# Properties stable under base change in algebraic geometry

I remember to have seen a big list in the EGA of properties $(P)$ such that: if $f : X \to Y$ has $(P)$ then, $f_{(S')} : X_{(S')}\to Y_{(S')}$ has $(P)$, where $f_{(S')}$ is the morphism $f$ after a base change $S'\to S$., etc. but I can't find it now...

Does anyone know where I can find such a list ?

(I am interested by $(P)$ = "to be a closed map")

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The property of being a closed map (in the sense of the image of closed sets being closed) is not stable under base change. Hence one introduces the stronger notion of "universally closed", which is a map all of whose base-changes are closed, and which is the key condition in the definition of proper morphisms (which are morphisms that are finite type, separated, and universally closed). – Emerton Apr 8 '10 at 16:08
Thanks for your comment Emerton. Fortunately, in my case, $f$ is a closed immersion ! – user2330 Apr 8 '10 at 16:12
You're in good shape then! As you no doubt know by now, these are stable under base change. – Emerton Apr 8 '10 at 17:34

## 3 Answers

One list I've seen is in Appendix C from a course 'Rational Points on Varieties' taught by Bjorn Poonen. Here's the link:

http://math.mit.edu/~poonen/papers/Qpoints.pdf

Also, the appendix to the book of Gortz and Wedhorn 'Algebraic Geometry 1: Schemes With Examples and Exercises' is a great reference.

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Thanks for this reference ! However, a reference to EGA should be better for me. – user2330 Apr 8 '10 at 16:11
I don't understand what you're looking for. Poonen's notes contain references to EGA proofs, including the case of a closed immersion. Professor Emerton's comment settles the closed map case, more information for which you could find in Hartshorne; for example where he defined proper maps somewhere in II.4. – Frank Apr 8 '10 at 16:16
OK, sorry... Indeed, Poonen's notes contains references to EGA. – user2330 Apr 8 '10 at 16:54

The "cheat sheets" on the bottom of this page might be helpful http://www.math.ucdavis.edu/~osserman/classes/256B/

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There are some lists like this in Section Tag 02WE.

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