In algebraic geometry, a morphism of schemes from to is called
quasi-separated if the diagonal map from to is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi-compact). A scheme is called quasi-separated if the morphism to Spec is quasi-separated. Quasi-separated
and
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 (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.
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 (resp., a morphism of schemes) is quasi-separated in the scheme-theoretic sense if and only if it is quasi-separated in the topological sense, see .
Examples
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If is a locally Noetherian scheme then any morphism from to any scheme is quasi-separated, and in particular is a quasi-separated scheme.
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Any separated scheme or morphism is quasi-separated.
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The line with two origins over a field is quasi-separated over the field but not separated.
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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 Gluing schemes together by identifying the nonzero points in each copy.
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The quotient of an algebraic space by an infinite discrete group free action 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 . 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.
