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 $\bigcap _{i \in I} U_ i$ 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. $\square$


Comments (2)

Comment #7455 by WhatJiaranEatsTonight on

I think here we shall add and but not .

There are also:

  • 2 comment(s) on Section 20.20: Vanishing on Noetherian topological spaces

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A38. Beware of the difference between the letter 'O' and the digit '0'.