The Stacks project

Lemma 99.3.6. In Situation 99.3.1. If $\mathcal{F}$ is of finite presentation and $f$ is quasi-compact and quasi-separated, then $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is limit preserving.

Proof. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ be a directed limit of affine $B$-schemes. We have to show that

\[ \mathit{Hom}(\mathcal{F}, \mathcal{G})(T) = \mathop{\mathrm{colim}}\nolimits \mathit{Hom}(\mathcal{F}, \mathcal{G})(T_ i) \]

Pick $0 \in I$. We may replace $B$ by $T_0$, $X$ by $X_{T_0}$, $\mathcal{F}$ by $\mathcal{F}_{T_0}$, $\mathcal{G}$ by $\mathcal{G}_{T_0}$, and $I$ by $\{ i \in I \mid i \geq 0\} $. See Remark 99.3.4. Thus we may assume $B = \mathop{\mathrm{Spec}}(R)$ is affine.

When $B$ is affine, then $X$ is quasi-compact and quasi-separated. Choose a surjective ├ętale morphism $U \to X$ where $U$ is an affine scheme (Properties of Spaces, Lemma 66.6.3). Since $X$ is quasi-separated, the scheme $U \times _ X U$ is quasi-compact and we may choose a surjective ├ętale morphism $V \to U \times _ X U$ where $V$ is an affine scheme. Applying Lemma 99.3.5 we see that $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is the equalizer of two maps between

\[ \mathit{Hom}(\mathcal{F}|_ U, \mathcal{G}|_ U) \quad \text{and}\quad \mathit{Hom}(\mathcal{F}|_ V, \mathcal{G}|_ V) \]

This reduces us to the case that $X$ is affine.

In the affine case the statement of the lemma reduces to the following problem: Given a ring map $R \to A$, two $A$-modules $M$, $N$ and a directed system of $R$-algebras $C = \mathop{\mathrm{colim}}\nolimits C_ i$. When is it true that the map

\[ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _{A \otimes _ R C_ i}(M \otimes _ R C_ i, N \otimes _ R C_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{A \otimes _ R C}(M \otimes _ R C, N \otimes _ R C) \]

is bijective? By Algebra, Lemma 10.127.5 this holds if $M \otimes _ R C$ is of finite presentation over $A \otimes _ R C$, i.e., when $M$ is of finite presentation over $A$. $\square$

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