The Stacks project

Lemma 74.42.6. In Situation 74.42.5. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Assume that the intersection of the supports of $\mathcal{F}$ and $\mathcal{G}$ is proper over $\mathop{\mathrm{Spec}}(A)$. Then the map

\[ \mathop{Mor}\nolimits _{\textit{Coh}(\mathcal{O}_ X)}(\mathcal{F}, \mathcal{G}) \longrightarrow \mathop{Mor}\nolimits _{\textit{Coh}(X, \mathcal{I})} (\mathcal{F}^\wedge , \mathcal{G}^\wedge ) \]

coming from ( is a bijection. In particular, ( is fully faithful.

Proof. Let $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F})$. This is a coherent $\mathcal{O}_ X$-module because its restriction of schemes ├ętale over $X$ is coherent by Modules, Lemma 17.21.5. By Lemma 74.42.4 the map

\[ \mathop{\mathrm{lim}}\nolimits _ n H^0(X, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) \to \mathop{Mor}\nolimits _{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge , \mathcal{F}^\wedge ) \]

is bijective. Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{H}$. It is clear that $Z$ is a closed subspace such that $|Z|$ is contained in the intersection of the supports of $\mathcal{F}$ and $\mathcal{G}$. Hence $Z \to \mathop{\mathrm{Spec}}(A)$ is proper by assumption (see Derived Categories of Spaces, Section 73.7). Write $\mathcal{H} = i_*\mathcal{H}'$ for some coherent $\mathcal{O}_ Z$-module $\mathcal{H}'$. We have $i_*(\mathcal{H}'/I^ n\mathcal{H}') = \mathcal{H}/I^ n\mathcal{H}$. Hence we obtain

\begin{align*} \mathop{\mathrm{lim}}\nolimits _ n H^0(X, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) & = \mathop{\mathrm{lim}}\nolimits _ n H^0(Z, \mathcal{H}'/\mathcal{I}^ n\mathcal{H}') \\ & = H^0(Z, \mathcal{H}') \\ & = H^0(X, \mathcal{H}) \\ & = \mathop{Mor}\nolimits _{\textit{Coh}(\mathcal{O}_ X)}(\mathcal{F}, \mathcal{G}) \end{align*}

the second equality by the theorem on formal functions (Cohomology of Spaces, Lemma 67.21.6). This proves the lemma. $\square$

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