Cohomology and projectivity of modules for finite group schemes

Abstract

Let G be a finite group scheme over a field k, that is, an affine group scheme whose coordinate ring k[G] is finite dimensional. The dual algebra k[G]* ≡ Homk(k[G], k) is then a finite dimensional cocommutative Hopf algebra. Indeed, there is an equivalence of categories between finite group schemes and finite dimensional cocommutative Hopf algebras (cf. [19]). Further the representation theory of G is equivalent to that of k[G]*. Many familiar objects can be considered in this context. For example, any finite group G can be considered as a finite group scheme. In this case, the algebra k[G]* is simply the group algebra kG. Over a field of characteristic p > 0, the restricted enveloping algebra u([gfr ]) of a p-restricted Lie algebra [gfr ] is a finite dimensional cocommutative Hopf algebra. Also, the mod-p Steenrod algebra is graded cocommutative so that some finite dimensional Hopf subalgebras are such algebras.Over the past thirty years, there has been extensive study of the modular representation theory (i.e. over a field of positive characteristic p > 0) of such algebras, particularly in regards to understanding cohomology and determining projectivity of modules. This paper is primarily interested in the following two questions:Questions1·1. Let G be a finite group scheme G over a field k of characteristic p > 0, and let M be a rational G-module.(a) Does there exist a family of subgroup schemes of G which detects whether M is projective?(b) Does there exist a family of subgroup schemes of G which detects whether a cohomology class z ∈ ExtnG(M, M) (for M finite dimensional) is nilpotent?

Infinite dimensional modules for Frobenius kernels

“…The purpose of this section is to prove a local criterion for projectivity: M is projective if and only if it is locally projective, which is the content of Theorem 1.6. This projectivity detection result was proved in [4] for unipotent infinitesimal group schemes.…”

Stratification for module categories of finite group schemes

“…The essential new ingredient is a "subgroup reduction principle", Theorem 3.7 which allows us to extend the detection theorem from the case of a connected finite group scheme to an arbitrary one. Theorem 3.7 relies on a remarkable result of Suslin (see also [2] for the special case of a unipotent finite group scheme) on detection of nilpotents in the cohomology ring H * (G, Λ) for a G-algebra Λ, generalising work of Quillen and Venkov for finite groups.…”

“…For finite groups, Dade's Lemma was generalised to infinite dimensional modules by Benson, Carlson, and Rickard [5]. For connected finite groups schemes the analogue of Dade's Lemma is that projectivity can be detected by restriction to one-parameter subgroups, and was proved in a series of papers by Suslin, Friedlander, and Bendel [38], Bendel [2], and Pevtsova [34,35].…”

Stratification and $$\pi $$ π -cosupport: finite groups

“…We note that G a(r) is the rth Frobenius kernel of the additive group scheme G a over k; see, for instance, [20, I.9.4] Example 1.5 (Quasi-elementary group schemes). Following Bendel [3], a group scheme over a field k of positive characteristic p is said to be quasi-elementary if it is isomorphic to G a(r) × (Z/p) s . Its group algebra structure is the same as that of an elementary abelian p-group.…”