The Stacks project

Lemma 20.42.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given a solid diagram of $\mathcal{O}_ X$-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 $\mathcal{O}_ X$-module and $p$ surjective, then a dotted arrow making the diagram commute exists locally on $X$.

Proof. We may assume $\mathcal{E} = \mathcal{O}_ X^{\oplus n}$ for some $n$. In this case finding the dotted arrow is equivalent to lifting the images of the basis elements in $\Gamma (X, \mathcal{F})$. This is locally possible by the characterization of surjective maps of sheaves (Sheaves, Section 6.16). $\square$


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