In algebraic geometry, a morphism of schemes f from X to Y is called quasi-separated if the diagonal map from X to X×YX 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 (1964, 1.2.1) as a generalization of separated morphisms.
The concept of quasi-separated morphisms does not usually appear in introductory courses on algebraic geometry, because all separated morphisms (and all morphisms of Noetherian schemes) are automatically quasi-separated. However quasi-separated morphisms are more 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.
- 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[x1,x2,....] 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 Gm by an infinite subgroup, or the quotient of the complex numbers by a lattice.