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24.16 Internal hom for sheaves of differential graded modules

We are going to need the sheafified version of the construction in Section 24.14. Let $(\mathcal{C}, \mathcal{O})$, $\mathcal{A}$, $\mathcal{M}$, $\mathcal{L}$ be as in Section 24.14. Then we define

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_\mathcal {A}(\mathcal{M}, \mathcal{L}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{gr}_\mathcal {A}(\mathcal{M}, \mathcal{L}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n_\mathcal {A}(\mathcal{M}, \mathcal{L}) \]

as a graded $\mathcal{O}$-module, see Section 24.7. In other words, a section $f$ of the $n$th graded piece $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n_\mathcal {A}(\mathcal{L}, \mathcal{M})$ over $U$ is a map of right $\mathcal{A}_ U$-module map $\mathcal{L}|_ U \to \mathcal{M}|_ U$ homogeneous of degree $n$. For such $f$ we set

\[ \text{d}(f) = \text{d}_\mathcal {M}|_ U \circ f - (-1)^ n f \circ \text{d}_\mathcal {L}|_ U \]

To make sense of this we think of $\text{d}_\mathcal {M}|_ U$ and $\text{d}_\mathcal {L}|_ U$ as graded $\mathcal{O}_ U$-module maps and we use composition of graded $\mathcal{O}_ U$-module maps. It is clear that $\text{d}(f)$ is homogeneous of degree $n + 1$ as a graded $\mathcal{O}_ U$-module map. Using the exact same computation as in Section 24.14 we see that $\text{d}(f)$ is $\mathcal{A}_ U$-linear.

As in Section 24.14 there is a composition map

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_\mathcal {A}(\mathcal{L}, \mathcal{M}) \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_\mathcal {A}(\mathcal{K}, \mathcal{L}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_\mathcal {A}(\mathcal{K}, \mathcal{M}) \]

where the left hand side is the tensor product of differential graded $\mathcal{O}$-modules defined in Section 24.15. This map is given by the composition map

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ m(\mathcal{L}, \mathcal{M}) \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n(\mathcal{K}, \mathcal{L}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{n + m}(\mathcal{K}, \mathcal{M}) \]

defined by simple composition (locally). Using the exact same computation as in Section 24.14 on local sections we see that the composition map is a morphism of differential graded $\mathcal{O}$-modules.

With these definitions we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A})}(\mathcal{L}, \mathcal{M}) = \Gamma (\mathcal{C}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_\mathcal {A}(\mathcal{L}, \mathcal{M})) \]

as graded $R$-modules compatible with composition.

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