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. Equivalently, the diagonal morphism has an affine image (hence quasicompact image, that is, the semiseparated scheme is automatically quasiseparated).

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.

Every semiseparated scheme is quasiseparated.

Literature

The notion is explained in

• R.W. Thomason, T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, in: The Grothendieck Festschrift, vol. III, in: Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435

See also Sect. 2 of

• L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, María J. Vale Gonsalves, The derived category of quasi- coherent sheaves and axiomatic stable homotopy, Adv. Math. 218 (4) (2008) 1224–1252

A very short exposition is at the end of

• Daniel Murfet, Concentrated schemes, pdf

category: algebraic geometry