Remark 22.37.10. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras which are derived equivalent, i.e., such that there exists an $R$-linear equivalence $D(A, \text{d}) \to D(B, \text{d})$ of triangulated categories. We would like to show that certain invariants of $(A, \text{d})$ and $(B, \text{d})$ coincide. In many situations one has more control of the situation. For example, it may happen that there is an equivalence of the form

for some differential graded $(A, B)$-bimodule $\Omega $ (this happens in the situation of Proposition 22.37.6 and is often true if the equivalence comes from a geometric construction). If also the quasi-inverse of our functor is given as

for a differential graded $(B, A)$-bimodule $\Omega '$ (and as before such a module $\Omega '$ often exists in practice). In this case we can consider the functor

on derived categories of bimodules (use Lemma 22.28.3 to turn bimodules into right modules). Observe that this functor sends the $(A, A)$-bimodule $A$ to the $(B, B)$-bimodule $B$. Under suitable conditions (e.g., flatness of $A$, $B$, $\Omega $ over $R$, etc) this functor will be an equivalence as well. If this is the case, then it follows that we have isomorphisms of Hochschild cohomology groups

For example, if $A = H^0(A)$, then $HH^0(A, \text{d})$ is equal to the center of $A$, and this gives a conceptual proof of the result mentioned in Remark 22.37.9. If we ever need this remark we will provide a precise statement with a detailed proof here.

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