Hochschild cohomology of Frobenius algebras

Abstract

Let k be a field and let A be a Frobenius algebra over k. Assume that the Nakayama automorphism of A associated to a Frobenius homomorphism of A has finite order m, and k has a m-th primitive root of unity. Then, A has a natural Z/mZ-gradation. Consider the decomposition of the Hochschild cohomology HH*(A), of A with coefficients in A, induced by this gradation. We prove that just the 0-degree component of HH*(A) is non trivial. Moreover, we prove that if A is a strongly Z/mZ-graded algebra, then Z/mZ acts on the Hochschild cohomology HH*(A_0), of the 0-degree component of A, and HH*(A) is the set of invariants of this action.