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?
The cohomology ring of a finite group scheme