The Stacks project

Lemma 86.20.6. 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

  1. if $Y$ is an affine formal algebraic space and $f : Y \to X$ is representable by algebraic spaces and étale, then $Y$ has property $P$,

  2. for some $\{ X_ i \to X\} _{i \in I}$ as in Definition 86.11.1 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 86.15.2) we can pick coverings $\{ X_{ij} \to X_ i \times _ X Y\} $ as in Definition 86.11.1. 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 86.19.10. 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 86.20.1, 86.20.2, 86.20.3, 86.20.4, and 86.20.5. $\square$

Comments (9)

Comment #1723 by Matthew Emerton on

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

Comment #1910 by Antoine Chambert-Loir on

TeX markup : in the statement of the proposition, enter

Comment #1911 by Antoine Chambert-Loir on

TeX markup : in the statement of the proposition, enter

Comment #1957 by Brian Conrad on

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 on

@#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 #2104 by on

@#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 on

@#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.

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