Let $k$ be a field, and let $A$ be a commutative $k$-algebra.
Let $M$ be an abelian group, and assume that it an $n$-$A$-module. That is: it has $n$ different $A$-module structures, and they are pairwise compatible. (For example, if $M_1,\dots,M_n$ are $A$-modules, then $M_1 \otimes_k \dots \otimes_k M_n$ is clearly an $n$-$A$-module).
Then, for each $i$, consider the $A$-modules
$$ \operatorname{Ext}^i_{A^{\otimes_k n}}(A, M) \,. $$
For $n=2$ this is just Hochschild cohomology. Here, $M$ has the structure of an $A^{\otimes_k n}$-module where the $i$-th $A$ operates on $M$ via the $i$-th action.
Has this been considered in the literature? are there any applications of this construction?
