Let $A$ be a finite-dimensional $k$-algebra and $U$ and $V$ two finite-dimensional projective $A$-modules (maybe neither the finiteness nor projectivity has to play a role, but these requirements are satisfied in my problem).

Now consider the $A$-bimodule \begin{align} M := \mathrm{Hom}_A (U,A) \otimes \mathrm{Hom}_A (A,V) \ , \end{align} where $\mathrm{Hom}_A (U,A)$ is a right $A$-module via right multiplication applied to $A$, and $\mathrm{Hom}_A (A,V)$ a left $A$-module by right action on $A$ in the source (again by right multiplication); this ensures $\mathrm{Hom}_A (A,V) =V$ as left $A$-modules.

I would like to compute the Hochschild homology $HH_*(A;M)$.

If you write down the corresponding Hochschild complex, there is an augmentation by $\mathrm{Hom}_A (U,V)$. Does that induce a quasi-isomorphism? Maybe such a result is known to experts?

Is there, more generally, a coefficient theorem I could use?

As usual, thank you for any hints.

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