The Stacks project

Lemma 48.9.1. With notation as above. The functor $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ is a right adjoint to the functor $i_* : \textit{Mod}(\mathcal{O}_ Z) \to \textit{Mod}(\mathcal{O}_ X)$. For $V \subset Z$ open we have

\[ \Gamma (V, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, \mathcal{F})) = \{ s \in \Gamma (U, \mathcal{F}) \mid \mathcal{I}s = 0\} \]

where $U \subset X$ is an open whose intersection with $Z$ is $V$.

Proof. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ Z$-modules. Then

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{i_*\mathcal{O}_ Z}(i_*\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{O}_ Z, \mathcal{F})) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ Z}(\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, \mathcal{F})) \]

The first equality by Modules, Lemma 17.22.3 and the second by the fully faithfulness of $i_*$, see Modules, Lemma 17.13.4. The description of sections is left to the reader. $\square$

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