## 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

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{gr}_\mathcal {A}(\mathcal{M}, \mathcal{L})$

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

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^ n_\mathcal {A}(\mathcal{M}, \mathcal{L}) \subset \prod \nolimits _{p + q = n} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}^{-q}, \mathcal{M}^ p)$

is the subsheaf consisting of those local sections $f = (f_{p, q})$ 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 $\mathcal{L}^{-q - i}$. As in Section 24.5 there is a composition map

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

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

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

defined by simple composition (locally).

With these definitions we have

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

as graded $R$-modules compatible with composition.

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