Graded group schemes

Abstract

We define graded group schemes and graded group varieties and develop their theory. We give a generalization of the result that connected graded bialgebras are graded Hopf algebra. Our result is given for a broader class of graded Hopf algebras: the coordinate rings of graded group varieties. We also give a classification for graded group algebras and graded group varieties. We proceed to using tools of representation theory to get a better understanding of the cohomology of graded group schemes. For that, we focus our attention on the case in which the base field is of characteristic p > 0. Using as inspiration the work on [SFB97b], [SFB97a] and [FP05], we define graded p-points and build the theory of graded 1-parameter subgroups. We give a natural homomorphism from the cohomology of a graded group scheme to the coordinate ring of graded 1-parameter subgroups and we show that it is an F-monomorphism.