Lemma 30.9.4 (01Y2)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 30: Cohomology of Schemes
Section 30.9: Coherent sheaves on locally Noetherian schemes
Lemma 30.9.4 (cite)

Lemma 30.9.4. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. The $\mathcal{O}_ X$-modules $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$ and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ are coherent.

Proof. It is shown in Modules, Lemma 17.22.6 that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is coherent. The result for tensor products is Modules, Lemma 17.16.6 $\square$

Comments (2)

Comment #5854 by Takagi Benseki（高城辨積） on December 18, 2020 at 09:02

The locally Noetherian condition is really necessary for $X$ ? 01CQ and 01CE have no such condition on $X$ .

Comment #5855 by Johan on December 18, 2020 at 14:36

The Stacks project only considers coherent modules on schemes which are locally Noetherian. See introduction to Section 30.9. The main reason for this is that it is too easy to make mistakes if you don't do this, especially since we are all used to using coherent modules over Noetherian schemes (because we first encounter them in this setting as for example in Harthorne's book). For example, we often silently use the result of Lemma 30.9.3 for coherent modules in algebraic geometry (which doesn't hold for general ringed spaces or schemes).

But, yes, the results do hold more generally for arbitrary ringed spaces (as is shown in 17.22.6 and 17.16.6) as you mention. For those people who want to use coherent modules over non-Noetherian schemes they should look there.

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