Definition 24.4.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of graded algebras on $(\mathcal{C}, \mathcal{O})$. A (right) *graded $\mathcal{A}$-module* or (right) *graded module* over $\mathcal{A}$ is given by a family $\mathcal{M}^ n$ indexed by $n \in \mathbf{Z}$ of $\mathcal{O}$-modules 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 $n$.

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

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