|
In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, a related concept is used. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
Topological spaces
Definition A function f : X → Y between two topological spaces is proper if and only if the preimage of every compact set in Y is compact in X. Partial plot of a function f. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
An equivalent, possibly more intuitive definition is as follows: We say an infinite sequence of points {pi} in a topological space X escapes to infinity if, for every compact set S ⊂ X, only finitely many points pi are in S. Then a map f : X → Y is proper if and only if, for every sequence of points {pi} that escapes to infinity in X, {f(pi)} escapes to infinity in Y.
Properties - A topological space is compact if and only if the map from that space to a single point is proper.
- Every continuous map from a compact space to a Hausdorff space is both proper and closed.
- If f : X → Y is a proper continuous map and Y is a compactly generated Hausdorff space (this includes Hausdorff spaces which are either first-countable or locally compact), then f is closed.
In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
In topology, an open map is a function between two topological spaces which maps open sets to open sets. ...
In topology, a compactly generated space is a topological space X satisfying the following condition: a subspace A is closed in X if and only if A ∩ K is closed in K for all compact subspaces K ⊆ X. Equivalently, one can replace closed with open in this definition. ...
In topology, a first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ...
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ...
Algebraic varieties and schemes
Definition A morphism f : X → Y of algebraic varieties or schemes is called universally closed if all its fiber products f × Id: X × Z → Y × Z are closed maps of the underlying topological spaces. A morphism f : X → Y of algebraic varieties or is called proper if it is separated and universally closed. A morphism of schemes is called proper if it is separated, of finite type and universally closed ([EGA] II, 5.4.1 [1]). One also says that X is proper over Y. A variety X over a field k is complete when the constant morphism from X to a point is proper. In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...
In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ...
In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...
This is a glossary of scheme theory. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y is a closed map, i. ...
A morphism f : X → Y of algebraic varieties over the field of complex numbers C induces a continuous function The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
between their sets of complex points with their complex topology (see GAGA). It can be shown that f is a proper morphism if and only if f(C) is a proper continuous function. In mathematics, algebraic geometry and analytic geometry are two closely related subjects. ...
Properness is a local property that is stable under base change. The composition of two proper morphisms is proper. Grothendiecks relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of objects explicitly depending on parameters, as the basic field of study, rather than a single such object. ...
In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...
Examples The projective space Pd over a field K is proper, over a point (that is, Spec(K)). In the more classical language, this is the same as saying that projective space is a complete variety. Projective morphisms are proper, but not all proper morphisms are projective. Closed immersions are proper. More generally, finite morphisms are proper. In mathematics, a projective space is a fundamental construction from any vector space. ...
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y is a closed map, i. ...
This is a glossary of scheme theory. ...
In mathematics, in algebraic geometry, a morphism of schemes is a finite morphism, if has an open cover by affine schemes such that for each , is an open affine subscheme , and the restriction of f to , which induces a map of rings makes a finitely generated module over . ...
Affine varieties of non-zero dimension are never proper. For example, it is not hard to see that the affine line A1 is not proper. In fact the map taking A1 to a point x is not universally closed. For example, the morphism f × Id: A1 × A1 → {x} × A1 is not closed since the image of the hyperbola uv = 1, which is closed in A1 × A1, is the affine line minus the origin and thus not closed. This article is about algebraic varieties. ...
In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ...
Valuative criterion of properness There is a very intuitive criterion for properness which goes back to Chevalley. It is commonly called the valuative criterion of properness. Let f: X → Y be a morphism of finite type, with X noetherian. Then f is proper if and only if for all valuation rings R with fields of fractions K all K-valued points x ∈ X(K) that map to a point f(x) that is defined over R there is a unique lift of x to . (EGA II, 7.3.8 [2]) Claude Chevalley (11 February 1909 - 28 June 1984) was a French mathematician with an austere style based on abstract algebra. ...
This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
In abstract algebra, local rings are certain rings that are comparatively simple and serve to describe the local behavior of functions defined on varieties or manifolds. ...
In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ...
For example, the projective line is proper over a field (or even over Z) since one can always scale homogeneous co-ordinates by their least common denominator. In mathematics, a projective line is a one-dimensional projective space. ...
In mathematics, homogeneous co-ordinates, introduced by August Ferdinand Möbius, make calculations possible in projective space just as Cartesian co-ordinates do in Euclidean space. ...
In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. ...
Stein factorization A deep property of proper morphisms is the existence of a Stein factorization, namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism. (EGA III, 4.3.3 [3]) In mathematics, in algebraic geometry, a morphism of schemes is a finite morphism, if has an open cover by affine schemes such that for each , is an open affine subscheme , and the restriction of f to , which induces a map of rings makes a finitely generated module over . ...
See also |
Feeds |
Contact