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separated morphism of schemes

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Definition

Definition

Let f:XYf : X \to Y be a morphism of schemes. Write Δ:XX× 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 XSpecX \to \operatorname{Spec} \mathbb{Z} is separated.

Proposition

Let XX be a scheme (resp. a locally noetherian scheme), f:XYf: 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 XX× 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 YXY'\to X which coincide at the generic point of YY' are equal.

  3. The diagonal morphism XX× 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 YY' are equal.

This is the valuative criterion of separatedness. See Hartshorne or EGA II for more details.

Remark

Therefore separated schemes are analogous to Hausdorff topological spaces (which are also sometimes called ‘separated’) and more generally of Hausdorff toposes. The characterization in terms of the diagonal map is precisely the same as that used for Hausdorff locales. See separated geometric morphism for more.

Properties

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Revised on May 21, 2014 22:54:05 by Urs Schreiber (82.136.246.44)