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The Stacks project

Lemma 26.19.4. The composition of quasi-compact morphisms is quasi-compact.

Proof. This follows from the definitions and Topology, Lemma 5.12.2. $\square$

Comments (2)

Comment #646 by Kestutis Cesnavicius on

Proof: Apply http://stacks.math.columbia.edu/tag/005B

Also, I find it counterintuitive that the section "Quasi-compact morphisms" is in the chapter "Schemes" rather than "Morphisms of schemes", but perhaps there is a good reason for this.

Comment #658 by on

The "reason" this lemma is in the chapter of schemes is that we need the notion on quasi-compact morphisms in order to talk about quasi-separation of schemes. Of course, in some sense there is no way to divide the material into stuff about properties, morphisms, etc and so the thing that is causing trouble is the attempt to do this.

Thanks for the suggested edit. It will be incroporated soon.

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