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 Spec may refer to:
*Specification (technical standard), an explicit set of requirements to be satisfied by a material, product, or service
**datasheet, or "spec sheet"
People
* Spec Harkness (1887-1952), American professional baseball pitcher
...
is quasi-separated. Quasi-separated
algebraic space In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively,
schemes are given by gluing together affine schemes using the Zariski topology, w ...
s and
algebraic stack
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's repr ...
s 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.
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 In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff. ...
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.
References
*{{EGA , book=IV-1
Algebraic geometry