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×_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.

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[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 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.