The Stacks project

[Page 168, Tohoku].

Lemma 20.20.3. Let $X$ be a topological space such that the intersection of any two quasi-compact opens is quasi-compact. Let $\mathcal{F} \subset \underline{\mathbf{Z}}$ be a subsheaf generated by finitely many sections over quasi-compact opens. Then there exists a finite filtration

\[ (0) = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ n = \mathcal{F} \]

by abelian subsheaves such that for each $0 < i \leq n$ there exists a short exact sequence

\[ 0 \to j'_!\underline{\mathbf{Z}}_ V \to j_!\underline{\mathbf{Z}}_ U \to \mathcal{F}_ i/\mathcal{F}_{i - 1} \to 0 \]

with $j : U \to X$ and $j' : V \to X$ the inclusion of quasi-compact opens into $X$.

Proof. Say $\mathcal{F}$ is generated by the sections $s_1, \ldots , s_ t$ over the quasi-compact opens $U_1, \ldots , U_ t$. Since $U_ i$ is quasi-compact and $s_ i$ a locally constant function to $\mathbf{Z}$ we may assume, after possibly replacing $U_ i$ by the parts of a finite decomposition into open and closed subsets, that $s_ i$ is a constant section. Say $s_ i = n_ i$ with $n_ i \in \mathbf{Z}$. Of course we can remove $(U_ i, n_ i)$ from the list if $n_ i = 0$. Flipping signs if necessary we may also assume $n_ i > 0$. Next, for any subset $I \subset \{ 1, \ldots , t\} $ we may add $\bigcup _{i \in I} U_ i$ and $\gcd (n_ i, i \in I)$ to the list. After doing this we see that our list $(U_1, n_1), \ldots , (U_ t, n_ t)$ satisfies the following property: For $x \in X$ set $I_ x = \{ i \in \{ 1, \ldots , t\} \mid x \in U_ i\} $. Then $\gcd (n_ i, i \in I_ x)$ is attained by $n_ i$ for some $i \in I_ x$.

As our filtration we take $\mathcal{F}_0 = (0)$ and $\mathcal{F}_ n$ generated by the sections $n_ i$ over $U_ i$ for those $i$ such that $n_ i \leq n$. It is clear that $\mathcal{F}_ n = \mathcal{F}$ for $n \gg 0$. Moreover, the quotient $\mathcal{F}_ n/\mathcal{F}_{n - 1}$ is generated by the section $n$ over $U = \bigcup _{n_ i \leq n} U_ i$ and the kernel of the map $j_!\underline{\mathbf{Z}}_ U \to \mathcal{F}_ n/\mathcal{F}_{n - 1}$ is generated by the section $n$ over $V = \bigcup _{n_ i \leq n - 1} U_ i$. Thus a short exact sequence as in the statement of the lemma. $\square$


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