The Stacks project

Lemma 22.28.2. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. Let $M$ be a right differential graded $B$-module. There is a $1$-to-$1$ correspondence between $(A, B)$-bimodule structures on $M$ compatible with the given differential graded $B$-module structure and homomorphisms

\[ A \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(M, M) \]

of differential graded $R$-algebras.

Proof. Let $\mu : A \times M \to M$ define a left differential graded $A$-module structure on the underlying complex of $R$-modules $M^\bullet $ of $M$. By Lemma 22.13.1 the structure $\mu $ corresponds to a map $\gamma : A \to \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , M^\bullet )$ of differential graded $R$-algebras. The assertion of the lemma is simply that $\mu $ commutes with the $B$-action, if and only if $\gamma $ ends up inside

\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(M, M) \subset \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , M^\bullet ) \]

We omit the detailed calculation. $\square$

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