I would like to prove that the coherence of a sheaf on a scheme is affine-local. For this, it is necessary to prove that for a coherent $A$-module $M$, its localization $M_{f}$ at $f\in A$ is a coherent $A_{f}$-module. But I don't see how to prove this---I cannot relate the map $A^p_f\to M_f$ to some appropriate map $A^p\to M$ for instance. Would someone help me with this problem?

Cf. We follow the definition of stacks project on coherence, i.e., the module itself is finitely generated, and any finitely generated submodule of it is finitely presented.

  • $\begingroup$ What does the superscript $p$ denote? $\endgroup$ – Mose Wintner Aug 4 '15 at 5:34
  • $\begingroup$ @Mose Wintner: It is a positive integer. Indeed, the map $A^{\oplus p}\to M$ was meant to give a finitely generated submodule of $M$. $\endgroup$ – Manan Aug 4 '15 at 5:36
  • $\begingroup$ There is no need to add «Self-answer» to the title, really... $\endgroup$ – Mariano Suárez-Álvarez Aug 4 '15 at 7:05

Let $M$ be a coherent $A$-module and $S\subset A$ a multiplicative set. Then $S^{-1}M$ is a coherent $S^{-1}A$-module.

If $N'$ is a finitely generated $S^{-1}A$-submodule of $S^{-1}M$, then there is a finitely generated $A$-submodule $N$ of $M$ such that $N'=S^{-1}N$. (Suppose that $N'$ is generated by $x_1/s_1,\dots,x_n/s_n$. Set $N=Ax_1+\cdots+Ax_n$. Then $N'=S^{-1}N$.) Since $N$ is finitely presented, by localizing at $S$ we get that $N'$ is also finitely presented.

  • $\begingroup$ Wow, thanks for the answer. This is exactly what I have been looking for! +I found a mistake on my attempt above after looking at your answer. $\endgroup$ – Manan Aug 4 '15 at 7:10

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