fact that $X^{\mathit{yon}} \otimes_A P$ is quasi-isomorphic to the $\tilde{A}$-module $P(\cdot,X)$, which has finite cohomology by assumption on $P$). Hence, we always get a functor $K_P: A^{\mathit{perf}} \rightarrow \tilde{A}^{\mathit{mod}}$. If $P$ is itself [*right perfect*]{}, which means that $P(\cdot,X)$ is itself a perfect $\tilde{A}$-module for any $X$, then $K_P$ takes $A^{\mathit{perf}}$ to $\tilde{A}^{\mathit{perf}}$.

Let’s suppose, for the sake of concreteness, that $P$ is right perfect. There are natural chain maps $$\label{eq:one-sided-deformation}
\mathit{CC}(A,A) \longrightarrow \mathit{hom}_{(A,\tilde{A})^{\mathit{mod}}}(P,P) \longleftarrow
\mathit{CC}(\tilde{A},\tilde{A}).
$$ Informally, one can think of these as follows. Given an infinitesimal deformation of either $A$ or $\tilde{A}$, they measure the failure of $P$ to remain a bimodule with respect to the deformed structure. This may remind the reader of , and indeed one can define a map (represented by the dashed arrow below) which fits into a homotopy commutative diagram $$\label{eq:grrr}
\xymatrix{
\mathit{CC}(A,A) \ar[d]^{\Gamma^{\mathit{perf}}} \ar[r] & \mathit{hom}_{(A,\tilde{A})^{\mathit{mod}}}(P,P) \ar@{-->}[d] & \ar[l] \ar[d]^{\Gamma^{\mathit{perf}}} \mathit{CC}(\tilde{A},\tilde{A}) \\
\mathit{CC}(A^{\mathit{perf}},A^{\mathit{perf}}) \ar[r]^-{(K_P)_*} &
\mathit{CC}(A^{\mathit{perf}},\tilde{A}^{\mathit{perf}}) & \ar[l]_-{(K_P)^*}
\mathit{CC}(\tilde{A}^{\mathit{perf}},\tilde{A}^{\mathit{perf}}).
}$$

Connections
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Fix a smooth affine algebraic curve ${\mathscr{S}}$ over $R$, with ${\mathscr{R}}= R[{\mathscr{S}}]$ its ring of functions. Recall that ${\mathscr{R}}$-modules correspond to quasi-coherent sheaves on ${\mathscr{S}}$; finitely generated modules to coherent sheaves; projective modules to vector bundles; and rank $1$ projective 