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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]].
| | 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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| 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.
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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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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
Feel free to visit my website :: hack clash of clans (http://prometeu.net)