Magnification - Revision history

en>JaconaFrere: Reverted 1 edit by 109.152.74.130 identified as test/vandalism using STiki

2014-09-22T21:40:05Z

Reverted 1 edit by 109.152.74.130 identified as test/vandalism using STiki

← Older revision		Revision as of 23:40, 22 September 2014
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	As we all know that higher heel sneakers are not fantastic for our ladies feet particularly the girls who are not adult because they are escalating up but haven't mature. Having said that even though lots of ladies know the damage the superior-heels will do for our toes, they also like to wear a pair of significant-heel because the superior-heels can make a feminine a lot more mature and magnificence. <br><br>		Eusebio Stanfill is what's displayed on my birth records although it is not the name on particular birth certificate. Idaho is our birth install. I work as an order clerk. As a man what Post really like is performing but I'm thinking with regards to starting something new. You can find my website here: http://prometeu.net<br><br>Feel free to visit my website :: hack clash of clans ([http://prometeu.net http://prometeu.net])

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en>Kylegodbey: /* Magnification as a number (optical magnification) */ It appeared to be a mistake made by a previous editor. There was text that made no sense hanging at the end of a sentence.

2014-02-11T14:15:39Z

Magnification as a number (optical magnification): It appeared to be a mistake made by a previous editor. There was text that made no sense hanging at the end of a sentence.

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182.178.78.79: /* Magnification as a number (optical magnification) */

2014-01-18T09:33:18Z

Magnification as a number (optical magnification)

← Older revision		Revision as of 11:33, 18 January 2014
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	~~The person who wrote~~ the ~~short post~~ is called ~~Roberto Ledbetter~~ and ~~his wife isn~~'~~t really like~~ it at all of the. In ~~his professional life he can~~ be a ~~people manager~~. He's ~~always loved living~~ for ~~Guam~~ and ~~he gives you everything~~ that ~~he needs there~~. ~~The precious hobby for him also his kids~~ is ~~farming but he~~'s ~~been bringing~~ on ~~new things~~ a ~~short time ago~~. He's ~~been working on~~ [~~http~~://~~www~~.~~dailymail~~.Co.uk/~~home~~/~~search~~.~~html?sel~~=~~site&searchPhrase~~=~~website website~~] for ~~some time now~~. ~~Check it available here~~: ~~http~~://~~circuspartypanama~~.~~com~~<br><br>~~Here~~ is ~~my page~~ [http://~~circuspartypanama~~.~~com clash~~ of ~~clans triche~~]		In [[algebraic geometry]], a '''proper morphism''' between [[scheme (mathematics)\|schemes]] is a scheme-theoretic analogue of a [[proper map]] between [[Complex analytic variety\|complex-analytic varieties]].

			A basic example is a [[complete variety]] (e.g., [[projective variety]]) in the following sense: a ''k''-variety ''X'' is complete in the classical definition if it is universally closed. A proper morphism is a generalization of this to schemes.

			A [[closed immersion]] is proper. A morphism is finite if and only if it is proper and quasi-finite.

			== Definition ==

			A [[morphism]] ''f'' : ''X'' → ''Y'' of [[algebraic variety\|algebraic varieties]] or more generally of [[Scheme (mathematics)\|schemes]], is called '''universally closed''' if for all morphisms ''Z'' → ''Y'', the projections for the [[fiber product]]
			:<math>X \times_Y Z \to Z</math>
			are [[closed map]]s of the underlying [[topological spaces]]. A [[morphism]] ''f'' : ''X'' → ''Y'' of [[algebraic variety\|algebraic varieties]] is called '''proper''' if it is [[separated morphism\|separated]] and universally closed. A morphism of schemes is called '''proper''' if it is separated, of [[morphism of finite type\|finite type]] and universally closed ([EGA] II, 5.4.1 [http://modular.fas.harvard.edu/scans/papers/grothendieck/PMIHES_1961__8__5_0.pdf]). One also says that ''X'' is proper over ''Y''. A variety ''X'' over a [[field (mathematics)\|field]] ''k'' is [[complete variety\|complete]] when the structural morphism from ''X'' to the spectrum of ''k'' is proper.

			== Examples ==
			The [[projective space]] '''P'''<sup>''d''</sup> 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 morphism]]s are proper, but not all proper morphisms are projective. For example, it can be shown that the scheme obtained by contracting two disjoint [[projective line]]s in some '''P'''<sup>3</sup> to one is a proper, but non-projective variety.<ref>{{Citation \| last1=Ferrand \| first1=Daniel \| title=Conducteur, descente et pincement \| year=2003 \| journal=[[Bulletin de la Société Mathématique de France]] \| issn=0037-9484 \| volume=131 \| issue=4 \| pages=553–585}}, 6.2</ref> [[Affine variety\|Affine varieties]] of non-zero dimension are never complete. More generally, it can be shown that affine proper morphisms are necessarily finite. For example, it is not hard to see that the [[affine line]] '''A'''<sup>1</sup> is not complete. In fact the map taking '''A'''<sup>1</sup> to a point ''x'' is not universally closed. For example, the morphism
			:<math>f \times \textrm{id}: \mathbb{A}^1 \times \mathbb{A}^1 \to \{x\} \times \mathbb{A}^1</math>
			is not closed since the image of the hyperbola ''uv'' = 1, which is closed in '''A'''<sup>1</sup> × '''A'''<sup>1</sup>, is the affine line minus the origin and thus not closed.

			==Properties and characterizations of proper morphisms==
			In the following, let ''f'' : ''X'' → ''Y'' be a morphism of schemes.
			* Properness is a [[local property of a scheme morphism\|local property]] on the base, i.e. if ''Y'' is covered by some open subschemes ''Y<sub>i</sub>'' and the restriction of ''f'' to all ''f<sup>-1</sup>(Y<sub>i</sub>)'' is proper, then so is ''f''.
			* Proper morphisms are [[stable under base change]] and composition.
			* [[Closed immersion]]s are proper.
			* More generally, [[finite morphism]]s are proper. This is a consequence of the [[going up and going down\|going up]] theorem.
			* Conversely, every [[quasi-finite morphism\|quasi-finite]], locally of finite presentation and proper morphism is finite. (EGA III, 4.4.2 in the noetherian case and EGA IV, 8.11.1 for the general case)
			* [[Stein factorization]] theorem states that any proper morphism to a locally noetherian scheme can be factorized into <math>X\to Z\to Y</math>, where the first morphism has geometrically connected fibers and the second on is finite.
			* Proper morphisms are closely related to [[projective morphism]]s: If ''f'' is proper over a [[noetherian scheme\|noetherian]] base ''Y'', then there is a morphism: ''g'': ''X' '' →''X'' which is an isomorphism when restricted to a suitable open dense subset: ''g''<sup>-1</sup>(''U'') ≅ ''U'', such that ''f' '' := ''fg'' is projective. This statement is called [[Chow's lemma]].
			*[[Nagata's compactification theorem]]<ref>B. Conrad, [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.190.9680&rep=rep1&type=pdf Deligne's notes on Nagata compactifications]</ref> says that a separated morphism of finite type between quasi-compact and quasi-separated schemes (e.g., noetherian schemes) factors as an open immersion followed by a proper morphism.
			* Proper morphisms between locally noetherian schemes or complex analytic spaces preserve coherent sheaves, in the sense that the [[higher direct image]]s ''R<sup>i</sup>f''<sub>∗</sub>(''F'') (in particular the [[direct image]] ''f''<sub>∗</sub>(''F'')) of a [[coherent sheaf]] ''F'' are coherent (EGA III, 3.2.1). This boils down to the fact that the cohomology groups of [[projective space]] over some [[field (mathematics)\|field]] ''k'' with respect to coherent sheaves are [[finitely generated module\|finitely generated]] over ''k'', a statement which fails for non-projective varieties: consider '''C'''<sup>∗</sup>, the [[punctured disc]] and its sheaf of [[holomorphic function]]s <math>\mathcal O</math>. Its sections <math>\mathcal O(\mathbb C^*)</math> is the ring of [[Laurent polynomial]]s, which is infinitely generated over '''C'''.
			There is also a slightly stronger statement of this:{{harv\|EGA III\|loc=3.2.4}} let <math>f: X \to S</math> be a morphism of finite type, ''S'' locally noetherian and <math>F</math> a <math>\mathcal{O}_X</math>-module. If the support of ''F'' is proper over ''S'', then for each <math>i \ge 0</math> the [[higher direct image]] <math>R^i f_ F</math> is coherent.:
			*{{harv\|SGA 1\|loc=XII}} If ''X'', ''Y'' are schemes of locally of finite type over the field of complex numbers <math>\mathbb{C}</math>, ''f'' induces a morphism of [[complex analytic space]]s
			*:<math>f(\mathbb{C}): X(\mathbb{C}) \to Y(\mathbb{C})</math>
			:between their sets of complex points with their complex topology. (This is an instance of [[Algebraic geometry and analytic geometry\|GAGA]].) Then ''f'' is a proper morphism defined above if and only if <math>f(\mathbb{C})</math> is a proper map in the sense of Bourbaki and is separated.<ref>{{harvnb\|SGA 1\|loc=XII Proposition 3.2}}</ref>
			* If ''f: X''→''Y'' and ''g:Y''→''Z'' are such that ''gf'' is proper and ''g'' is separated, then ''f'' is proper. This can for example be easily proven using the following criterion

			=== Valuative criterion of properness ===

			[[Image:Valuative criterion of properness.png\|thumb\|300px\|Valuative criterion of properness]] There is a very intuitive criterion for properness which goes back to [[Claude Chevalley\|Chevalley]]. It is commonly called the '''valuative criterion of properness'''. Let ''f'': ''X'' → ''Y'' be a morphism of finite type of [[noetherian scheme]]s. Then ''f'' is proper if and only if for all [[discrete valuation ring]]s ''R'' with [[field of fractions\|fields of fractions]] ''K'' and for any ''K''-valued point ''x'' ∈ ''X''(''K'') that maps to a point ''f''(''x'') that is defined over ''R'', there is a unique lift of ''x'' to <math>\overline{x} \in X(R)</math>. (EGA II, 7.3.8). Noting that ''Spec K'' is the [[generic point]] of ''Spec R'' and discrete valuation rings are precisely the [[regular ring\|regular]] [[local ring\|local]] one-dimensional rings, one may rephrase the criterion: given a regular curve on ''Y'' (corresponding to the morphism ''s : Spec R → Y'') and given a lift of the generic point of this curve to ''X'', ''f'' is proper if and only if there is exactly one way to complete the curve.

			Similarly, ''f'' is separated if and only if in all such diagrams, there is at most one lift <math>\overline{x} \in X(R)</math>.

			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]].

			== Proper morphism of formal schemes ==
			Let <math>f: \mathfrak{X} \to \mathfrak{S}</math> be a morphism between [[locally noetherian formal scheme]]s. We say ''f'' is '''proper''' or <math>\mathfrak{X}</math> is '''proper''' over <math>\mathfrak{S}</math> if (i) ''f'' is an [[adic morphism]] (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map <math>f_0: X_0 \to Y_0</math> is proper, where <math>X_0 = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/I), S_0 = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/K), I = f^*(K) \mathcal{O}_\mathfrak{X}</math> and ''K'' is the ideal of definition of <math>\mathfrak{S}</math>.{{harv\|EGA III\|loc=3.4.1}} The definition is independent of the choice of ''K''. If one lets
			<math>X_n = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/I^{n+1}), S_n = (\mathfrak{X}, \mathcal{O}_\mathfrak{X}/K^{n+1})</math>, then<math>f_n: X_n \to S_n</math> is proper.

			For example, if <math>g: Y \to Z</math> is a proper morphism, then its extension <math>\widehat{g}: \widehat{Y} \to \widehat{Z}</math> between formal completions is proper in the above sense.

			As before, we have the coherence theorem: let <math>f: \mathfrak{X} \to \mathfrak{S}</math> be a proper morphism between locally noetherian formal schemes. If ''F'' is a coherent <math>\mathcal{O}_\mathfrak{X}</math>-module, then the higher direct images <math>R^i f_* F</math> are coherent.

			== See also ==
			* [[Proper base change theorem]]
			* [[Stein factorization]]

			==References==
			{{reflist}}
			* {{Citation \| last1=Grothendieck \| first1=Alexandre \| author1-link=Alexandre Grothendieck \| last2=Dieudonné \| first2=Jean \| author2-link=Jean Dieudonné \| title=Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes \| url=http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1961__8_ \| year=1961 \| journal=[[Publications Mathématiques de l'IHÉS]] \| issn=1618-1913 \| volume=8 \| pages=5–222 \| doi=10.1007/BF02699291}}, section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)
			* {{Citation \| last1=Grothendieck \| first1=Alexandre \| author1-link=Alexandre Grothendieck \| last2=Dieudonné \| first2=Jean \| author2-link=Jean Dieudonné \| title=Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie \| url=http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1966__28_ \| year=1966 \| journal=[[Publications Mathématiques de l'IHÉS]] \| issn=1618-1913 \| volume=28 \| pages=5–255}}, section 15.7. (generalisations of valuative criteria to not necessarily noetherian schemes)
			* {{Citation \| last1=Hartshorne \| first1=Robin \| author1-link= Robin Hartshorne \| title=[[Algebraic Geometry (book)\|Algebraic Geometry]] \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-0-387-90244-9 \| id={{MathSciNet \| id = 0463157}} \| year=1977}}

			==External links==
			*{{springer \|id=P/p075450\|title=Proper morphism\|author=V.I. Danilov}}

			[[Category:Morphisms of schemes]]

en>Gelingvistoj: /* Sheldon's Calculations */ grammar

2012-08-26T05:33:54Z

Sheldon's Calculations: grammar

New page

The person who wrote the short post is called Roberto Ledbetter and his wife isn't really like it at all of the. In his professional life he can be a people manager. He's always loved living for Guam and he gives you everything that he needs there. The precious hobby for him also his kids is farming but he's been bringing on new things a short time ago. He's been working on [http://www.dailymail.Co.uk/home/search.html?sel=site&searchPhrase=website website] for some time now. Check it available here: http://circuspartypanama.com<br><br>Here is my page [http://circuspartypanama.com clash of clans triche]