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  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.