The Stacks project

Definition 59.15.1. Let $T$ be a scheme. An fpqc covering of $T$ is a family $\{ \varphi _ i : T_ i \to T\} _{i \in I}$ such that

  1. each $\varphi _ i$ is a flat morphism and $\bigcup _{i\in I} \varphi _ i(T_ i) = T$, and

  2. for each affine open $U \subset T$ there exists a finite set $K$, a map $\mathbf{i} : K \to I$ and affine opens $U_{\mathbf{i}(k)} \subset T_{\mathbf{i}(k)}$ such that $U = \bigcup _{k \in K} \varphi _{\mathbf{i}(k)}(U_{\mathbf{i}(k)})$.

Comments (3)

Comment #5892 by David Holmes on

This seems to be a duplicate of tag 022B. Different phrasing but same content. Maybe you're aware of this and it's intentional, but I found it slightly confusing at first.

Comment #6095 by on

Yes, hmm, going to leave as is for now. Note that a reason for the difference between this defintiion and Definition 34.9.1 is that the discussion in Remark 59.15.2 points out that part (1) corresponds to the letters fp (fidelement plat) and part (2) corresponds to the letters qc (quasi-compact) in the four letters fpqc.

Comment #9764 by Matthieu Romagny on

It seems that the surjectivity condition in (1) is implied by (2).

There are also:

  • 4 comment(s) on Section 59.15: The fpqc topology

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View Definition 59.15.1 as pdf