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In [[algebraic geometry]], a '''regular map''' between [[Algebraic variety#Affine_varieties|affine varieties]] is a mapping which is given by polynomials. To be explicit, suppose ''X'' and ''Y'' are [[subvarieties]] (or algebraic subsets) of '''A'''<sup>''n''</sup> resp. '''A'''<sup>''m''</sup>. A regular map ''f'' from ''X'' to ''Y'' has the form <math>f = (f_1, \dots, f_m)</math> where the <math>f_i</math> are in <math>k[x_1, \dots, x_n]/I</math>, ''I'' the ideal defining ''X'', so that the image <math>f(X)</math> lies in ''Y''; i.e., satisfying the defining equations of ''Y''. <ref>This is perhaps the simplest definition and agrees with the more traditional definitions; cf. {{harvnb|Milne|loc=Proposition 3.16}}</ref> | |||
<!-- This doesn't make much sense since we haven't defined regular on an open set.-->More generally, a map ƒ:''X''→''Y'' between two (quasi-projective) [[abstract variety|varieties]] is '''regular at a point''' ''x'' if there is a neighbourhood ''U'' of ''x'' and a neighbourhood ''V'' of ƒ(''x'') such that the restricted function ƒ:''U''→''V'' is regular. Then ƒ is called '''regular''', if it is regular at all points of ''X''. | |||
and | In the particular case that '''Y''' equals '''A'''<sup>1</sup> the map ƒ:''X''→'''A'''<sup>1</sup> is called a '''regular function''', and correspond to [[scalar function]]s in differential geometry. In other words, a scalar function is regular at a point ''x'' if, in a neighborhood of ''x'', it is a [[rational function]] (i.e., a fraction of polynomials) such that the denominator does not vanish at ''x''. The [[ring of regular functions]] (that is the [[coordinate ring]] or more abstractly the ring of global sections of the structure sheaf) is a fundamental object in affine algebraic geometry. The only regular function on a connected [[projective variety]] is constant; thus, in the projective case, one usually considers the global sections of a line bundle (or divisor) instead. (this can be viewed as an algebraic analogue of [[Liouville's theorem (complex analysis)|Liouville's theorem]] in [[complex analysis]]) | ||
In fact taking the [[function field of an algebraic variety|function field]] ''k''(''V'') of an [[irreducible variety|irreducible]] [[algebraic curve]] ''V'', the functions ''F'' in the function field may all be realised as morphisms from ''V'' to the [[projective line]] over ''k''. The image will either be a single point, or the whole projective line (this is a consequence of the [[completeness of projective varieties]]). That is, unless ''F'' is actually constant, we have to attribute to ''F'' the value ∞ at some points of ''V''. Now in some sense ''F'' is no worse behaved at those points than anywhere else: ∞ is just the chosen [[point at infinity]] on the projective line, and by using a [[Möbius transformation]] we can move it anywhere we wish. But it is in some way inadequate to the needs of geometry to use only the affine line as target for functions, since we shall end up only with constants. | |||
Regular maps are, by definition, [[morphisms]] in the [[Algebraic geometry#The category of affine varieties|category of algebraic varieties]]. In particular, a regular map between affine varieties corresponds contravariantly in one-to-one to a [[ring homomorphism]] between the coordinate rings. | |||
A regular map whose inverse is also regular is called '''biregular''', and are [[isomorphism]]s in the category of algebraic varieties. A morphism between algebraic varieties that is a homeomorphism between the underlying topological spaces need not be an isomorphism (a counterexample is given by a [[Frobenius morphism]] <math>t \mapsto t^p</math>.) | |||
Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective space – the weaker condition of a [[rational map]] and [[birational]] maps are frequently used as well. | |||
A regular map between [[complex algebraic variety|complex algebraic varieties]] is a [[holomorphic map]]. (There is actually a slight technical difference: a regular map is a holomorphic map with [[removable singularity|removable singularities]], but the distinction is usually ignored in practice.) In particular, a regular map into the complex numbers is just a usual [[holomorphic function]] (complex-analytic function). | |||
== See also == | |||
* [[Algebraic function]] | |||
* [[Étale morphism]]s – The algebraic analogue of [[local diffeomorphism]]s. | |||
==References== | |||
{{reflist}} | |||
*{{cite book | |||
| author = [[Robin Hartshorne]] | |||
| year = 1997 | |||
| title = [[Hartshorne's Algebraic Geometry|Algebraic Geometry]] | |||
| publisher = [[Springer Science+Business Media|Springer-Verlag]] | |||
| isbn = 0-387-90244-9 | |||
}} | |||
*{{cite book | |||
| author = [[Igor Shafarevich]] | |||
| year = 1995 | |||
| title = Basic Algebraic Geometry I: Varieties in Projective Space | |||
| edition = 2nd | |||
| publisher = [[Springer Science+Business Media|Springer-Verlag]] | |||
| isbn = 0-387-54812-2 | |||
}} | |||
*Milne, [http://www.jmilne.org/math/CourseNotes/ag.html Algebraic geometry] | |||
[[Category:Algebraic varieties]] | |||
[[Category:Types of functions]] | |||
[[Category:Functions and mappings]] | |||
Revision as of 09:46, 9 November 2013
In algebraic geometry, a regular map between affine varieties is a mapping which is given by polynomials. To be explicit, suppose X and Y are subvarieties (or algebraic subsets) of An resp. Am. A regular map f from X to Y has the form where the are in , I the ideal defining X, so that the image lies in Y; i.e., satisfying the defining equations of Y. [1]
More generally, a map ƒ:X→Y between two (quasi-projective) varieties is regular at a point x if there is a neighbourhood U of x and a neighbourhood V of ƒ(x) such that the restricted function ƒ:U→V is regular. Then ƒ is called regular, if it is regular at all points of X.
In the particular case that Y equals A1 the map ƒ:X→A1 is called a regular function, and correspond to scalar functions in differential geometry. In other words, a scalar function is regular at a point x if, in a neighborhood of x, it is a rational function (i.e., a fraction of polynomials) such that the denominator does not vanish at x. The ring of regular functions (that is the coordinate ring or more abstractly the ring of global sections of the structure sheaf) is a fundamental object in affine algebraic geometry. The only regular function on a connected projective variety is constant; thus, in the projective case, one usually considers the global sections of a line bundle (or divisor) instead. (this can be viewed as an algebraic analogue of Liouville's theorem in complex analysis)
In fact taking the function field k(V) of an irreducible algebraic curve V, the functions F in the function field may all be realised as morphisms from V to the projective line over k. The image will either be a single point, or the whole projective line (this is a consequence of the completeness of projective varieties). That is, unless F is actually constant, we have to attribute to F the value ∞ at some points of V. Now in some sense F is no worse behaved at those points than anywhere else: ∞ is just the chosen point at infinity on the projective line, and by using a Möbius transformation we can move it anywhere we wish. But it is in some way inadequate to the needs of geometry to use only the affine line as target for functions, since we shall end up only with constants.
Regular maps are, by definition, morphisms in the category of algebraic varieties. In particular, a regular map between affine varieties corresponds contravariantly in one-to-one to a ring homomorphism between the coordinate rings.
A regular map whose inverse is also regular is called biregular, and are isomorphisms in the category of algebraic varieties. A morphism between algebraic varieties that is a homeomorphism between the underlying topological spaces need not be an isomorphism (a counterexample is given by a Frobenius morphism .)
Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective space – the weaker condition of a rational map and birational maps are frequently used as well.
A regular map between complex algebraic varieties is a holomorphic map. (There is actually a slight technical difference: a regular map is a holomorphic map with removable singularities, but the distinction is usually ignored in practice.) In particular, a regular map into the complex numbers is just a usual holomorphic function (complex-analytic function).
See also
- Algebraic function
- Étale morphisms – The algebraic analogue of local diffeomorphisms.
References
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- ↑ This is perhaps the simplest definition and agrees with the more traditional definitions; cf. Template:Harvnb