Lemma 17.17.10. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a flat $\mathcal{O}_ X$-module. Let $U \subset X$ be open and let

$\mathcal{O}_ U \xrightarrow {(f_1, \ldots , f_ n)} \mathcal{O}_ U^{\oplus n} \xrightarrow {(s_1, \ldots , s_ n)} \mathcal{F}|_ U$

be a complex of $\mathcal{O}_ U$-modules. For every $x \in U$ there exists an open neighbourhood $V \subset U$ of $x$ and a factorization

$\mathcal{O}_ V^{\oplus n} \xrightarrow {A} \mathcal{O}_ V^{\oplus m} \xrightarrow {(t_1, \ldots , t_ m)} \mathcal{F}|_ V$

of $(s_1, \ldots , s_ n)|_ V$ such that $A \circ (f_1, \ldots , f_ n)|_ V = 0$.

Proof. Let $\mathcal{I} \subset \mathcal{O}_ U$ be the sheaf of ideals generated by $f_1, \ldots , f_ n$. Then $\sum f_ i \otimes s_ i$ is a section of $\mathcal{I} \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U$ which maps to zero in $\mathcal{F}|_ U$. As $\mathcal{F}|_ U$ is flat the map $\mathcal{I} \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U \to \mathcal{F}|_ U$ is injective. Since $\mathcal{I} \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U$ is the sheaf associated to the presheaf tensor product, we see there exists an open neighbourhood $V \subset U$ of $x$ such that $\sum f_ i|_ V \otimes s_ i|_ V$ is zero in $\mathcal{I}(V) \otimes _{\mathcal{O}(V)} \mathcal{F}(V)$. Unwinding the definitions using Algebra, Lemma 10.107.10 we find $t_1, \ldots , t_ m \in \mathcal{F}(V)$ and $a_{ij} \in \mathcal{O}(V)$ such that $\sum a_{ij}f_ i|_ V = 0$ and $s_ i|_ V = \sum a_{ij}t_ j$. $\square$

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.

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 08BK. Beware of the difference between the letter 'O' and the digit '0'.