The Stacks project

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

is a fpqc covering, but not according to this definition (here denotes the local ring at the origin).

Comment #6280 by Owen on

but Lemma 34.9.2 cannot be true… the only open of that contains all the closed points of is itself, and this is not an affine variety ().

Comment #6281 by on

The map in definition needn't be injective.

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  • 8 comment(s) on Section 34.9: The fpqc topology

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