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The Stacks project

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


Comments (1)

Comment #878 by on

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


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