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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.

Warning

The definition of a separated scheme is formally similar to the definition of a Hausdorff space which says that the diagonal Δ(X)X×X\Delta(X) \subseteq X \times X is closed; the same pattern is followed in the definition of a Hausdorff locale, Hausdorff topos, etc. More generally, the definition of a separated morphism of schemes is formally similar to e.g. a separated geometric morphism. This leads to these properties having similar formal properties. Nevertheless, because finite products and pullbacks in these categories do not necessarily agree, these notions of separation also vary. For example, the underlying topological space of a separated scheme is typically not Hausdorff.

Properties

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Last revised on October 1, 2018 at 15:12:34. See the history of this page for a list of all contributions to it.