On the cohomology of Frobenius algebras

Abstract

Let R be a commutative ring and let A be a finitely generated free Frobenius algebra over R. In [4], Nakayama constructed a complete (Hochschild) cohomology theory of A by means of his automorphism. The purpose of the paper is to investigate some structures of the complete cohomology groups H’(A, -) (r E Z) of A, in particular a periodicity with respect to r; A is said to have periodic cohomology of period d # 0 if H”(A, M) 2: Hn+d(A, M) holds for all two-sided A-modules M and all integers n. One of the most important parts of the cohomology theory of finite groups may be the periodicity of the cohomology. In such a case a fundamental approach is one with a cup product on the Tate cohomology (cf. [2,3,5]). Now [3, Chapter XI, Exercises 1 and 21 gives a prototype, one of the ‘products of the second kind’, of the cup product which seems to be appropriate to the complete (Hochschild) cohomology in dealing with its periodicity. Modifying this, in parallel with the cohomology theory of finite groups, we proceed to a general theory which is needed here and we will prove a basic theorem for the periodicity of the cohomology. In Section 1 we explain the complete cohomology theory of Frobenius algebras and dimension-shifting of the cohomology in a bit more detail after Nakayama [4]. In Section 2 we introduce the definition of the cup product on the complete cohomology and construct it explicitly. In the latter half we prove two basic properties: anti-commutativity and associativity. In Section 3, along the same lines as Cartan and Eilenberg [3, Chapter XII, Section 61, we prove the