Cohomology and prjectivity of modules for finite group schemes 
Christopher P. Bendel

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. 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(g) of a p-restricted Lie algebra g 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: 

Questions.  Let G be a finite group scheme 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 in Ext_{G_n}(M,M)  (for M finite dimensional) 
is nilpotent? 

It is shown here that when the connected component of G is unipotent there 
is a family of subgroup schemes (with simple structure) that provides an 
affirmative answer to both questions. These are referred to as elementary 
group schemes. 

Definition. Given a field k of characteristic p > 0 and a pair of non-negative 
integers r and s, define the elementary group scheme Er,s to be the product 
G_{a(r)} x E_s over k, where G_{a(r)} denotes the rth Frobenius kernel of the 
additive group scheme G_a, and E_s is an elementary abelian p-group of rank s 
(considered as a finite group scheme). The groups G_{a(0)} and E_0 are 
identified with the trivial group. 

The main results are the following two theorems: 

Theorem. Let k be a field of characteristic p > 0, G be a finite group scheme 
over k, and A be an associative, unital rational G-algebra. Suppose further 
that the (infinitesimal) connected component of the identity, G_0, of G is 
unipotent.  If z in H^n(G,A) satisfies the property that for any field 
extension K / k and any closed subgroup scheme (E_{r,s})_K of G_K  the 
cohomology class f*(z) in H^n((E_{r,s})_K, A_K) is nilpotent, then z is 
itself nilpotent. 

Theorem. Let k be a field of characteristic p > 0, G be a finite group 
scheme over k, and M be a rational G-module. Suppose further that the 
(infinitesimal) connected component of the identity, G_0, of G is unipotent 
and that all the points of G are k-rational.  Then M is projective as a 
G-module if and only if for every field extension K / k and closed subgroup 
scheme H of G_K with H = (E_{r,s})_K the restriction of M_K to H is 
projective.