On the Cohomology Groups of an Associative Algebra

Abstract

The cohomology theory of associative algebras is concerned with the m-linear mappings of an algebra W into a two-sided W-module A. In this theory, the additive (2(m):$) of the m-linear mappings of 2I into $ plays a r6le analogous to that of the of m-dimensional cochains in combinatorial topology. A linear mapping of (2[(m):93) into (2(m+1):3) analogous to the coboundary operator of combinatorial topology and leading to the notion of $cohomology group has been defined by Eilenberg and MacLane'. The special cases of dimension one and two (linear and bilinear mappings of S into a two-sided 21-module) have appeared before in connection with the so-called first and second lemmas of Whitehead2. In a sense, the cohomology theory of associative algebras is degenerate: the 1-dimensional cohomology groups already determine the others. In fact, if $ is any two-sided Et-module, one can construct another two-sided S-module, (21:3), such that (for m > 2) the m-dimensional $-cohomology of 2t is isomorphic with the (m l)-dimensional (21:$)-cohomology of 2[ (Theorem 3.1). The present paper is concerned primarily with the connections between the structure of an algebra and the vanishing of its cohomology groups. It is shown that an algebra is separable if and only if its cohomology groups vanish (Theorem 4.1). This is a generalization of a result obtained previously for the case of a non-modular ground field3. The 2-dimensional cohomology groups of an algebra are directly connected with the extensions of A, i.e. algebras e3 of which 21 is a homomorphic image. In particular, the condition that 2-dimensional cohomology groups of 21 vanish signifies that every extension of 2f has the form e= * + S, where 2[* is a subalgebra isomorphic with 21 and S is the kernel of the homomorphism of e3 onto 51 (Theorem 6.1). This is connected with the (generalized) third structure theorem of Wedderburn which may be stated by saying, that the 2-dimensional cohomology groups of a separable algebra More generally, one would be interested in the structural significance of the condition Cm: all m-dimensional cohomology groups vanish. Theorem 3.1 implies that C.+, is a consequence of Cm, for m > 1. But it is an open question whether or not Cm and C.+, are equivalent.