Lemma 30.4.1 (08DR): Induction Principle

Lemma 30.4.1 (Induction Principle). Let $X$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property of the quasi-compact opens of $X$. Assume that

$P$ holds for every affine open of $X$,
if $U$ is quasi-compact open, $V$ affine open, $P$ holds for $U$, $V$, and $U \cap V$, then $P$ holds for $U \cup V$.

Then $P$ holds for every quasi-compact open of $X$ and in particular for $X$.

Proof. First we argue by induction that $P$ holds for separated quasi-compact opens $W \subset X$. Namely, such an open can be written as $W = U_1 \cup \ldots \cup U_ n$ and we can do induction on $n$ using property (2) with $U = U_1 \cup \ldots \cup U_{n - 1}$ and $V = U_ n$. This is allowed because $U \cap V = (U_1 \cap U_ n) \cup \ldots \cup (U_{n - 1} \cap U_ n)$ is also a union of $n - 1$ affine open subschemes by Schemes, Lemma 26.21.7 applied to the affine opens $U_ i$ and $U_ n$ of $W$. Having said this, for any quasi-compact open $W \subset X$ we can do induction on the number of affine opens needed to cover $W$ using the same trick as before and using that the quasi-compact open $U_ i \cap U_ n$ is separated as an open subscheme of the affine scheme $U_ n$. $\square$

Comments (6)

Comment #6800 by Anya on December 12, 2021 at 17:28

It seems to me that this section really lacks a reference to Bondal and Van den Bergh. Same as section 09IS, which also looks like a rewriting of that paper. Of course, the word "reduction" they use is replaced by "induction" here, but certainly this is not a good enough reason to leave out the reference.

Comment #6801 by Johan on December 12, 2021 at 19:02

OK, let me know what lemma/proposition/theorem in their paper this corresponds to and I will put in the reference. We reference their paper in 9 spots, see for example the introductory text in Section 36.15 which contains Theorem 36.15.3 you refer to. The Stacks project never claims originality or priority; any time somebody suggests a precise, correct reference, it will be added quickly. Thanks!

Comment #6802 by Johan on December 12, 2021 at 19:14

OK, I found it: Proposition 3.3.1 in Bondal and van den Bergh. I will add it the next time I go through all the comments.

Comment #6804 by Anya on December 13, 2021 at 02:54

I see, thank you. For 09IS even the name of the file (?) says theorem-bondal-van-den-bergh in perfect.tex, so it just seemed a bit odd to me that there is no explicit reference.

Comment #6805 by Pieter Belmans on December 13, 2021 at 05:03

It is possible to suggest references for every single result in the Stacks project. E.g. better referencing of EGA and SGA would be desirable, but the same applies to results in papers (like Bondal--Van den Bergh). If you want you can suggest these citations (at the level of lemmas etc.) on each tag relevant tag page.

Comment #6945 by Johan on January 20, 2022 at 20:15

@#6800: thanks and fixed here.

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5 comment(s) on Section 30.4: Quasi-coherence of higher direct images

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