Hochschild cohomology of Frobenius algebras

Abstract

Let k be a field, A a finite dimensional Frobenius k-algebra and φ: A→ A, the Nakayama automorphism of A with respect to a Frobenius homomorphism φ: A→ A,. Assume that φ has finite order m and that k has a primitive m-th root of unity w. Consider the decomposition A = A_0 ⊕...⊕ A_{m-1} of A, obtained defining A_i = {a ∈ A:φ(a) = w^i a}, and the decomposition HH^*(A) = ⊕_{i=0}^{m-1} HH_i^*(A) of the Hochschild cohomology of A, obtained from the decomposition of A. In this paper we prove that HH^*(A) = HH^*_0(A) and that if decomposition of A is strongly Z/m Z-graded, then Z/mZ acts on HH^*(A_0) and HH^*(A) = HH_0^*(A) = HH^*(A_0)^{Z/mZ}.