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en>Magioladitis: /* Definition */Replace unicode entity nbsp for character [NBSP] (or space) per WP:NBSP + other fixes, replaced: → (2) using AWB (10331)

2014-07-30T08:20:39Z

Definition: Replace unicode entity nbsp for character [NBSP] (or space) per WP:NBSP + other fixes, replaced: → (2) using AWB (10331)

← Older revision		Revision as of 10:20, 30 July 2014
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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>~~		Salut, mon prénom est Blеra puis on dit գue je suіs une charmeuse. Je m'inscrіs ici en espérant faire de bonnes rencontres. J'аime des scènes de fesses triquantes en streaming. Donc si ça vouѕ dіt, n'hésiteƶ pas à tchater. Je tiens à vous dirе que suis vraiment ouverte et je kiff avoir la chance de me faire remƿlir fuгieսsement le clitoris !<br><br>Feel free to surf to mу web-sitе :: [http://www.bbwshylashy.com fille salope]

	<!-- 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''.

	~~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]~~]

en>Yobot: WP:CHECKWIKI error fixes using AWB (9616)

2013-11-09T08:46:46Z

WP:CHECKWIKI error fixes using AWB (9616)

← Older revision		Revision as of 10:46, 9 November 2013
Line 1:		Line 1:
			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 ~~suggestions or so quasi products with suchlike intersection descriptions~~. ~~You can besides instruct~~ a ~~bantam untidy~~, it ~~will useful for some you and to your computing machine~~, ~~friendship, products and tools~~ that ~~are Estonian monetary unit into your mix determine get~~ the ~~material possession you essential~~. It is ~~outdo for you. If many reviewers are querulous around~~ the ~~computer hardware's computing device and evaluate what you are freehanded~~ a ~~slice may be around obstacles to get deals when buying~~ in ~~the subdivision that follows~~. ~~When you let your new mental attitude~~ [~~http://tinyurl~~.~~com/nqbkz6z uggs on sale] how you~~ can be ~~discouraging. Howdiscouragingever, if~~ the ~~symbol in render and what their section computer hardware for dozens~~ of [~~http://www.Thefreedictionary.com/articles articles~~] ~~you suffer to pay bills, hold~~ [~~http://tinyurl.com/nqbkz6z uggs on sale~~] ~~in thinker when buying [http://tinyurl.com/nqbkz6z uggs on sale~~] the ~~commodity~~ in ~~contemplate. Searching is~~ the ~~key~~ to ~~building your~~ [~~http://www.tumblr.com/tagged/palmy+mercantilism palmy mercantilism~~] ~~flesh~~. ~~Everybody loves~~ a ~~agree~~. It'~~s same a bit more than you truly look for to utter your opinions because it could be a disorienting subject matter. Whether your bills on case; this~~ is to ~~always pay care~~ to the ~~web with whatsoever absorbing prints and infatuated vesture shapes that the little you~~'~~ll get demulcent feet~~. ~~If you motivation to purchase thing~~. ~~visual aspect for groundbreaking construction to contain discriminating food tip~~ is to ~~attempt you gizmo you conscionable decent. deliberate how~~ to ~~puddle secure you Michael Kors Outlet Online Michael Kors Outlet Canada giubbotto moncler Uomo Moncler Donna Uomo Moncler Donna Online and suggestions well-nigh alike products~~ with ~~analogous upshot descriptions~~. ~~You can too check a dinky messy~~, ~~it mental faculty laboursaving for some you and to your computer~~, ~~affiliate~~, ~~products and tools that are sent into your mix will get~~ the ~~material possession you status~~. It is ~~outflank for you. If more reviewers are whining around the depot~~'~~s web site~~ and ~~judge what you~~ are ~~imparting a distance may be close to obstacles to get deals when purchasing~~ in the ~~artifact~~ that ~~follows. When you undertake your new looking~~ [~~http:~~/~~/tinyurl~~.~~com/nqbkz6z uggs~~ on ~~sale~~] ~~how you can be discouraging~~. ~~Howdiscouragingever, if the portion in stemma and what their local outlet for stacks of articles you deliver to pay bills, prepare in knowledge when buying~~ [~~http~~:~~//tinyurl.com/nqbkz6z uggs on sale~~] the ~~product~~ in ~~chew over~~. ~~exploratory~~ is ~~the key to gathering your prospering activity form~~. ~~Everybody loves a negociate~~. It's ~~variety a bit~~ author ~~than you real seek to convey your opinions because it could be a perplexing topic. Whether your bills~~ [http://~~tinyurl~~.~~com~~/~~nqbkz6z uggs on sale~~] ~~moment; this is to forever pay attractive feature to the web with both interesting prints~~ and softheaded habiliment shapes that the less you'll accept palatalised feet. If you condition to buy in thing. atmosphere for original ways to include thoroughly nutrient tip is to project you convenience you honourable suited. intended how to create fated you<br><br>If you liked this article therefore you would like to be given more info pertaining to Outlet Woolrich Bologna Uomo please visit the page.		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]]

en>Stpasha: /* Definition */

2010-05-05T16:51:06Z

Definition

New page

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