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24.5 The graded category of sheaves of graded modules

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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