The Stacks project

Lemma 59.73.7. Let $X$ be a quasi-compact and quasi-separated scheme. The category of constructible abelian sheaves is exactly the category of abelian sheaves of the form

\[ \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{j = 1, \ldots , m} j_{V_ j!}\underline{\mathbf{Z}/m_ j\mathbf{Z}}_{V_ j} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} j_{U_ i!}\underline{\mathbf{Z}/n_ i\mathbf{Z}}_{U_ i} \right) \]

with $V_ j$ and $U_ i$ quasi-compact and quasi-separated objects of $X_{\acute{e}tale}$ and $m_ j$, $n_ i$ positive integers. In fact, we can even assume $U_ i$ and $V_ j$ affine.

Proof. This follows from Lemma 59.73.6 applied with $\Lambda = \mathbf{Z}/n\mathbf{Z}$ and the fact that, since $X$ is quasi-compact, every constructible abelian sheaf is annihilated by some positive integer $n$ (details omitted). $\square$

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