Lemma 52.19.6. In Situation 52.16.1 let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module and $d \geq 1$. Assume

1. $A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$,

2. the completion $\mathcal{F}^\wedge$ of $\mathcal{F}$ satisfies the strict $(1, 1 + d)$-inequalities, and

3. for $x \in U$ with $\overline{\{ x\} } \cap Y \subset Z$ we have $\text{depth}(\mathcal{F}_ x) \geq 2$.

Then the map

$\mathop{\mathrm{Hom}}\nolimits _ U(\mathcal{G}, \mathcal{F}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\textit{Coh}(U, I\mathcal{O}_ U)}(\mathcal{G}^\wedge , \mathcal{F}^\wedge )$

is bijective for every coherent $\mathcal{O}_ U$-module $\mathcal{G}$.

Proof. Set $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}, \mathcal{F})$. Using Cohomology of Schemes, Lemma 30.11.2 or More on Algebra, Lemma 15.23.10 we see that the completion of $\mathcal{H}$ satisfies the strict $(1, 1 + d)$-inequalities and that for $x \in U$ with $\overline{\{ x\} } \cap Y \subset Z$ we have $\text{depth}(\mathcal{H}_ x) \geq 2$. Details omitted. Thus by Lemma 52.19.5 we have

$\mathop{\mathrm{Hom}}\nolimits _ U(\mathcal{G}, \mathcal{F}) = H^0(U, \mathcal{H}) = \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Coh}(U, I\mathcal{O}_ U)} (\mathcal{G}^\wedge , \mathcal{F}^\wedge )$

See Cohomology of Schemes, Lemma 30.23.5 for the final equality. $\square$

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