Lemma 67.4.9 (04ZH)—The Stacks project

Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.4: Separation axioms
Lemma 67.4.9 (cite)

numberstags

Lemma 67.4.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

If $Y$ is separated and $f$ is separated, then $X$ is separated.
If $Y$ is quasi-separated and $f$ is quasi-separated, then $X$ is quasi-separated.
If $Y$ is locally separated and $f$ is locally separated, then $X$ is locally separated.
If $Y$ is separated over $S$ and $f$ is separated, then $X$ is separated over $S$.
If $Y$ is quasi-separated over $S$ and $f$ is quasi-separated, then $X$ is quasi-separated over $S$.
If $Y$ is locally separated over $S$ and $f$ is locally separated, then $X$ is locally separated over $S$.

Proof. Parts (4), (5), and (6) follow immediately from Lemma 67.4.8 and Spaces, Definition 65.13.2. Parts (1), (2), and (3) reduce to parts (4), (5), and (6) by thinking of $X$ and $Y$ as algebraic spaces over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Properties of Spaces, Definition 66.3.1. $\square$

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