separated morphism of schemes in nLab

Context

Algebraic geometry

• Definition
• Warning
• Properties
• Related concepts

Definition

Let f:X→Yf : X \to Y be a morphism of schemes. Write Δ:X→X× YX\Delta : X \to X \times_Y X for the diagonal morphism.

• The morphism ff is called separated if Δ(X)\Delta(X) is a closed subspace of X× YXX \times_Y X.

• A scheme XX is called separated if the terminal morphism X→SpecℤX \to \operatorname{Spec} \mathbb{Z} is separated.

Proposition

Let XX be a scheme (resp. a locally noetherian scheme), f:X→Yf: X\to Y a morphism of schemes (resp. a morphism locally of finite type). The following conditions are equivalent.

1. ff is separated.

2. The diagonal morphism X→X× YXX\to X\times_Y X is quasicompact, and for every affine scheme Y′=SpecAY' = Spec A in which AA is a valuation ring (resp. a discrete valuation ring), any two morphisms from Y′→XY'\to X which coincide at the generic point of Y′Y' are equal.

3. The diagonal morphism X→X× YXX\to X\times_Y X is quasicompact, and for every affine scheme of the form Y′=SpecAY' = Spec A in which AA is a valuation ring (resp. a discrete valuation ring), any two sections of X′=X(Y′)X' = X(Y') which coincide at the generic point of Y′Y' are equal.