Lemma 59.88.5 (0F06)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.88: Base change for higher direct images
Lemma 59.88.5 (cite)

Lemma 59.88.5. Let $T$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property for quasi-compact and quasi-separated schemes over $T$. Assume

If $T'' \to T'$ is a thickening of quasi-compact and quasi-separated schemes over $T$, then $P(T'')$ if and only if $P(T')$.
If $T' = \mathop{\mathrm{lim}}\nolimits T_ i$ is a limit of an inverse system of quasi-compact and quasi-separated schemes over $T$ with affine transition morphisms and $P(T_ i)$ holds for all $i$, then $P(T')$ holds.
If $Z \subset T'$ is a closed subscheme with quasi-compact complement $V \subset T'$ and $P(T')$ holds, then either $P(V)$ or $P(Z)$ holds.

Then $P(T)$ implies $P(\mathop{\mathrm{Spec}}(K))$ for some morphism $\mathop{\mathrm{Spec}}(K) \to T$ where $K$ is a field.

Proof. Consider the set $\mathfrak T$ of closed subschemes $T' \subset T$ such that $P(T')$. By assumption (2) this set has a minimal element, say $T'$. By assumption (1) we see that $T'$ is reduced. Let $\eta \in T'$ be the generic point of an irreducible component of $T'$. Then $\eta = \mathop{\mathrm{Spec}}(K)$ for some field $K$ and $\eta = \mathop{\mathrm{lim}}\nolimits V$ where the limit is over the affine open subschemes $V \subset T'$ containing $\eta $. By assumption (3) and the minimality of $T'$ we see that $P(V)$ holds for all these $V$. Hence $P(\eta )$ by (2) and the proof is complete. $\square$

The Stacks project

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