Please skip this section. This section is the analogue of Differential Graded Algebra, Example 22.25.6. For our conventions on graded categories, please see Differential Graded Algebra, Section 22.25.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of graded algebras on $(\mathcal{C}, \mathcal{O})$. We will construct a graded category $\textit{Mod}^{gr}(\mathcal{A})$ over $R = \Gamma (\mathcal{C}, \mathcal{O})$ whose associated category $(\textit{Mod}^{gr}(\mathcal{A}))^0$ is the category of graded $\mathcal{A}$-modules. As objects of $\textit{Mod}^{gr}(\mathcal{A})$ we take right graded $\mathcal{A}$-modules (see Section 24.4). Given graded $\mathcal{A}$-modules $\mathcal{L}$ and $\mathcal{M}$ we set

$\mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}(\mathcal{L}, \mathcal{M}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits ^ n(\mathcal{L}, \mathcal{M})$

where $\mathop{\mathrm{Hom}}\nolimits ^ n(\mathcal{L}, \mathcal{M})$ is the set of right $\mathcal{A}$-module maps $f : \mathcal{L} \to \mathcal{M}$ which are homogeneous of degree $n$. More precisely, $f$ is given by a family of maps $f : \mathcal{L}^ i \to \mathcal{M}^{i + n}$ for $i \in \mathbf{Z}$ compatible with the multiplication maps. In terms of components, we have that

$\mathop{\mathrm{Hom}}\nolimits ^ n(\mathcal{L}, \mathcal{M}) \subset \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{L}^{-q}, \mathcal{M}^ p)$

(observe reversal of indices) is the subset consisting of those $f = (f_{p, q})$ such that

$f_{p, q}(m a) = f_{p - i, q + i}(m)a$

for local sections $a$ of $\mathcal{A}^ i$ and $m$ of $\mathcal{L}^{-q - i}$. For graded $\mathcal{A}$-modules $\mathcal{K}$, $\mathcal{L}$, $\mathcal{M}$ we define composition in $\textit{Mod}^{gr}(\mathcal{A})$ via the maps

$\mathop{\mathrm{Hom}}\nolimits ^ m(\mathcal{L}, \mathcal{M}) \times \mathop{\mathrm{Hom}}\nolimits ^ n(\mathcal{K}, \mathcal{L}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^{n + m}(\mathcal{K}, \mathcal{M})$

by simple composition of right $\mathcal{A}$-module maps: $(g, f) \mapsto g \circ f$.

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