǫ) * (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 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 ( is projective (the Steinberg module for SL 2(1) ) but S p−1 is not itself projective. Hence, V (G) Sp−1 is a proper non-trivial subvariety of V (G). Using the notation introduced in Example 1.12(4), we have (see [26, 6.10]). V (G) Sp−1 can be described as the subscheme of V (G) defined by the equations λ1 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,

Global p-nilpotent operators
In this section, we introduce in Definition 2.1 and study the global p-nilpotent operator , 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 as formalized in Definition 2.3. Before giving definitions, we mention as motivation the example G = G ×2 a (1) . In this case, the group algebra kG equals k[x, y]/(x p , y p ), the scheme of 1-parameter subgroups equals In this special case, Θ G takes the form If M is a kG-module, then the k[s, t]-linear operator Θ M is given by "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
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 U G,r : G a(r),A / / G A , the subgroup corresponding to the identity map on A. The subgroup U G,r induces a map on coordinate algebras Definition 2.1. Let G be an algebraic affine group scheme over k. We define to be the k-linear, p-nilpotent functional defined by the composition (2.1.1) As an element of Hom(k , Θ G,r can be equivalently defined as Thus, Θ G,r as given in (2.1.1) satisfies the property that its composition with The following proposition justifies using the simplified notation Recall the canonical projection p r ′ ,r : G a(r ′ ) ։ G a(r) , and the induced closed Proposition 2.2. Let G be an an infinitesimal group scheme and let r ′ ≥ r. Let be the surjective homomorphism corresponding to the canonical embedding i r,r ′ : Proof. Consider the composition 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 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 which implies the equality of maps of group algebras 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 , 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 .
Remark 2.4. The fact that Θ M is p-nilpotent follows immediately from (2.1.2) since u p r−1 = 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 : determines the pairing of k-schemes As a pairing of representable functors of commutative k-algebras A, this pairing sends (1) Let G = GL m(1) ≡ gl m , with group algebra kgl m = u(gl m ). Then the composition of For a general p-restricted Lie algebra g, we have where the sum is over basis elements X ∈ g # with image 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 Observe that this formula agrees with the presentation of a "generic" π-point for u(sl 2 ) as given in [17, 2.5].
is graded in such a way that x i has degree p i (see Proposition 2.11 below). We com- ] 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 : is given by the "generic" additive polynomial: . . , 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: By [25, 1.4], v i can expressed in terms of the algebraic generators u j of kG a(r) via the following formulae 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) )]: 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 ×r n ]/I, a quotient of the coordinate algebra of the variety M ×r n 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 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 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 when taking products of matrix coefficients, one uses the usual rule for matrix multiplication); 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 , and set 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 i 1,2 X j 2,1 (X 1,1 − 1) ℓ for i + j + ℓ ≤ p is given by 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)). (1) . Then G corresponds to the abelian p-nilpotent Lie algebra g ⊕r a , and kG = k[u 0 , . . . , 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 = 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 g as the k[N p (g)]-linear endomorphism is given by the complicated, but explicit formula (2.6.3). We conclude that 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 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) h ℓ act trivially on M . Hence, the map is given by the matrix When viewing group schemes as functors, it is often convenient to think of 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.
Consequently, the following square of k-linear maps commutes: Thus, for any rational G-module M we have a compatibility of coactions on M : 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 . 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: Concatenating (2.9.4) on the right with the commutative square of linear maps and with the inclusions on the left, we obtain a commutative diagram: 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 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 Hence, the coordinate functions of the matrix α i have grading p i and, therefore, 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 ] is a map of graded algebras.
Proposition 2.11. For any algebraic affine group scheme G and integer r > 0, the k-linear map If G is infinitesimal, then this is equivalent to the following: . . + 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 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: 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: 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: 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), α = (α 0 , . . . , α r−1 ) ∈ V r (GL N )(S). Then a • α = (aα 0 , a p α 1 , . . . , a p r−1 α r−1 ), so exp 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, α) ∈ (G a(r) × V r (GL N ))(S), (2.11.2) becomes Commutativity of (2.11.3) is implied by the evident equality exp a•α (s) = exp α (as).
Consequently, we have a commutative diagram on coordinate algebras corresponding to (2.11.1): ] 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 The composition of the lower horizontal and left vertical maps of (2.11.4) sends λ to On the other hand, the composition of the right vertical and upper horizontal maps of (2.11.4) sends λ to 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 . 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 r+1) ]. Yet this map clearly multiplies degree by p.

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.
be the associated 1-parameter subgroup (for r ≥ ht(G)). We define the local pnilpotent operator at v, θ v , to be 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 α for the local p-nilpotent operator at α = (α 0 , . . . , α r−1 ) ∈ V (GL n(r) ) ≃ where k(α) is the residue field of 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 We refer to the p-tuple For simplicity, we introduce the following notation. We refer to this Jordan type as the local Jordan type of M at v ∈ V (G). 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 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 α ∈ k(α) GL n(r) of (3.1.1) on M equals the coefficient of T p r−1 of 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 and thus the comodule structure of M k(α) for k(α)[G a(r) ] is given by The proposition follows from the fact that u r−1 : k(α)[G a(r) ] → k(α) is given by reading off the coefficient of T p r−1 ∈ k(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 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 ad K → g ad K , JType(g ad , θ X ) = JType(X). (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 where β ℓ are matrices determined by α i as in Proposition 1.7, we conclude JType(V n , θ α ) = JType(α r−1 ).
Then JType(V 2 , θ α ) equals the Jordan type of the matrix We extend Example 3.5(3) by considering tensor powers V ⊗d n of the canonical rational representation of GL n restricted to GL n (2) . In this example, the role of both entries of the pair α = (α 0 , α 1 ) is non-trivial.
Example 3.6. Consider the N = n d -dimensional rational GL n -module M = V ⊗d n 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 i1 ⊗ · · · ⊗ e i d ; 1 ≤ i j ≤ n}, where {e i ; 1 ≤ i ≤ n} is a basis for V n . Let {X i1,j1;...,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 .
To simplify matters even further, consider the special case (α 0 ) 2 = 0. For 1 ≤ d < p, θ (α0,α1) on M is given by the N × N -matrix For d = p, the action of θ (α0,α1) on M is given by the N × N -matrix 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 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, 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 α on a k GL n(r) -module M and θ 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 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 so that F s can be viewed as a self-map of 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).
The following definition introduces interesting classes of kG-modules which have special local behavior.
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 operatorsũ i , 0 ≤ i < p on M . Moreover, for a = (0, . . . , 1, 0, . . . , 0), with 1 at the i th spot, JType(M, θ 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 a = (0, . . . , 1, 0, . . . , 0), then eachũ 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.   (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 < √ 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.

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 : 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 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 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. 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 is given on points by sending x ∈ P(G) to the equivalence class of the π-point µ v, * • ǫ for any v ∈ V (G)\{0} projecting to x. In particular, equivalence classes of generic π-points of G are represented by (µ v, * • ǫ) as v ∈ V (G) runs through the (scheme-theoretic) generic points of V (G).
(sending u to θ Ki ) 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. 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 .  the map obtained by tensoring (4.6.1) with O P(G) (n). For any point x ∈ P(G), we use the notation 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 and similarly for kernels.
In the next section, we shall be particularly interested in kernels and images of Θ G . The following proposition relates the fibers of the kernel and image of the global p-nilpotent operator Θ 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.
Then we define Let v ∈ V (G) be any point projecting onto x. We have k(x) = k(v). By Def. 3.1, We observe that ΘG s (x) = ΘG(v) s(v) for any v projecting onto x (and, in particular, is independent of the choice of v). Therefore, The equalities(4.8.2) and (4.8.1) now follow. Remark 4.10. Proposition 4.8 immediately generalizes to Θ j G 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: In what follows, we shall use the following abbreviations: 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{ Θ j G , M} (and thus Ker{ Θ j G , 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. Proof. Assume that the function x → dim k(x) ( M ⊗ OX k(x)) is constant on each connected component of X. To prove that M 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 M ) 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 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 ⊗ Rp k(p) ≃ M ⊗ Rp 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 ⊂ ( Recall that Q is a free module so that We shall find it convenient to "localize" the notion of a kG-module of constant j-rank given in Definition 3.12 as follows.
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 Θ j U : M |U → M(jp r−1 ) |U be the restriction to U of the j th iterate of Θ G on M = M ⊗ O X as given in (4.6.1). Then the following are equivalent for some fixed j, 1 ≤ j < p: Moreover, each of these conditions implies that 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 ) vanishes on locally free sheaves, taking the O P(G) -dual of a short exact sequence of vector bundles (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 ∈ H n (u(g), k) ≃ Ext 1 u(g) (Ω n−1 (k), k), we consider the extension of u(g)-modules [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 p 0 , . . . , u p p−1 ). 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 .
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 Let G = G a(r) , and set I equal to the augmentation ideal of kG ≃ k[u 0 , . . . , u p−1 ]/(u p 0 , . . . , u p p−1 ). Consider any ideal J of kG with the property that I i ⊂ J for some i, i ≤ p − 1. Note that for any a ∈ A r , and any j ≤ p − i, . Indeed, since I i is a module of constant Jordan type, it suffices to check the statement for θ 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{θ j a : J → J}) = pj for any j ≤ p − i and any 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 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.
(1) For 0 ≤ λ ≤ 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: [21, 7.2] or [2, §1]). Recall that Θ G = xe + yf + zh (see Example 2.8(1)). Hence, the operator is represented by the matrix For p ≤ m ≤ 2p − 2, V m has a decomposition series which can be represented as follows: S µ e e e e e e e S μ~} 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) One may readily determine the rank of various bundles of P(G) associated to modules of constant Jordan type using the next proposition. . Then for any j, 1 ≤ j < p, In particular, Proof. The formula (5.10.1) is the formula for the rank of u j on the k[u]/u p -module . 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{ Θ j G , 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. . We see that these modules are precisely those whose associated vector bundles are trivial vector bundles. Proof. If M has a constant j-image property then Im{ Θ j G , M} is a free O X -module generated by I(j). Conversely, assume that Im{ Θ j G , 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 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{ Θ G , M # } is locally free of rank 2 but not free, since the global sections have dimension one. On the other hand, Im{ Θ 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{ Θ G , M} is free of rank 2 whereas Im{ Θ 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 Θ p G = 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: 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 In particular, V is projective as a k[t]/t p -module if and only if V [1] Proof. The sheaf M [1] is locally free by Lemma 5. 15  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 A similar statement applies with θ v in place of Θ G . For any scheme Y and any complex of O Y -modules there is a natural pairing induced by the evident pairing for any x ∈ X. By naturality, the specialization of f at a point x corresponds to the map f x : N k(x) ) # 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, is an isomorphism for any x ∈ X. Therefore, Consideration of M [1] leads to another characterization of projective kG-modules. Proof. To prove that Γ(P(G), Ker{ Θ j G , 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 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: , 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 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{ Θ 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  ). Hence, the inequality dim H 0 (G, M ) < p i=1 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 pmodule, then M is free as an R[t]/t p -module.
Proof. Let m 1 , . . . , m s ∈ M be such that m 1 , . . . , 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 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.
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 .
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 [1] (x) = 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 OX (M [1] ) ∩ Spec(A F ) 0 . Since localization is exact, this is equivalent to [1] ) = 0, and thus that v / ∈ Supp AF ((M ⊗ A F ) [1] ).
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) ≡ sl 2 in the following proposition. Proposition 6.3. Let G = sl 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 where i : P 1 → P(G) is the isomorphism (5.8.1).
By Theorem 4.13 (8) and Nakayama's Lemma, it suffices to prove that 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).
We shall omit the subscript G is κ G when the group scheme is clear from the context. Note that since Θ p = 0, κ p returns the entire class [Q ⊗ O P(G) ].
The second conclusion of the following Proposition is in sharp contrast with Proposition 6.5.
Proposition 6.7. Let G = sl 2 , and denote by St the Steinberg module for sl 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, 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 np 2 +1 = y p−1 a n . Everything below the middle layer which is in Ker Θ E also lies in Im Θ p−1 E . One verifies that the quotient Ker{Θ E : Ω n k ⊗ k[s, t] → Ω n k ⊗ k[s, t]}/ Im{Θ p−1 E : Ω n k ⊗ k[s, t] → Ω n k ⊗ k[s, t]} is generated by s . Arguing as in the proof of Prop. 6.13, we conclude that the corresponding locally free sheaf of rank 1 is O P 1 (− 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 ×2 a (1) , and let f : P(E) → P(F ) be the map induced by the embedding i. Since (Ω n E k) ↓ F ≃ Ω n F k ⊕ proj, we conclude that H [1] ((Ω n E k) ↓ F ) ≃ H [1] (Ω n F k). Proposition 5.3 implies an isomorphism f * (H [1] (Ω n E k)) ≃ H [1] ((Ω n E k) ↓ F ). Hence, f * (H [1] (Ω n E k)) ≃ H [1] (Ω n F k). The proposition now follows from the observation that f : P(F ) ≃ P 1 / / P(E) ≃ P r−1 induces an isomorphism on Picard groups via f * .