Relations Between the Cohomology of an Algebraic Group and Its Infinitesimal Subgroups

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. IntroductionIn this paper we relate Lie-algebra and Hochschild cohomologies. The aim is to carry information from Lie cohomology to algebraic group cohomology along the chain of infinitesimal neighborhoods of the identity. (The ith infinitesimal neighborhood G' is the kernel of the ith power of the Frobenius morphism of G.)The vehicle for transmitting information from the cohomology group H2(G'-', V) of G'-' with values in a G-module V to the cohomology group H2(G', V) is an exact sequence of 2-cohomologies, applied to the exact sequence 1-G G'-GG'-11 (Theorem 5.3). This is used to show that the composition factors of the G-module H2(G',k) are Frobenius powers of the composition factors of H2(G 1, k) (Corollary 5.4).The main illustration of the transmission process comes in showing that the group cohomology H2(G, k) is (0) at the trivial module k when the Lie cohomology is (0) at k (Theorem 6.3). To get the process started, H2(G ', k) and H2(L, k) are related in Section 8. There, when H2(L, k) is (0), H2(G', k) is shown to be isomorphic as a G-module to the dual of the adjoint representation of G on L. This computation is made in the category of p-Lie algebras.In order to make the passage to p-Lie algebras, we show in Section 7 that, for Va a G-module and Vaj its first neighborhood, the groups H2(G 1, V,) and H2(G', Va') are isomorphic. G1 and Va' lie in a category which is equivalent to the category of p-Lie algebras, and we are free to pass to the latter category.In showing that H2(G, k) is (0) when H2(L, k) is (0), we conclude from the knowledge accumulated above about the composition factors of H2(G', k) that the canonical image of H2(G, k) in H2(G, k) is (0) for each i. At the same time, we show that the canonical map from H2(G, k) to lim H2(G',k) is injective Manuscript