Lemma 87.20.6 (0AKX)—The Stacks project

Lemma 87.20.6. Let $S$ be a scheme. Let

$P \in \left\{ \begin{matrix} countably\ indexed, \\ countably\ indexed\ and\ classical, \\ weakly\ adic,\ adic*,\ Noetherian \end{matrix} \right\}$

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

if $Y$ is an affine formal algebraic space and $f : Y \to X$ is representable by algebraic spaces and étale, then $Y$ has property $P$ ,
for some $\{ X_ i \to X\} _{i \in I}$ as in Definition 87.11.1 each $X_ i$ has property $P$ .

Proof. It is clear that (1) implies (2). Assume (2) and let $Y \to X$ be as in (1). Since the fibre products $X_ i \times _ X Y$ are formal algebraic spaces (Lemma 87.15.2) we can pick coverings $\{ X_{ij} \to X_ i \times _ X Y\}$ as in Definition 87.11.1. Since $Y$ is quasi-compact, there exist $(i_1, j_1), \ldots , (i_ n, j_ n)$ such that

$X_{i_1 j_1} \amalg \ldots \amalg X_{i_ n j_ n} \longrightarrow Y$

is surjective and étale. Then $X_{i_ kj_ k} \to X_{i_ k}$ is representable by algebraic spaces and étale hence $X_{i_ kj_ k}$ has property $P$ by Lemma 87.19.10. Then $X_{i_1 j_1} \amalg \ldots \amalg X_{i_ n j_ n}$ is an affine formal algebraic space with property $P$ (small detail omitted on finite disjoint unions of affine formal algebraic spaces). Hence we conclude by applying one of Lemmas 87.20.1, 87.20.2, 87.20.3, 87.20.4, and 87.20.5. $\square$

Comments (9)

Comment #1723 by Matthew Emerton on December 10, 2015 at 05:55

In line 2 of the proof, I think $X$ and $Y$ should be switched in the fibre product.

Comment #1762 by Johan on December 15, 2015 at 19:08

Thanks, fixed here.

Comment #1910 by Antoine Chambert-Loir on April 06, 2016 at 20:02

TeX markup : in the statement of the proposition, enter

$P \in \{\text{countably indexed}, \text{adic*}, \text{Noetherian\}\}$

Comment #1911 by Antoine Chambert-Loir on April 06, 2016 at 20:02

TeX markup : in the statement of the proposition, enter

$P \in \{\text{countably indexed}, \text{adic*}, \text{Noetherian}\}$

Comment #1957 by Brian Conrad on May 01, 2016 at 04:08

Just after the displayed expression in the proof, it may help the reader to replace "is surjective" with "is surjective and etale".

Comment #1983 by Johan on May 03, 2016 at 12:53

@#1911 This is a problem with the rendering of mathematics in the browser and not a problem with the underlying LaTeX. So I'm ignoring this for now.

Comment #2011 by Johan on May 05, 2016 at 14:16

@#1957 OK, thanks, fixed here.

Comment #2104 by Pieter Belmans on July 07, 2016 at 08:27

@#1983 Actually Antoine is right, you shouldn't write English in math mode and hope for the best (although it turns out pretty well in this case, except for the weird kerning in Noetherian). I just checked, enclosing the three words using \text as Antoine suggests gives the desired rendering in HTML (i.e. the parser can already handle this).

Comment #2130 by Johan on July 21, 2016 at 14:25

@#2104: Of course he is right, but I am right also: there is a backslash followed by a space in the latex code which should always produce a space in the display of said latex code.

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).

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