The Stacks project

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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