nLab quasiseparated morphism

Definition

Let $f : X \to Y$ be a morphism of schemes. Let $\Delta_f: X \to X \times_Y X$ be the diagonal map. We say that $f$ is quasi-separated if $\Delta_f$ is a quasicompact morphism.

A scheme $X$ is quasi-separated if the morphism $X \to Spec\, \mathbf{Z}$ is quasi-separated, i.e. $\Delta:X\to X\times X$ is quasicompact. Every quasi-separated scheme is semiseparated.

Properties

Every separated morphism of schemes is quasi-separated; every monomorphism of schemes is separated hence also quasi-separated.

Related concepts

• quasicompact morphism

• quasicompact quasiseparated morphism

• morphism of finite presentation

References

• MathOverflow why-does-finitely-presented-imply-quasi-separated
• Daniel Murfet, Concentrated schemes, pdf, an expositional digest from EGA

category: algebraic geometry