\text{$j$ odd.}
\end{cases}$$ We refer to [@seidel04 Section 4] for full details, including the definition of the remaining maps $\mu^d_M$. By the same kind of argument as in , idempotent summands of objects in $A^{\mathit{tw}}$ (or indeed in $A^{\mathit{mod}}$) can also be represented in $A^{\mathit{mod}}$. One defines $A^{\mathit{perf}}$, the category of [*perfect modules*]{}, to be the full subcategory of $A^{\mathit{mod}}$ consisting of all objects that are quasi-isomorphic to an idempotent summand of an object in $A^{\mathit{tw}}$. It is easy to see that $A^{\mathit{perf}}$ is again proper.

Hochschild cohomology
---------------------

Let $\mathit{CC}(A,A)$ be the (reduced) Hochschild complex of $A$, and $\mathit{HH}(A,A)$ its cohomology. A Hochschild cochain $g$ is a sequence of multilinear maps $$g^d: \mathit{hom}_A(X_{d-1},X_d) \otimes \cdots \otimes \mathit{hom}_A(X_0,X_1) \longrightarrow
\mathit{hom}_A(X_0,X_d)[|g|-d],$$ $d \geq 0$, which vanish if one of the inputs is an identity morphism. The Hochschild differential is $$\begin{aligned}
(\partial g)^d(a_d,\dots,a_1) = \sum_{i,j} (-1)^{(|g|-1)(|a_1|+\cdots+|a_i|-i)}
\mu^{d-j+1}_A(a_d,\dots,g^j(a_{i+j},\dots,a_{i+1}),\dots,a_1) \\
+ \sum_{i,j} (-1)^{|g|+|a_1|+\cdots+|a_i|-i} g^{d-j+1}(a_d,\dots,\mu_A^j(a_{i+j},\dots,a_{i+1}),\dots,a_1).
\end{aligned}$$ We need to remind the reader briefly of the (partial) functoriality properties of Hochschild cohomology. Let $G: A \rightarrow \tilde{A}$ be a (strictly unital) $A_\infty$-functor. Then there are canonical chain maps $$\label{eq:functoriality}
\mathit{CC}(A,A) \xrightarrow{G_*} \mathit{CC}(A,\tilde{A}) \xleftarrow{G^*} \mathit{CC}(\tilde{A},\tilde{A}),$$ where the middle term is the Hochschild cochain complex of $A$ with 