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Kernels of epimorphisms and extensions of flat sheaves of modules over a ringed space are again flat.

Lemma 17.17.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let

\[ 0 \to \mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F}_0 \to 0 \]

be a short exact sequence of $\mathcal{O}_ X$-modules.

  1. If $\mathcal{F}_2$ and $\mathcal{F}_0$ are flat so is $\mathcal{F}_1$.

  2. If $\mathcal{F}_1$ and $\mathcal{F}_0$ are flat so is $\mathcal{F}_2$.

Proof. Since exactness and flatness may be checked at the level of stalks this follows from Algebra, Lemma 10.39.13. $\square$

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