# Quasi-separated morphism

In algebraic geometry, a morphism of schemes *f* from *X* to *Y* is called **quasi-separated** if the diagonal map from *X* to *X* × _{Y}*X* is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi-compact). A scheme *X* is called quasi-separated if the morphism to Spec **Z** is quasi-separated. Quasi-separated algebraic spaces and algebraic stacks and morphisms between them are defined in a similar way, though some authors include the condition that *X* is quasi-separated as part of the definition of an algebraic space or algebraic stack *X*. Quasi-separated morphisms were introduced by (Grothendieck Dieudonné) as a generalization of separated morphisms.
All separated morphisms (and all morphisms of Noetherian schemes) are automatically quasi-separated. Quasi-separated morphisms are important for algebraic spaces and algebraic stacks, where many natural morphisms are quasi-separated but not separated.

The condition that a morphism is quasi-separated often occurs together with the condition that it is quasi-compact.

## Examples

- If
*X*is a locally Noetherian scheme then any morphism from*X*to any scheme is quasi-separated, and in particular*X*is a quasi-separated scheme. - Any separated scheme or morphism is quasi-separated.
- The line with two origins over a field is quasi-separated over the field but not separated.
- If
*X*is an "infinite dimensional vector space with two origins" over a field*K*then the morphism from*X*to spec*K*is not quasi-separated. More precisely*X*consists of two copies of Spec*K*[*x*_{1},*x*_{2},....] glued together by identifying the nonzero points in each copy. - The quotient of an algebraic space by an infinite discrete group acting freely is often not quasi-separated. For example, if
*K*is a field of characteristic 0 then the quotient of the affine line by the group**Z**of integers is an algebraic space that is not quasi-separated. This algebraic space is also an example of a group object in the category of algebraic spaces that is not a scheme; quasi-separated algebraic spaces that are group objects are always group schemes. There are similar examples given by taking the quotient of the group scheme*G*_{m}by an infinite subgroup, or the quotient of the complex numbers by a lattice.

## References

- Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie".
*Publications Mathématiques de l'IHÉS***20**. doi:10.1007/bf02684747. http://www.numdam.org/item/PMIHES_1964__20__5_0.

Original source: https://en.wikipedia.org/wiki/Quasi-separated morphism.
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