I am sorry if this is a naive question.

Let $k$ be a field, and let $A$ be a finitely generated commutative $k$-algebra.

Let $M$ be a finite $A$-module.

Consider the Hochschild cohomologies of $A$ with coefficients in $M$.

Obviously, they are finitely generated modules over the enveloping algebra $A\otimes_k A$. But are they finitely generated over $A$ as well?

Thanks for any comments or remarks.