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Rational cohomology of unipotent group schemes

Research paper by Eric M. Friedlander

Indexed on: 15 Feb '17Published on: 15 Feb '17Published in: arXiv - Mathematics - Representation Theory



Abstract

We investigate the rational cohomology algebra of various unipotent group schemes defined over an algebraically closed field $k$ of characteristic $p > 0$. For the $r$-th Fobenius kernel $U_{(r)}$ of many unipotent algebraic groups $U$, we construct an algebra $\overline{S}^*(U_{(r)})$ given by explicit generators and relations together with a map $\overline {\eta}_{U,r}: \overline{S}^*(U_{(r)}) \to H^*(U_{(r)},k)$ of graded $k$-algebras which is an isomorphism modulo nilpotents. To achieve our computations, we refine more qualitative and more generally applicable results of A. Suslin, C. Bendel, and the author. Our techniques incorporate the interplay between Lyndon-Hochschild-Serre spectral sequences applied to the descending central series of $U$ and spectral sequences introduced by H. Andersen and J. Jantzen. Our work is motivated by the challenge of understanding the stabilization with respect to $r$ of the cohomology $H^*(U_{(r)},k)$ and the natural map $H^*(U,k) \to \varprojlim_r H^*(U_{(r)},k)$.