The Stacks project

Lemma 21.44.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Given a solid diagram of $\mathcal{O}_ U$-modules

\[ \xymatrix{ \mathcal{E} \ar@{..>}[dr] \ar[r] & \mathcal{F} \\ & \mathcal{G} \ar[u]_ p } \]

with $\mathcal{E}$ a direct summand of a finite free $\mathcal{O}_ U$-module and $p$ surjective, then there exists a covering $\{ U_ i \to U\} $ such that a dotted arrow making the diagram commute exists over each $U_ i$.

Proof. We may assume $\mathcal{E} = \mathcal{O}_ U^{\oplus n}$ for some $n$. In this case finding the dotted arrow is equivalent to lifting the images of the basis elements in $\Gamma (U, \mathcal{F})$. This is locally possible by the characterization of surjective maps of sheaves (Sites, Section 7.11). $\square$

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