Lemma 59.73.2 (03SA)—The Stacks project

Lemma 59.73.2. Let $X$ be a quasi-compact and quasi-separated scheme.

Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$ . Then $\mathcal{F}$ is a filtered colimit of constructible sheaves of sets.
Let $\mathcal{F}$ be a torsion abelian sheaf on $X_{\acute{e}tale}$ . Then $\mathcal{F}$ is a filtered colimit of constructible abelian sheaves.
Let $\Lambda$ be a Noetherian ring and $\mathcal{F}$ a sheaf of $\Lambda$ -modules on $X_{\acute{e}tale}$ . Then $\mathcal{F}$ is a filtered colimit of constructible sheaves of $\Lambda$ -modules.

Proof. Let $\mathcal{B}$ be the collection of quasi-compact and quasi-separated objects of $X_{\acute{e}tale}$ . By Modules on Sites, Lemma 18.30.7 any sheaf of sets is a filtered colimit of sheaves of the form

$\text{Coequalizer}\left( \xymatrix{ \coprod \nolimits _{j = 1, \ldots , m} h_{V_ j} \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{i = 1, \ldots , n} h_{U_ i} } \right)$

with $V_ j$ and $U_ i$ quasi-compact and quasi-separated objects of $X_{\acute{e}tale}$ . By Lemmas 59.73.1 and 59.71.6 these coequalizers are constructible. This proves (1).

Let $\Lambda$ be a Noetherian ring. By Modules on Sites, Lemma 18.30.7 $\Lambda$ -modules $\mathcal{F}$ is a filtered colimit of modules of the form

$\mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{j = 1, \ldots , m} j_{V_ j!}\underline{\Lambda }_{V_ j} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} j_{U_ i!}\underline{\Lambda }_{U_ i} \right)$

with $V_ j$ and $U_ i$ quasi-compact and quasi-separated objects of $X_{\acute{e}tale}$ . By Lemmas 59.73.1 and 59.71.6 these cokernels are constructible. This proves (3).

Proof of (2). First write $\mathcal{F} = \bigcup \mathcal{F}[n]$ where $\mathcal{F}[n]$ is the $n$ -torsion subsheaf. Then we can view $\mathcal{F}[n]$ as a sheaf of $\mathbf{Z}/n\mathbf{Z}$ -modules and apply (3). $\square$

The Stacks project

Comments (2)

Post a comment