nLab semiseparated scheme

Definition

An algebraic scheme is semiseparated if it has a basis of topology by affine open subsets which is closed under finite intersections.

A cover $\{U_i\}_{i\in I}$ of a scheme by affine open subsets is semiseparated (also spelled semi-separated, some say semiseparating or semi-separating) if $U_i\cap U_j$ is also affine for every pair $(i,j)\in I\times I$.

A scheme is semiseparated iff it has an affine cover which is semiseparated.

Properties

An open subset of a semiseparated scheme is semiseparated.

Literature

The notion is explained in

• Thomason, Trobaugh, in Grothendieck Festschrift 1989

A very short exposition is at the end of

• Daniel Murfet, Concentrated schemes, pdf

category: algebraic geometry