Let be a morphism of schemes. Write for the diagonal morphism.
The morphism is called separated if is a closed subspace of .
A scheme is called separated if the terminal morphism is separated.
Let be a scheme (resp. a locally noetherian scheme), a morphism of schemes (resp. a morphism locally of finite type). The following conditions are equivalent.
is separated.
The diagonal morphism is quasicompact, and for every affine scheme in which is a valuation ring (resp. a discrete valuation ring), any two morphisms from which coincide at the generic point of are equal.
The diagonal morphism is quasicompact, and for every affine scheme of the form in which is a valuation ring (resp. a discrete valuation ring), any two sections of which coincide at the generic point of are equal.
This is the valuative criterion of separatedness. See Hartshorne or EGA II for more details.
The definition of a separated scheme is formally similar to the definition of a Hausdorff space which says that the diagonal 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.
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Last revised on August 24, 2024 at 14:05:25. See the history of this page for a list of all contributions to it.