The Stacks project

Lemma 26.7.6. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. Kernels and cokernels of maps of quasi-coherent $\mathcal{O}_ X$-modules are quasi-coherent.

Proof. This follows from the exactness of the functor $\widetilde{\ }$ since by Lemma 26.7.1 we know that any map $\psi : \widetilde{M} \to \widetilde{N}$ comes from an $R$-module map $\varphi : M \to N$. (So we have $\mathop{\mathrm{Ker}}(\psi ) = \widetilde{\mathop{\mathrm{Ker}}(\varphi )}$ and $\mathop{\mathrm{Coker}}(\psi ) = \widetilde{\mathop{\mathrm{Coker}}(\varphi )}$.) $\square$

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