nLab quasi-separated morphism



Let f:XYf : X \to Y be a morphism of schemes. Let Δ f:XX× YX\Delta_f: X \to X \times_Y X be the diagonal map. We say that ff is quasi-separated if Δ f\Delta_f is a quasicompact morphism.

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


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


Last revised on August 14, 2014 at 19:42:45. See the history of this page for a list of all contributions to it.