The Stacks project

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 $s_ i|_ V = \sum a_{ij}t_ j$. $\square$

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