Quasi-separated morphism

In algebraic geometry, a morphism of schemes from to is called quasi-separated if the Diagonal [morphism (algebraic geometry)|diagonal map] from to is quasi-compact. A scheme is called quasi-separated if the morphism to Spec 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 is quasi-separated as part of the definition of an algebraic space or algebraic stack. Quasi-separated morphisms were introduced by as a generalization of separated morphisms.
All separated morphisms 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.

Topological description

We say a topological space is quasi-separated if the intersection of two open quasi-compact subsets of is quasi-compact. We say that a continuous map of topological spaces from to is quasi-separated if the inverse image along of every open quasi-separated subset of is quasi-separated. Then a scheme is quasi-separated in the scheme-theoretic sense if and only if it is quasi-separated in the topological sense, see.

Examples

If is a locally Noetherian scheme then any morphism from to any scheme is quasi-separated, and in particular 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 is an "infinite dimensional vector space with two origins" over a field then the morphism from to spec is not quasi-separated. More precisely consists of two copies of Spec 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 is a field of characteristic then the quotient of the affine line by the group 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 by an infinite subgroup, or the quotient of the complex numbers by a lattice.