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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