Definition 34.9.1 (022B)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 34: Topologies on Schemes
Section 34.9: The fpqc topology
Definition 34.9.1 (cite)

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Definition 34.9.1. Let $T$ be a scheme. An fpqc covering of $T$ is a family of morphisms $\{ f_ i : T_ i \to T\} _{i \in I}$ of schemes such that each $f_ i$ is flat and such that for every affine open $U \subset T$ there exists $n \geq 0$, a map $a : \{ 1, \ldots , n\} \to I$ and affine opens $V_ j \subset T_{a(j)}$, $j = 1, \ldots , n$ with $\bigcup _{j = 1}^ n f_{a(j)}(V_ j) = U$.

Comments (4)

Comment #6276 by Owen on June 05, 2021 at 20:44

$\textstyle(\operatorname{Spec}\mathcal O_{\mathbf{A}^2,0})\coprod(\mathbf{A}^2\smallsetminus\{0\})\to\mathbf{A}^2$ is a fpqc covering, but not according to this definition (here $\mathcal O_{\mathbf{A}^2,0}$ denotes the local ring at the origin).

Comment #6278 by Johan on June 09, 2021 at 17:59

Lemma 34.9.2 shows it is an fpqc covering.

Comment #6280 by Owen on June 10, 2021 at 14:48

but Lemma 34.9.2 cannot be true… the only open of $\mathbf{A}^2\smallsetminus\{0\}$ that contains all the closed points of $\mathbf{A}^2\smallsetminus\{0\}$ is $\mathbf{A}^2\smallsetminus\{0\}$ itself, and this is not an affine variety ( $\mathbf{A}^2:=\mathbf{A}^2_k$ ).

Comment #6281 by Johan on June 10, 2021 at 21:57

The map $a$ in definition needn't be injective.

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