Lemma 87.16.1 (0AJ5)—The Stacks project

Processing math: 100%

Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.16: Separation axioms for formal algebraic spaces
Lemma 87.16.1 (cite)

next lemma

numberstags

Lemma 87.16.1. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$ . The following are equivalent

the reduction of $X$ (Lemma 87.12.1) is a quasi-separated algebraic space,
for $U \to X$ , $V \to X$ with $U$ , $V$ quasi-compact schemes the fibre product $U \times _ X V$ is quasi-compact,
for $U \to X$ , $V \to X$ with $U$ , $V$ affine the fibre product $U \times _ X V$ is quasi-compact.

Proof. Observe that $U \times _ X V$ is a scheme by Lemma 87.11.2. Let $U_{red}, V_{red}, X_{red}$ be the reduction of $U, V, X$ . Then

$U_{red} \times _{X_{red}} V_{red} = U_{red} \times _ X V_{red} \to U \times _ X V$

is a thickening of schemes. From this the equivalence of (1) and (2) is clear, keeping in mind the analogous lemma for algebraic spaces, see Properties of Spaces, Lemma 66.3.3. We omit the proof of the equivalence of (2) and (3). $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment