The Stacks project

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

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

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

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

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

as graded $R$-modules compatible with composition.