Geometry of Infinitesimal Group Schemes

We consider aane group schemes G over a eld k of characteristic p > 0. Equivalently , we consider nitely generated commutative k-algebras kG] (the coordinate algebra of G) endowed with the structure of a Hopf algebra. The group scheme G is said to be nite if kG] is nite dimensional (over k) and a nite group scheme is said to be innnitesimal if the ((nite dimensional) algebra kG] is local. A rational G-module is a k-vector space endowed with the structure of a comodule for the Hopf algebra kG]. The abelian category of rational G-modules has enough injectives, so that Ext i G (M; N) is well deened for any pair of rational G-modules M; N and any non-negative integer i. Unlike the situation in characteristic 0, this category has many non-trivial extensions reeected by the cohomology groups we study. We sketch recent results concerning the cohomology algebras H (G; k) and the H (G; k)-modules Ext G (M; M) for innnitesimal group schemes G and nite dimensional rational G-modules M. These results, obtained with Andrei Suslin and others, are inspired by analogous results for nite groups. Indeed, we anticipate but have yet to realize a common generalization to the context of nite group schemes of our results and those for nite groups established by D. Quillen Q1], J. Carlson C], G. Avrunin and L. Scott A-S], and others. Although there is considerable parallelism between the contexts of nite groups and innnitesimal group schemes, new techniques have been required to work with innnitesimal group schemes. Since the geometry rst occuring in the context of nite groups occurs more naturally and with more structure in these recent developments, we expect these developments to ooer new insights into the representation theory of nite groups. The most natural examples of innnitesimal group schemes arise as Frobenius kernels of aane algebraic groups G over k (i.e., aane group schemes whose coordinate algebras are reduced). Recall that the Frobenius map