separated scheme

A scheme X is defined to be a separated scheme if the morphism


into the fibre product X×SpecX which is induced by the identity mapsMathworldPlanetmath i:XX in each coordinate is a closed immersion.

Note the similarityMathworldPlanetmath to the definition of a Hausdorff topological space. In the situation of topological spacesMathworldPlanetmath, a space X is Hausdorff if and only if the diagonal morphism XX×X is a closed embeddingMathworldPlanetmath of topological spaces. The definition of a separated scheme is very similarMathworldPlanetmath, except that the topological product is replaced with the scheme fibre product.

More generally, if X is a scheme over a base scheme Y, the scheme X is defined to be separated over Y if the diagonal embedding


is a closed immersion.

Title separated scheme
Canonical name SeparatedScheme
Date of creation 2013-03-22 12:50:25
Last modified on 2013-03-22 12:50:25
Owner djao (24)
Last modified by djao (24)
Numerical id 6
Author djao (24)
Entry type Definition
Classification msc 14A15
Defines separated