Hochschild cohomology of Frobenius algebras

Let k k be a field, A A a finite-dimensional Frobenius k k -algebra and ρ : A → A \rho \colon A\to A , the Nakayama automorphism of A A with respect to a Frobenius homomorphism φ : A → k \varphi \colon A\to k . Assume that ρ \rho has finite order m m and that k k has a primitive m m -th root of unity w w . Consider the decomposition A = A 0 ⊕ ⋯ ⊕ A m − 1 A = A_0\oplus \cdots \oplus A_{m-1} of A A , obtained by defining A i = { a ∈ A : ρ ( a ) = w i a } A_i = \{a\in A:\rho (a) = w^i a\} , and the decomposition H H ∗ ( A ) = ⨁ i = 0 m − 1 H H i ∗ ( A ) \mathsf {HH}^*(A) = \bigoplus _{i=0}^{m-1} \mathsf {HH}_i^*(A) of the Hochschild cohomology of A A , obtained from the decomposition of A A . In this paper we prove that H H ∗ ( A ) = H H 0 ∗ ( A ) \mathsf {HH}^*(A) = \mathsf {HH}^*_0(A) and that if the decomposition of A A is strongly Z / m Z \mathbb {Z}/m\mathbb {Z} -graded, then Z / m Z \mathbb {Z}/m\mathbb {Z} acts on H H ∗ ( A 0 ) \mathsf {HH}^*(A_0) and H H ∗ ( A ) = H H 0 ∗ ( A ) = H H ∗ ( A 0 ) Z / m Z \mathsf {HH}^*(A) = \mathsf {HH}_0^*(A) = \mathsf {HH}^*(A_0)^{\mathbb {Z}/m \mathbb {Z}} .