Lemma 37.3.1 (09ZV)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.3: Morphisms of thickenings
Lemma 37.3.1 (cite)

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Lemma 37.3.1. Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism of thickenings. Then

$f$ is an affine morphism if and only if $f'$ is an affine morphism,
$f$ is a surjective morphism if and only if $f'$ is a surjective morphism,
$f$ is quasi-compact if and only if $f'$ quasi-compact,
$f$ is universally closed if and only if $f'$ is universally closed,
$f$ is integral if and only if $f'$ is integral,
$f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,
$f$ is universally injective if and only if $f'$ is universally injective,
$f$ is universally open if and only if $f'$ is universally open,
$f$ is quasi-affine if and only if $f'$ is quasi-affine, and
add more here.

Proof. Observe that $S \to S'$ and $X \to X'$ are universal homeomorphisms (see for example Morphisms, Lemma 29.45.6). This immediately implies parts (2), (3), (4), (7), and (8). Part (1) follows from Lemma 37.2.3 which tells us that there is a 1-to-1 correspondence between affine opens of $S$ and $S'$ and between affine opens of $X$ and $X'$. Part (9) follows from Limits, Lemma 32.11.5 and the remark just made about affine opens of $S$ and $S'$. Part (5) follows from (1) and (4) by Morphisms, Lemma 29.44.7. Finally, note that

\[ S \times _ X S = S \times _{X'} S \to S \times _{X'} S' \to S' \times _{X'} S' \]

is a thickening (the two arrows are thickenings by Lemma 37.2.4). Hence applying (3) and (4) to the morphism $(S \subset S') \to (S \times _ X S \to S' \times _{X'} S')$ we obtain (6). $\square$

The Stacks project

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