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| 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]].
| | 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> |
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| 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.
| | So are there any large heels would not do damage to our youthful girl's toes. That signifies our small female will sense at ease when sporting the footwear. Currently I might like to introduce some footwear which is suited for the tiny woman.<br><br>one. Half heel shoes<br>If you want to be mature and comfortable when sporting a pair of large-heel, the thick heel shoes is the ideal decision considering that the 50 percent heel can launch your height. Seem at the adhering to photo, with the total Korean design the piscine mouth shoe has turn out to be the most common sneakers in the summer. It is a pair of sneakers with a vibrant shade thick heel. In the sizzling summer months this piscine mouth shoe can will make you the two class and neat. <br><br>two. Thick large heel<br>The shoes are 50 percent sweet and half experienced are always the common shoes for the girls who are nonetheless in the Ivory Tower. A pair of sweet woman-like shoes can be worn in every single year. There is wonderful butterfly tie in the center of the footwear upper, it tends to make the large-heel has a minimal sweet sensation in the similar time. <br><br>A bit of fur all around the heel can keep your toes warm in the cold time. Moreover, the thick large heel can equally fulfill the young girl's need of putting on superior heel shoes and come to be comfy when going for walks. It is a not lousy selection!<br><br>three. Wedge heel shoes<br>White color wedge heel sneakers are kinds of romantic sneakers which are really suited for the very little ladies who are continue to dreaming of princess and prince. The wedge heel is a superior choice for the slender higher heel and will do fewer hurt to the little girl's feet.<br><br>Furthermore, the shiny coloration is really quick to match with any clothing. The most significant level is the shoe is a pair light-weight sneakers that you would not come to feel hefty when you are going for walks.<br><br>four. UGG superior heel boot.<br>Each grownup lady could have a stunning boot, so does the very little female, they also want a good and really boot in the wintertime. Effectively. The UGG boot has develop into the scorching wind in this 12 months. <br><br>When is the New 12 months is coming, the temperature turn out to be colder and colder. A boot both equally attractive and warm is in urgent require of women. The UGG high heel boot is the best alternative for the winter season because it was produced of sheepskin and pure wool.<br><br>If you loved this short article and you would like to receive details about [http://tinyurl.com/k7shbtq http://tinyurl.com/k7shbtq] generously visit the web-site. |
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| A [[closed immersion]] is proper. A morphism is finite if and only if it is proper and quasi-finite.
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| == Definition ==
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| 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]]
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| :<math>X \times_Y Z \to Z</math>
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| 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.
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| == Examples ==
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| 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
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| :<math>f \times \textrm{id}: \mathbb{A}^1 \times \mathbb{A}^1 \to \{x\} \times \mathbb{A}^1</math>
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| 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.
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| ==Properties and characterizations of proper morphisms==
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| In the following, let ''f'' : ''X'' → ''Y'' be a morphism of schemes.
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| * 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''.
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| * Proper morphisms are [[stable under base change]] and composition.
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| * [[Closed immersion]]s are proper.
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| * More generally, [[finite morphism]]s are proper. This is a consequence of the [[going up and going down|going up]] theorem.
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| * 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)
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| * [[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.
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| * 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]].
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| *[[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.
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| * 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'''.
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| *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.:
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| *{{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
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| *:<math>f(\mathbb{C}): X(\mathbb{C}) \to Y(\mathbb{C})</math>
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| :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>
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| * 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
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| === Valuative criterion of properness ===
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| [[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.
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| 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>.
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| 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]].
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| == Proper morphism of formal schemes ==
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| 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
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| <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.
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| 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.
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| 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.
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| == See also ==
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| * [[Proper base change theorem]]
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| * [[Stein factorization]]
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| ==References==
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| {{reflist}}
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| * {{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)
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| * {{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)
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| * {{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}}
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| ==External links==
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| *{{springer |id=P/p075450|title=Proper morphism|author=V.I. Danilov}}
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| [[Category:Morphisms of schemes]]
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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.
So are there any large heels would not do damage to our youthful girl's toes. That signifies our small female will sense at ease when sporting the footwear. Currently I might like to introduce some footwear which is suited for the tiny woman.
one. Half heel shoes
If you want to be mature and comfortable when sporting a pair of large-heel, the thick heel shoes is the ideal decision considering that the 50 percent heel can launch your height. Seem at the adhering to photo, with the total Korean design the piscine mouth shoe has turn out to be the most common sneakers in the summer. It is a pair of sneakers with a vibrant shade thick heel. In the sizzling summer months this piscine mouth shoe can will make you the two class and neat.
two. Thick large heel
The shoes are 50 percent sweet and half experienced are always the common shoes for the girls who are nonetheless in the Ivory Tower. A pair of sweet woman-like shoes can be worn in every single year. There is wonderful butterfly tie in the center of the footwear upper, it tends to make the large-heel has a minimal sweet sensation in the similar time.
A bit of fur all around the heel can keep your toes warm in the cold time. Moreover, the thick large heel can equally fulfill the young girl's need of putting on superior heel shoes and come to be comfy when going for walks. It is a not lousy selection!
three. Wedge heel shoes
White color wedge heel sneakers are kinds of romantic sneakers which are really suited for the very little ladies who are continue to dreaming of princess and prince. The wedge heel is a superior choice for the slender higher heel and will do fewer hurt to the little girl's feet.
Furthermore, the shiny coloration is really quick to match with any clothing. The most significant level is the shoe is a pair light-weight sneakers that you would not come to feel hefty when you are going for walks.
four. UGG superior heel boot.
Each grownup lady could have a stunning boot, so does the very little female, they also want a good and really boot in the wintertime. Effectively. The UGG boot has develop into the scorching wind in this 12 months.
When is the New 12 months is coming, the temperature turn out to be colder and colder. A boot both equally attractive and warm is in urgent require of women. The UGG high heel boot is the best alternative for the winter season because it was produced of sheepskin and pure wool.
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