REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 17, Pages 469–507 (September 5, 2013) S 1088-4165(2013)00440-5
COHOMOLOGICAL FINITE GENERATION FOR RESTRICTED LIE SUPERALGEBRAS AND FINITE SUPERGROUP SCHEMES
CHRISTOPHER M. DRUPIESKI
Abstract. We prove that the cohomology ring of a finite-dimensional re- stricted Lie superalgebra over a field of characteristic p>2 is a finitely- generated algebra. Our proof makes essential use of the explicit projective resolution of the trivial module constructed by J. Peter May for any graded restricted Lie algebra. We then prove that the cohomological finite generation problem for finite supergroup schemes over fields of odd characteristic reduces to the existence of certain conjectured universal extension classes for the gen- eral linear supergroup GL(m|n) that are similar to the universal extension classes for GLn exhibited by Friedlander and Suslin.
1. Introduction Motivated by the desire to apply geometric methods in the study of cohomol- ogy and modular representation theory, much attention has been paid in the past decades to the qualitative properties of the cohomology rings associated to various classes of Hopf algebras. Specifically, given a Hopf algebra A over the field k,ifthe cohomology ring H•(A, k) is a finitely-generated k-algebra, then one can establish, via a now well-tread path, a theory of support varieties that associates a geometric invariant to each A-module M. One of the most general results in this direction is due to Friedlander and Suslin [11]. Building on the earlier work of Venkov [24] and Evens [8] for finite groups, as well as the work of Friedlander and Parshall [9, 10] for restricted Lie algebras, Friedlander and Suslin proved cohomological finite gen- eration for any finite-dimensional cocommuative Hopf algebra over k,byusingthe fact that the category of finite-dimensional cocommuative Hopf algebras over k is naturally equivalent to the category of finite k-group schemes. Their argument involved embedding a finite group scheme in some general linear group, and then exhibiting certain universal extension classes for GLn that provided the necessary generators for the cohomology ring. More generally, Etingof and Ostrik [7] have conjectured that the Ext-algebra of an arbitrary finite tensor category is a finitely generated algebra. This conjecture encompasses the cohomology rings of arbitrary finite-dimensional Hopf algebras and of more general finite-dimensional algebras. In this paper we prove that the cohomology ring of an arbitrary finite-dimensional restricted Lie superalgebra is a finitely generated algebra. Similar results have also been claimed by Liu [14] and by Bagci [1], though their proofs are incomplete; see Remarks 3.5.6 and 3.6.1. We then begin the process of verifying Etingof and Ostrik’s conjecture for the class of finite tensor categories that arise as module cate- gories for finite-dimensional cocommuative Hopf superalgebras. Since the category of finite-dimensional cocommuative Hopf superalgebras is naturally equivalent to
Received by the editors January 9, 2013 and, in revised form, May 8, 2013. 2010 Mathematics Subject Classification. Primary 17B56, 20G10; Secondary 17B55.