algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a

morphism of schemes In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generali ...

from to is called quasi-separated if the diagonal map from to is

quasi-compact In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...

(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 The Standard Performance Evaluation Corporation (SPEC) is a non-profit consortium that establishes and maintains standardized benchmarks and performance evaluation tools for new generations of computing systems. SPEC was founded in 1988 and i ...

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, ...

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 re ...

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 In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and ...

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 category theory, a branch of mathematics, group objects are certain generalizations of group (mathematics), groups that are built on more complicated structures than Set (mathematics), sets. A typical example of a group object is a topological gr ...

in the

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

of algebraic spaces that is not a scheme; quasi-separated algebraic spaces that are group objects are always

group scheme In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups hav ...

s. 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 Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an or ...

References

*{{EGA , book=IV-1 Algebraic geometry