quasi-separated morphism
In algebraic geometry, a morphism of schemes {{math|f}} from {{math|X}} to {{math|Y}} is called quasi-separated if the diagonal map from {{math|X}} to {{math|X ×Y X}} is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi-compact). A scheme {{math|X}} is called quasi-separated if the morphism to Spec {{math|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 {{math|X}} is quasi-separated as part of the definition of an algebraic space or algebraic stack {{math|X}}. Quasi-separated morphisms were introduced by {{harvtxt|Grothendieck|Dieudonné|1964|loc=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 {{math|X}} is a locally Noetherian scheme then any morphism from {{math|X}} to any scheme is quasi-separated, and in particular {{math|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 {{math|X}} is an "infinite dimensional vector space with two origins" over a field {{math|K}} then the morphism from {{math|X}} to spec {{math|K}} is not quasi-separated. More precisely {{math|X}} consists of two copies of Spec {{math|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 {{math|K}} is a field of characteristic {{math|0}} then the quotient of the affine line by the group {{math|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 {{math|Gm}} by an infinite subgroup, or the quotient of the complex numbers by a lattice.
References
- {{EGA | book=IV-1}}