Lemma 59.73.8 (09Z4)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.73: More on constructible sheaves
Lemma 59.73.8 (cite)

Lemma 59.73.8. Let $X$ be a quasi-compact and quasi-separated scheme. Let $\Lambda $ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of sets, abelian groups, or $\Lambda $-modules on $X_{\acute{e}tale}$. Let $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ be a filtered colimit of sheaves of sets, abelian groups, or $\Lambda $-modules. Then

\[ \mathop{\mathrm{Mor}}\nolimits (\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Mor}}\nolimits (\mathcal{F}, \mathcal{G}_ i) \]

in the category of sheaves of sets, abelian groups, or $\Lambda $-modules on $X_{\acute{e}tale}$.

Proof. The case of sheaves of sets. By Lemma 59.73.5 it suffices to prove the lemma for $h_ U$ where $U$ is a quasi-compact and quasi-separated object of $X_{\acute{e}tale}$. Recall that $\mathop{\mathrm{Mor}}\nolimits (h_ U, \mathcal{G}) = \mathcal{G}(U)$. Hence the result follows from Sites, Lemma 7.17.7.

In the case of abelian sheaves or sheaves of modules, the result follows in the same way using Lemmas 59.73.7 and 59.73.6. For the case of abelian sheaves, we add that $\mathop{\mathrm{Mor}}\nolimits (j_{U!}\underline{\mathbf{Z}/n\mathbf{Z}}, \mathcal{G})$ is equal to the $n$-torsion elements of $\mathcal{G}(U)$. $\square$

The Stacks project

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