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 $X \times Y X$ 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 (1964, 1.2.1) 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. MR 0173675.

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