In the "The Rising Sea" by Vakil one can find the base change theorem for proper morphisms over a locally Noetherian base (28.1.6). He later indicates (28.2.M) how one could exchange the locally Noetherian condition by finitely presented using a result of Grothendieck. And indeed, it does not seem too hard to show this.

I am wondering if this version of the theorem is written down anywhere else in the literature (with or without a proof), as I seem unable to find it. I see the stronger version getting applied quite often (most recently in Olsson's book "Algebraic Spaces and Stacks" proof of lemma 8.4.6) but always without a proper reference (Olsson references Hartshorne, who only shows the statement for projective morphisms).

I could imagine that there is a formulation for algebraic spaces or stacks such that the theorem for schemes is just a special case. The closest thing I could find was theorem 1.9 / 1.10 in "Compatifying the Picard Scheme" by Altman and Kleiman, but this also does not seem to imply the base change theorem as stated by Vakil. It just seems really odd to me that such a prominent theorem can not be found in the Stacks Project or any other reputable source (without implying that Vakil is not reputable...)

Edit: To add to the list below: In Conrad, Brian, Grothendieck duality and base change, Lecture Notes in Mathematics. 1750. Berlin: Springer. x, 296 p. (2000). ZBL0992.14001. in Chapter 5.1 one can find an argument how to remove the Noetherian condition. But he only does so for a slightly weaker statement.

  • $\begingroup$ Is Hall, Jack, Cohomology and base change for algebraic stacks, Math. Z. 278 (2014) useful to you ? $\endgroup$ May 5, 2021 at 7:12
  • $\begingroup$ How to reduce to the Noetherian case is detailed in [FGA explained, Proposition 4.37], but the statement there may not be as general as you need. $\endgroup$ May 5, 2021 at 7:19
  • $\begingroup$ @MatthieuRomagny As far as I can tell he only seems to proof a similar theorem to base change (theorem A) using a Noetherian condition. $\endgroup$ May 5, 2021 at 7:28
  • $\begingroup$ @OlivierBenoist They seem to use the same argument as indicated by Vakil. Thank you for the reference, I did not know this one. $\endgroup$ May 5, 2021 at 7:30
  • $\begingroup$ Right -- the point is, you seemed to require a generalization of the "historical" base change theorem in two directions (remove noetherian assumptions ; pass from schemes to stacks) and I was pointing out that for the stack direction, it's in the literature. $\endgroup$ May 5, 2021 at 7:40

1 Answer 1


There is a fairly general version of base change for schemes in Lipman's "yellow book":

Lipman, Joseph; Hashimoto, Mitsuyasu: Foundations of Grothendieck duality for diagrams of schemes. Lecture Notes in Mathematics, 1960. Springer-Verlag, Berlin, 2009.

Also available at:


Concretely, Theorem (3.10.3) establishes the base-change theorem for an independent square of quasi-compact and quasi-separated maps of quasi-separated schemes. The proof does not use noetherian schemes.

  • $\begingroup$ This seems to be quite a bit more general but there is still an assumption on the base scheme. If there are no more answers till tomorrow I will accept your reply. (I think it is funny that the author remarks that his theorem 3.10.3 is basic but "difficult,if not impossible, to find elsewhere") $\endgroup$ May 5, 2021 at 13:30
  • $\begingroup$ @FabianRuoff For some reason, the basics of schemes are not treated in detail in the habitual courses. Most expositions stick to the Noetherian hypothesis while there are tools to get rid of the hypothesis most of the time since EGA IV. $\endgroup$
    – Leo Alonso
    May 5, 2021 at 16:26

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