Lemma 17.17.8. Let (X, \mathcal{O}_ X) be a ringed space. Let
be a short exact sequence of \mathcal{O}_ X-modules.
If \mathcal{F}_2 and \mathcal{F}_0 are flat so is \mathcal{F}_1.
If \mathcal{F}_1 and \mathcal{F}_0 are flat so is \mathcal{F}_2.
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
be a short exact sequence of \mathcal{O}_ X-modules.
If \mathcal{F}_2 and \mathcal{F}_0 are flat so is \mathcal{F}_1.
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 Konrad Voelkel on