The Stacks project

Lemma 59.73.6. Let $X$ be a quasi-compact and quasi-separated scheme. Let $\Lambda $ be a Noetherian ring. The category of constructible sheaves of $\Lambda $-modules is exactly the category 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 $X_{\acute{e}tale}$. In fact, we can even assume $U_ i$ and $V_ j$ affine.

Proof. In the proof of Lemma 59.73.2 we have seen modules of this form are constructible. Since the category of constructible modules is abelian (Lemma 59.71.6) it suffices to prove that given a constructible module $\mathcal{F}$ there is a surjection

\[ \bigoplus \nolimits _{i = 1, \ldots , n} j_{U_ i!}\underline{\Lambda }_{U_ i} \longrightarrow \mathcal{F} \]

for some affine objects $U_ i$ in $X_{\acute{e}tale}$. By Modules on Sites, Lemma 18.30.7 there is a surjection

\[ \Psi : \bigoplus \nolimits _{i \in I} j_{U_ i!}\underline{\Lambda }_{U_ i} \longrightarrow \mathcal{F} \]

with $U_ i$ affine and the direct sum over a possibly infinite index set $I$. For every finite subset $I' \subset I$ set

\[ T_{I'} = \text{Supp}(\mathop{\mathrm{Coker}}( \bigoplus \nolimits _{i \in I'} j_{U_ i!}\underline{\Lambda }_{U_ i} \longrightarrow \mathcal{F})) \]

By the very definition of constructible sheaves, the set $T_{I'}$ is a constructible subset of $X$. We want to show that $T_{I'} = \emptyset $ for some $I'$. Since every stalk $\mathcal{F}_{\overline{x}}$ is a finite type $\Lambda $-module and since $\Psi $ is surjective, for every $x \in X$ there is an $I'$ such that $x \not\in T_{I'}$. In other words we have $\emptyset = \bigcap _{I' \subset I\text{ finite}} T_{I'}$. Since $X$ is a spectral space by Properties, Lemma 28.2.4 the constructible topology on $X$ is quasi-compact by Topology, Lemma 5.23.2. Thus $T_{I'} = \emptyset $ for some $I' \subset I$ finite as desired. $\square$

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