Keystone effect - Revision history

46.64.61.239: Unsplit paragraph.

2014-08-24T09:16:46Z

Unsplit paragraph.

← Older revision		Revision as of 11:16, 24 August 2014
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	~~The title of the author is Garland~~. ~~My house~~ is ~~now in Kansas. One of his favorite hobbies is taking part in crochet~~ but he ~~hasn't made a dime with~~ it. Managing people is what I do in my working day ~~job~~.<br><br>My blog post ... auto warranty ([http://~~Bayi~~.~~gmy.com.tr~~/~~UserProfile~~/~~tabid~~/57/~~userId~~/~~6646/Default.aspx great post to read~~])		Hi there, I am Yoshiko Villareal but I by no means truly favored that name. What he really enjoys doing is to perform handball but he is having difficulties to find time for it. Managing people is what I do in my working day occupation. Years ago we moved to Kansas.<br><br>My blog post ... extended auto warranty - [http://Schnitzls-superdudes.de/index/users/view/id/18715 linked web page] -

en>Wavelength: inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

2014-02-27T20:14:01Z

inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

← Older revision		Revision as of 22:14, 27 February 2014
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	In [[mathematics]], the '''standard conjectures''' about algebraic cycles are several [[conjecture]]s describing the relationship of [[algebraic cycle]]s and [[Weil cohomology theory\|Weil cohomology theories]]. One of the original applications of these conjectures, envisaged by [[Alexander Grothendieck]], was to prove that his construction of [[motive (algebraic geometry)\|pure motives]] gave an [[abelian category]] that is [[semisimple category\|semisimple]]. Moreover, as he pointed out, the standard conjectures also imply the hardest part of the [[Weil conjectures]], namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by [[Pierre Deligne]]; for details on the link between Weil and standard conjectures, see {{harvtxt\|Kleiman\|1968}}. The standard conjectures remain open problems, so that their application gives only [[conditional proof]]s of results. In quite a few cases, including that of the Weil conjectures, other methods have been found to prove such results unconditionally.		The title of the author is Garland. My house is now in Kansas. One of his favorite hobbies is taking part in crochet but he hasn't made a dime with it. Managing people is what I do in my working day job.<br><br>My blog post ... auto warranty ([http://Bayi.gmy.com.tr/UserProfile/tabid/57/userId/6646/Default.aspx great post to read])

	The classical formulations of the standard conjectures involve a fixed Weil cohomology theory ''H''. All of the conjectures deal with "algebraic" cohomology classes, which means a morphism on the cohomology of a smooth projective variety

	~~:H(X) → H(X)~~

	~~induced by an algebraic cycle with rational coefficients on the product ''X'' × ''X'' via the ''cycle class map ''(which~~ is ~~part of the structure of a Weil cohomology theory)~~.

	~~== Lefschetz type standard conjecture ==~~
	~~''Lefschetz type standard conjecture'', also called conjecture B:~~ One of ~~the axioms of a Weil theory~~ is ~~the so-called~~ '~~'hard Lefschetz theorem'' (or axiom): for~~ a ~~fixed smooth [[hyperplane section]]~~

	~~:''W'' = ''H'' ∩ ''X'',~~

	~~for ''H'' some hyperplane in the ambient projective space '''P'''<sup>N</sup> containing the given smooth projective variety ''X'', the Lefschetz operator~~

	~~:''L'' : ''H<sup>i''</sup>(''X'') → ''H''<sup>''i''+2</sup>,~~

	~~which is defined by intersecting cohomology classes~~ with ~~''W'' gives an isomorphism~~

	~~:''L<sup>n-i</sup>: H<sup>i</sup>(X) → H<sup>2n-i</sup>(X)'' (''i'' ≦ ''n = dim X'')~~.

	~~Define~~

	~~:''Λ : H<sup>i</sup>(X)'' → ''H<sup>i-2</sup>(X)'' for 'i'' ≦ ''n''~~

	~~be the composition~~

	~~:''(L<sup>n-i+2</sup>)<sup>-1</sup> <math>\circ L \circ </math> (L<sup>n-i</sup>)''~~

	~~and~~

	~~:''Λ : H<sup>2n-i+2</sup>(X)'' → ''H<sup>2n-i</sup>(X)''~~

	by

	~~:''(L<sup>n-i</sup>) <math>\circ L \circ</math> (L<sup>n-i+2</sup>)<sup>-1</sup>''.~~

	~~The Lefschetz conjecture states that the ''Lefschetz operator'' ''Λ''~~ is ~~induced by an algebraic cycle.~~

	~~== Künneth type standard conjecture ==~~
	The Lefschetz conjecture implies the ''Künneth type standard conjecture'', also called conjecture C: It is conjectured that the projectors ''H''<sup>&lowast;</sup>(''X'') ↠ ''H''<sup>i</sup>(''X'') ↣ ''H''<sup>&lowast;</sup>(''X'') are algebraic, i.e. ~~induced by a cycle ''π~~<~~sup~~>i<~~/sup~~>~~'' ⊂ ''X × X'' with rational coefficients. This implies that every pure motive ''M'' decomposes in graded pieces of pure weights (see [[motive (algebraic geometry)\|motives]]).~~
	~~The conjecture is known to hold for curves, surfaces and abelian varieties.~~

	~~== Conjecture ''D'' (numerical equivalence vs. homological equivalence) ==~~
	''Conjecture D'' states that numerical equivalence and homological equivalence agree. (It implies in particular the latter does not depend on the choice of the Weil cohomology theory). This conjecture implies the Lefschetz conjecture. If the Hodge standard conjecture holds, then the Lefschetz conjecture and Conjecture D are equivalent.

	~~== The Hodge standard conjecture ==~~
	The Hodge standard conjecture is modelled on the [[Hodge index theorem]]. It states the definiteness (positive or negative, according to the dimension) of the cup product pairing on primitive algebraic cohomology classes. ~~If it holds, then the Lefschetz conjecture implies Conjecture ''D''~~. ~~In characteristic zero the Hodge standard conjecture holds, being a consequence of [[Hodge theory]]~~. ~~In positive characteristic the Hodge standard conjecture is known only for surfaces.~~

	~~The Hodge standard conjecture is not to be confused with the ''[[Hodge conjecture]]'' which states that for smooth projective varieties over '''C''', every rational ''~~(~~p,p)''-class is algebraic. The Hodge conjecture implies the Lefschetz conjecture and conjecture D for varieties over fields of characteristic zero. Likewise for fields of finite characteristic the~~ [~~[Tate conjectures]] in [[etale cohomology\|ℓ-adic cohomology]] imply the Lefschetz conjecture.~~

	~~== References ==~~
	*{{citation \| first = A. \| last = Grothendieck \| authorlink = Alexander Grothendieck \| url = http://~~www~~.~~math~~.~~jussieu~~.fr/~~~leila~~/~~grothendieckcircle~~/~~StandardConjs~~.pdf \| contribution = Standard Conjectures on Algebraic Cycles \| title = Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) \| publisher = Oxford University Press \| year = 1969 \| pages = 193–199 \| mr = 0268189}}.
	*{{citation \| last1=Kleiman \| first1=Steven L. \| authorlink=Steven Kleiman \| title=Dix exposés sur la cohomologie des schémas \| publisher=North-Holland \| location=Amsterdam \| mr = 0292838 \| year=1968 \| chapter=Algebraic cycles and the Weil conjectures \| pages=359–386}}.
	*{{citation \| last1=Kleiman \| first1=Steven L. \| authorlink=Steven Kleiman \| contribution=The standard conjectures \| title = Motives (Seattle, WA, 1991) ~~\| year = 1994 \| pages = 3–20 \| series = Proceedings of Symposia in Pure Mathematics \| volume = 55 \| publisher = American Mathematical Society \| mr = 1265519}}.~~

	~~[[Category:Algebraic geometry]]~~
	~~[[Category:Conjectures]]~~

en>Rgdboer: /* See also */ fx lk

2013-08-29T23:19:08Z

See also: fx lk

← Older revision		Revision as of 01:19, 30 August 2013
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	[~~http://automobiles.Honda.com/certified-pre-owned/warranty.aspx auto warranty~~] ~~Hello friend. Let me introduce myself. I am Luther Aubrey. Managing people is how I make cash and it~~'s ~~something I really appreciate~~. ~~Alabama~~ is ~~exactly where~~ he and ~~his spouse reside~~ and ~~he has extended car warranty every thing~~ that ~~he requirements~~ [~~http://www~~.~~friscowebsolutions~~.~~com/UserProfile/tabid/379/userId/203/Default~~.~~aspx car warranty] there~~. One of ~~my preferred hobbies~~ is [~~https~~://~~Www~~.~~Geico.com~~/~~getaquote~~/~~auto~~/~~mechanical~~-~~breakdown~~-~~insurance~~/ ~~tenting]~~ and ~~now I~~'~~m attempting to [http~~://~~www~~.~~motortrend~~.~~com~~/~~new_cars~~/~~auto_warranty~~/ ~~earn cash] with it~~.<br><br>~~Feel free~~ to ~~surf~~ to ~~my webpage;~~ [~~http://brazil~~.~~amor~~-~~amore~~.~~com/nilindon car warranty~~] - [http://www.~~carelion~~.~~com~~/~~UserProfile~~/~~tabid~~/~~61/userId/107768/Default~~.~~aspx Read Much more~~] -		In [[mathematics]], the '''standard conjectures''' about algebraic cycles are several [[conjecture]]s describing the relationship of [[algebraic cycle]]s and [[Weil cohomology theory\|Weil cohomology theories]]. One of the original applications of these conjectures, envisaged by [[Alexander Grothendieck]], was to prove that his construction of [[motive (algebraic geometry)\|pure motives]] gave an [[abelian category]] that is [[semisimple category\|semisimple]]. Moreover, as he pointed out, the standard conjectures also imply the hardest part of the [[Weil conjectures]], namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by [[Pierre Deligne]]; for details on the link between Weil and standard conjectures, see {{harvtxt\|Kleiman\|1968}}. The standard conjectures remain open problems, so that their application gives only [[conditional proof]]s of results. In quite a few cases, including that of the Weil conjectures, other methods have been found to prove such results unconditionally.

			The classical formulations of the standard conjectures involve a fixed Weil cohomology theory ''H''. All of the conjectures deal with "algebraic" cohomology classes, which means a morphism on the cohomology of a smooth projective variety

			:H(X) → H(X)

			induced by an algebraic cycle with rational coefficients on the product ''X'' × ''X'' via the ''cycle class map ''(which is part of the structure of a Weil cohomology theory).

			== Lefschetz type standard conjecture ==
			''Lefschetz type standard conjecture'', also called conjecture B: One of the axioms of a Weil theory is the so-called ''hard Lefschetz theorem'' (or axiom): for a fixed smooth [[hyperplane section]]

			:''W'' = ''H'' ∩ ''X'',

			for ''H'' some hyperplane in the ambient projective space '''P'''<sup>N</sup> containing the given smooth projective variety ''X'', the Lefschetz operator

			:''L'' : ''H<sup>i''</sup>(''X'') → ''H''<sup>''i''+2</sup>,

			which is defined by intersecting cohomology classes with ''W'' gives an isomorphism

			:''L<sup>n-i</sup>: H<sup>i</sup>(X) → H<sup>2n-i</sup>(X)'' (''i'' ≦ ''n = dim X'').

			Define

			:''Λ : H<sup>i</sup>(X)'' → ''H<sup>i-2</sup>(X)'' for 'i'' ≦ ''n''

			be the composition

			:''(L<sup>n-i+2</sup>)<sup>-1</sup> <math>\circ L \circ </math> (L<sup>n-i</sup>)''

			and

			:''Λ : H<sup>2n-i+2</sup>(X)'' → ''H<sup>2n-i</sup>(X)''

			by

			:''(L<sup>n-i</sup>) <math>\circ L \circ</math> (L<sup>n-i+2</sup>)<sup>-1</sup>''.

			The Lefschetz conjecture states that the ''Lefschetz operator'' ''Λ'' is induced by an algebraic cycle.

			== Künneth type standard conjecture ==
			The Lefschetz conjecture implies the ''Künneth type standard conjecture'', also called conjecture C: It is conjectured that the projectors ''H''<sup>&lowast;</sup>(''X'') ↠ ''H''<sup>i</sup>(''X'') ↣ ''H''<sup>&lowast;</sup>(''X'') are algebraic, i.e. induced by a cycle ''π<sup>i</sup>'' ⊂ ''X × X'' with rational coefficients. This implies that every pure motive ''M'' decomposes in graded pieces of pure weights (see [[motive (algebraic geometry)\|motives]]).
			The conjecture is known to hold for curves, surfaces and abelian varieties.

			== Conjecture ''D'' (numerical equivalence vs. homological equivalence) ==
			''Conjecture D'' states that numerical equivalence and homological equivalence agree. (It implies in particular the latter does not depend on the choice of the Weil cohomology theory). This conjecture implies the Lefschetz conjecture. If the Hodge standard conjecture holds, then the Lefschetz conjecture and Conjecture D are equivalent.

			== The Hodge standard conjecture ==
			The Hodge standard conjecture is modelled on the [[Hodge index theorem]]. It states the definiteness (positive or negative, according to the dimension) of the cup product pairing on primitive algebraic cohomology classes. If it holds, then the Lefschetz conjecture implies Conjecture ''D''. In characteristic zero the Hodge standard conjecture holds, being a consequence of [[Hodge theory]]. In positive characteristic the Hodge standard conjecture is known only for surfaces.

			The Hodge standard conjecture is not to be confused with the ''[[Hodge conjecture]]'' which states that for smooth projective varieties over '''C''', every rational ''(p,p)''-class is algebraic. The Hodge conjecture implies the Lefschetz conjecture and conjecture D for varieties over fields of characteristic zero. Likewise for fields of finite characteristic the [[Tate conjectures]] in [[etale cohomology\|ℓ-adic cohomology]] imply the Lefschetz conjecture.

			== References ==
			*{{citation \| first = A. \| last = Grothendieck \| authorlink = Alexander Grothendieck \| url = http://www.math.jussieu.fr/~leila/grothendieckcircle/StandardConjs.pdf \| contribution = Standard Conjectures on Algebraic Cycles \| title = Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) \| publisher = Oxford University Press \| year = 1969 \| pages = 193–199 \| mr = 0268189}}.
			*{{citation \| last1=Kleiman \| first1=Steven L. \| authorlink=Steven Kleiman \| title=Dix exposés sur la cohomologie des schémas \| publisher=North-Holland \| location=Amsterdam \| mr = 0292838 \| year=1968 \| chapter=Algebraic cycles and the Weil conjectures \| pages=359–386}}.
			*{{citation \| last1=Kleiman \| first1=Steven L. \| authorlink=Steven Kleiman \| contribution=The standard conjectures \| title = Motives (Seattle, WA, 1991) \| year = 1994 \| pages = 3–20 \| series = Proceedings of Symposia in Pure Mathematics \| volume = 55 \| publisher = American Mathematical Society \| mr = 1265519}}.

			[[Category:Algebraic geometry]]
			[[Category:Conjectures]]

en>AnomieBOT: Dating maintenance tags: {{How to}}

2012-02-03T08:40:40Z

Dating maintenance tags: {{How to}}

New page

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