Definition 24.13.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. A (right) *differential graded $\mathcal{A}$-module* or (right) *differential graded module* over $\mathcal{A}$ is a cochain complex $\mathcal{M}^\bullet $ endowed with $\mathcal{O}$-bilinear maps

called the multiplication maps with the following properties

multiplication satisfies $(xa)a' = x(aa')$,

the identity section $1$ of $\mathcal{A}^0$ acts as the identity on $\mathcal{M}^ n$ for all $n$,

for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $x \in \mathcal{M}^ n(U)$, and $a \in \mathcal{A}^ m(U)$ we have

\[ \text{d}^{n + m}(xa) = \text{d}^ n(x)a + (-1)^ n x\text{d}^ m(a) \]

We often say “let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module” to indicate this situation. A *homomorphism of differential graded $\mathcal{A}$-modules* from $\mathcal{M}$ to $\mathcal{N}$ is a map $f : \mathcal{M}^\bullet \to \mathcal{N}^\bullet $ of complexes of $\mathcal{O}$-modules compatible with the multiplication maps. The category of (right) differential graded $\mathcal{A}$-modules is denoted $\textit{Mod}(\mathcal{A}, \text{d})$.

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