The Stacks project

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

  1. $P$ holds for every affine open of $X$,

  2. 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 $U_ n$. $\square$

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