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quasi-separated morphism

Contents

Definition

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 XoSpecZX\o 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.

Properties

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

References

Revised on November 25, 2013 01:30:56 by Urs Schreiber (89.204.137.196)