Hochschild cohomology of Frobenius algebras

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