## 24.7 Internal hom for sheaves of graded modules

We urge the reader to skip this section.

We are going to need the sheafified version of the construction in Section 24.5. Let $(\mathcal{C}, \mathcal{O})$, $\mathcal{A}$, $\mathcal{M}$, $\mathcal{L}$ be as in Section 24.5. Then we define

as the graded $\mathcal{O}$-module whose degree $n$ term

is the subsheaf consisting of those local sections $f = (f_{p, q})$ such that

for local sections $a$ of $\mathcal{A}^ i$ and $m$ of $\mathcal{L}^{-q - i}$. As in Section 24.5 there is a composition map

where the left hand side is the tensor product of graded $\mathcal{O}$-modules defined in Section 24.6. This map is given by the composition map

defined by simple composition (locally).

With these definitions we have

as graded $R$-modules compatible with composition.

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