Lemma 26.21.6. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

The morphism $f$ is quasi-separated.

For every pair of affine opens $U, V \subset X$ which map into a common affine open of $S$ the intersection $U \cap V$ is a finite union of affine opens of $X$.

There exists an affine open covering $S = \bigcup _{i \in I} U_ i$ and for each $i$ an affine open covering $f^{-1}U_ i = \bigcup _{j \in I_ i} V_ j$ such that for each $i$ and each pair $j, j' \in I_ i$ the intersection $V_ j \cap V_{j'}$ is a finite union of affine opens of $X$.

## Comments (2)

Comment #1051 by Wessel Bindt on

Comment #1061 by Johan on

There are also: