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24.6 Tensor product for sheaves of graded modules

Please skip this section. This section is the analogue of part of Differential Graded Algebra, Section 22.12.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a right graded $\mathcal{A}$-module and let $\mathcal{N}$ be a left graded $\mathcal{A}$-module. Then we define the tensor product $\mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ to be the graded $\mathcal{O}$-module whose degree $n$ term is

\[ (\mathcal{M} \otimes _\mathcal {A} \mathcal{N})^ n = \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{r + s + t = n} \mathcal{M}^ r \otimes _\mathcal {O} \mathcal{A}^ s \otimes _\mathcal {O} \mathcal{N}^ t \longrightarrow \bigoplus \nolimits _{p + q = n} \mathcal{M}^ p \otimes _\mathcal {O} \mathcal{N}^ q \right) \]

where the map sends the local section $x \otimes a \otimes y$ of $\mathcal{M}^ r \otimes _\mathcal {O} \mathcal{A}^ s \otimes _\mathcal {O} \mathcal{N}^ t$ to $xa \otimes y - x \otimes ay$. With this definition we have that $(\mathcal{M} \otimes _\mathcal {A} \mathcal{N})^ n$ is the sheafification of the presheaf $U \mapsto (\mathcal{M}(U) \otimes _{\mathcal{A}(U)} \mathcal{N}(U))^ n$ where the tensor product of graded modules is as defined in Differential Graded Algebra, Section 22.12.

If we fix the left graded $\mathcal{A}$-module $\mathcal{N}$ we obtain a functor

\[ - \otimes _\mathcal {A} \mathcal{N} : \textit{Mod}(\mathcal{A}) \longrightarrow \text{Gr}(\textit{Mod}(\mathcal{O})) = \text{graded }\mathcal{O}\text{-modules} \]

For the notation $\text{Gr}(-)$ please see Homology, Definition 12.16.1. The graded category of graded $\mathcal{O}$-modules is denoted $\text{Gr}^{gr}(\textit{Mod}(\mathcal{O}))$, see Differential Graded Algebra, Example 22.25.5. The functor above can be upgraded to a functor of graded categories

\[ - \otimes _\mathcal {A} \mathcal{N} : \textit{Mod}^{gr}(\mathcal{A}) \longrightarrow \text{Gr}^{gr}(\textit{Mod}(\mathcal{O})) \]

by sending homomorphisms of degree $n$ from $\mathcal{M} \to \mathcal{M}'$ to the induced map of degree $n$ from $\mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ to $\mathcal{M}' \otimes _\mathcal {A} \mathcal{N}$.

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