Lemma 52.19.6. In Situation 52.16.1 let \mathcal{F} be a coherent \mathcal{O}_ U-module and d \geq 1. Assume
A is I-adically complete, has a dualizing complex, and \text{cd}(A, I) \leq d,
the completion \mathcal{F}^\wedge of \mathcal{F} satisfies the strict (1, 1 + d)-inequalities, and
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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