Differential group delay - Revision history

24.131.80.54: fix link

2014-08-14T00:53:47Z

fix link

← Older revision		Revision as of 02:53, 14 August 2014
Line 1:		Line 1:
	In [~~[algebraic geometry]], a branch of [[mathematics]], an '''adequate equivalence relation''' is an equivalence relation on [[algebraic cycles~~]~~] of smooth [[projective varieties]] used to obtain a well-working theory of such cycles, and in particular, well-defined [[intersection theory\|intersection products]]. Samuel formalized~~ the ~~concept of an adequate equivalence relation in 1958~~.<~~ref~~>~~{{citation \| last=Samuel \| first=C. \| title=Relations d'équivalence en géométrie algébrique \| journal=Proc. ICM 1958 \| publisher=Cambridge Univ. Press \| year=1960 \| pages=470–487}}~~<~~/ref~~> ~~Since then it has become central to theory~~ of ~~motives~~. ~~For every adequate equivalence relation, one may define~~ the ~~category of [[motive (algebraic geometry)\|pure motives]] with respect~~ to ~~that relation.~~		Uggs Classic Cardy [http://tinyurl.com/kecvhhb ugg boots sale] have been popular ever since they produced the line.<br><br>They are made of high quality materials that provide maximum style and comfort. Uggs Classic Cardy uses the best quality wool and sheepskin that provides extra comfort to the feet. They are extremely light weight hence does not cause any kind of pressure or uneasiness to your feet.<br><br>Uggs Classic Cardy Boots are exceptionally flexible that allows you to wear them in all kinds of seasons. The insoles are made from high quality sheepskin material that keeps your feet comfortable and dry at all times. This also prevents bad odor to a great extent.<br><br>There are various type of Uggs Classic Cardy Boots available today. They come in a variety of shapes, designs and color therefore if you are a man or a woman; you are bound to find the right kind of Uggs Classic Cardy.<br><br>There are numerous ways or styles through which you can wear these boots. Some of the ways you can wear Uggs Classic Cardy Boots are buttoned, cuffed and slouched. You will be absolutely amazed by how flexible these boots can be.<br>[http://Search.huffingtonpost.com/search?q=Metallic+Black&s_it=header_form_v1 Metallic Black] Gold, Metallic Charcoal Silver, Rose, Cream, Indigo and Oatmeal are just some of the colors you will find in Uggs Classic Cardy Boots.<br><br>Uggs Classic Cardy [http://tinyurl.com/kecvhhb ugg boots sale] are extremely reasonable and can be afforded by one and all. You will never experience such comfort with any other boots available in the market today. Each of these [http://tinyurl.com/kecvhhb ugg boots sale] is carefully constructed in order to make sure your feet are not harmed or bought discomfort in any way.<br><br>The new 2010 collection of Uggs Classic Cardy Boots is simply mind blowing and will captivate your heart and mind in no time at all. Uggs Classic Cardy can also be a great gift to the person you love. Since Thanksgiving and Christmas is on its way, these [http://tinyurl.com/kecvhhb ugg boots sale] will certainly bring a big smile to the receiver.<br><br>These boots can be easily stored for months and no harm will be bought to it no matter what.<br>Uggs Classic Cardy Boots can easily withstand all sorts of weather conditions and no special care has to be taken to make sure its long life. Uggs Classic Cardy Boots are exclusively available at selected stores and if you order now, you can save a lot of money.<br><br>So be smart and get your favorite pair of Uggs Classic Cardy [http://tinyurl.com/kecvhhb ugg boots sale] today.

	~~Possible (and useful) adequate equivalence relations include ''rational'', ''algebraic'', ''homological'' and ''numerical equivalence''~~. They are ~~called "adequate" because dividing out by the equivalence relation is functorial, i.e. push-forward (with change~~ of co-dimension) and pull-back of cycles is well-defined. Codimension one cycles modulo rational equivalence form the classical group of [[Divisor (algebraic geometry)\|divisor]]s. All cycles modulo rational equivalence form the [[Chow ring]].

	~~== Definition ==~~
	~~Let ''Z~~<~~sup~~><~~/sup~~>~~(X)'' := '''Z'''[''X''] be the free abelian group on the algebraic cycles~~ of ~~''X''~~. ~~Then an adequate equivalence relation is~~ a ~~family of [[equivalence relation]]s, ''∼~~<~~sub~~>X<~~/sub~~>~~'' on ''Z<sup></sup>(X)'', one for each smooth projective~~ variety ~~''X''~~, ~~satisfying~~ the ~~following three conditions:~~
	~~# (Linearity) The equivalence relation is compatible with addition~~ of ~~cycles~~.
	~~# ([[Chow's moving lemma\|Moving lemma]]) If~~ <~~math~~>~~\alpha, \beta \in Z^{}(X)~~<~~/math~~> are ~~cycles on ''X''~~, ~~then there exists a cycle <math>\alpha' \in Z^{}(X)</math> such that <math>\alpha</math> ''~<sub>X</sub>'' <math>\alpha'</math>~~ and ~~<math>\alpha'</math> intersects <math>\beta</math> properly~~.
	~~# (Push-forwards) Let <math>\alpha \in Z^{}(X)</math> and <math>\beta \in Z^{}(X \times Y)</math>~~ be ~~cycles such that <math>\beta</math> intersects <math>\alpha \times Y</math> properly~~. If <~~math~~>~~\alpha</math> ''~<sub>X</sub> 0'', then <math>(\pi_Y)_{*}(\beta \cdot (\alpha \times Y))</math> ''~<sub>Y</sub> 0'', where <math>\pi_Y : X \times Y \to Y</math> is the projection.~~

	~~The push-forward cycle in the last axiom is often denoted~~
	~~:<math>\beta(\alpha)~~ :~~= (\pi_Y)_{*}(\beta \cdot (\alpha \times Y))<~~/~~math>~~
	~~If <math>\beta<~~/~~math> is the graph of a function, then this reduces to the push-forward of the function~~. ~~The generalizations of functions from ''X'' to ''Y'' to cycles on ''X × Y'' are known as [[correspondence (mathematics)\|correspondences]]~~. ~~The last axiom allows us to push forward cycles by a correspondence.~~

	== ~~Examples of equivalence relations ==~~

	~~The most common equivalence relations~~, ~~listed from strongest to weakest~~, are ~~gathered below~~ in ~~a table~~.
	~~{\| class="wikitable" style="text-align:center"~~
	\|-
	~~! !! definition !! remarks~~
	\|-
	~~! rational equivalence~~
	~~\| ''Z ∼~~<~~sub~~>~~rat~~<~~/sub~~> ~~Z' '' if there is a cycle ''V'' on ''X × ''~~[~~[projective line\|'''P'''<sup>1<~~/~~sup>]] [[Flat morphism\|flat]] over '''P'''<sup>1<~~/~~sup>, such that [''V ∩ X × {0}''] - [''V ∩ X × {∞}''] = [''Z''] - [''Z' ''~~].
	~~\|\| the finest adequate equivalence relation. "∩" denotes intersection~~ in the ~~cycle-theoretic sense (i~~.~~e. with multiplicities) and~~ [''.''] ~~denotes the cycle associated~~ to ~~a subscheme~~. ~~see also [[Chow ring]]~~
	\|-
	~~! algebraic equivalence~~
	~~\| ''Z ∼~~<~~sub~~>~~alg~~<~~/sub~~> ~~Z' '' if there~~ is ~~a [[curve]] ''C''~~ and a ~~cycle ''V'' on ''X × C'' flat over ''C'', such that [''V ∩ X × {c}''] - [''V ∩ X × {d}''] = [''Z''] - [''Z' ''] for two points ''c''~~ and ~~''d''~~ on ~~the curve.~~
	~~\|\| strictly stronger than homological equivalence~~, ~~see also~~ [~~[Néron–Severi group]]~~
	\|-
	~~! smash-nilpotence equivalence~~
	~~\| ''Z ∼<sub>sn<~~/~~sub> Z' '' if ''Z - Z' '' is smash-nilpotent on ''X'', that is, if <math>(Z-Z')^{\otimes n}<~~/~~math> ''∼<sub>rat<~~/~~sub> 0'' on ''X''~~<~~sup~~>n<~~/sup~~> for ~~''n >> 0''.~~
	~~\|\| introduced by Voevodsky in 1995~~.<~~ref~~>~~{{citation \| first=V. \| last=Voevodsky \| title=A nilpotence theorem for cycles algebraically equivalent~~ to ~~0 \| journal=Int~~. ~~Math. Res. Notices \| volume=4 \| year=1995 \| pages=1–12}}</ref>~~
	\|-
	~~! homological equivalence~~
	~~\| for a given [[Weil cohomology theory\|Weil cohomology]] ''H''~~, ~~''Z ∼<sub>hom</sub> Z' '' if the image of the cycles under the cycle class map agrees~~
	~~\|\| depends~~ a ~~priori~~ of ~~the choice of ''H'', not assuming the [[standard conjectures on algebraic cycles\|standard conjecture]] ''D''~~
	\|-
	~~! numerical equivalence~~
	~~\| ''Z ∼~~<~~sub~~>~~num~~<~~/sub~~> ~~Z' '' if ''deg(Z ∩ T) = deg(Z' ∩ T)'', where ''T'' is any cycle such that ''dim T = codim Z'' (The intersection is a linear combination of points~~ and ~~we add the intersection multiplicities at each point to~~ get ~~the degree.)~~
	~~\|\| the coarsest equivalence relation~~
	\|}

	~~== Notes ==~~
	~~<references />~~

	~~== References==~~
	* {{Citation \| last1=Kleiman \| first1=Steven L. \| editor1-last=Oort \| editor1-first=F. \| title=Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970) \| publisher=Wolters-Noordhoff \| location=Groningen \| mr=0382267 \| year=1972 \| chapter=Motives \| pages=53–82}}
	* {{Citation \| last=Jannsen \| first=U. \| title=Equivalence relations on algebraic cycles \| journal=The Arithmetic and Geometry of ~~Algebraic Cycles, NATO, 200 \| publisher=Kluwer Ac. Publ. Co. \| year=2000 \| pages=225–260}}~~

	~~{{DEFAULTSORT:Adequate Equivalence Relation}}~~
	[~~[Category~~:~~Algebraic geometry]~~]

en>BattyBot: removed Template:Multiple issues & general fixes using AWB (8062)

2012-07-01T13:18:31Z

removed Template:Multiple issues & general fixes using AWB (8062)

New page

In [[algebraic geometry]], a branch of [[mathematics]], an '''adequate equivalence relation''' is an equivalence relation on [[algebraic cycles]] of smooth [[projective varieties]] used to obtain a well-working theory of such cycles, and in particular, well-defined [[intersection theory|intersection products]]. Samuel formalized the concept of an adequate equivalence relation in 1958.<ref>{{citation | last=Samuel | first=C. | title=Relations d'équivalence en géométrie algébrique | journal=Proc. ICM 1958 | publisher=Cambridge Univ. Press | year=1960 | pages=470–487}}</ref> Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the category of [[motive (algebraic geometry)|pure motives]] with respect to that relation.

Possible (and useful) adequate equivalence relations include ''rational'', ''algebraic'', ''homological'' and ''numerical equivalence''. They are called "adequate" because dividing out by the equivalence relation is functorial, i.e. push-forward (with change of co-dimension) and pull-back of cycles is well-defined. Codimension one cycles modulo rational equivalence form the classical group of [[Divisor (algebraic geometry)|divisor]]s. All cycles modulo rational equivalence form the [[Chow ring]].

== Definition ==
Let ''Z*(X)'' := '''Z'''[''X''] be the free abelian group on the algebraic cycles of ''X''. Then an adequate equivalence relation is a family of [[equivalence relation]]s, ''∼X'' on ''Z*(X)'', one for each smooth projective variety ''X'', satisfying the following three conditions:
# (Linearity) The equivalence relation is compatible with addition of cycles.
# ([[Chow's moving lemma|Moving lemma]]) If <math>\alpha, \beta \in Z^{*}(X)</math> are cycles on ''X'', then there exists a cycle <math>\alpha' \in Z^{*}(X)</math> such that <math>\alpha</math> ''~X'' <math>\alpha'</math> and <math>\alpha'</math> intersects <math>\beta</math> properly.
# (Push-forwards) Let <math>\alpha \in Z^{*}(X)</math> and <math>\beta \in Z^{*}(X \times Y)</math> be cycles such that <math>\beta</math> intersects <math>\alpha \times Y</math> properly. If <math>\alpha</math> ''~X 0'', then <math>(\pi_Y)_{*}(\beta \cdot (\alpha \times Y))</math> ''~Y 0'', where <math>\pi_Y : X \times Y \to Y</math> is the projection.

The push-forward cycle in the last axiom is often denoted
:<math>\beta(\alpha) := (\pi_Y)_{*}(\beta \cdot (\alpha \times Y))</math>
If <math>\beta</math> is the graph of a function, then this reduces to the push-forward of the function. The generalizations of functions from ''X'' to ''Y'' to cycles on ''X × Y'' are known as [[correspondence (mathematics)|correspondences]]. The last axiom allows us to push forward cycles by a correspondence.

== Examples of equivalence relations ==

The most common equivalence relations, listed from strongest to weakest, are gathered below in a table.
{| class="wikitable" style="text-align:center"
|-
! !! definition !! remarks
|-
! rational equivalence
| ''Z ∼rat Z' '' if there is a cycle ''V'' on ''X × ''[[projective line|'''P'''1]] [[Flat morphism|flat]] over '''P'''1, such that [''V ∩ X × {0}''] - [''V ∩ X × {∞}''] = [''Z''] - [''Z' ''].
|| the finest adequate equivalence relation. "∩" denotes intersection in the cycle-theoretic sense (i.e. with multiplicities) and [''.''] denotes the cycle associated to a subscheme. see also [[Chow ring]]
|-
! algebraic equivalence
| ''Z ∼alg Z' '' if there is a [[curve]] ''C'' and a cycle ''V'' on ''X × C'' flat over ''C'', such that [''V ∩ X × {c}''] - [''V ∩ X × {d}''] = [''Z''] - [''Z' ''] for two points ''c'' and ''d'' on the curve.
|| strictly stronger than homological equivalence, see also [[Néron–Severi group]]
|-
! smash-nilpotence equivalence
| ''Z ∼sn Z' '' if ''Z - Z' '' is smash-nilpotent on ''X'', that is, if <math>(Z-Z')^{\otimes n}</math> ''∼rat 0'' on ''X''n for ''n >> 0''.
|| introduced by Voevodsky in 1995.<ref>{{citation | first=V. | last=Voevodsky | title=A nilpotence theorem for cycles algebraically equivalent to 0 | journal=Int. Math. Res. Notices | volume=4 | year=1995 | pages=1–12}}</ref>
|-
! homological equivalence
| for a given [[Weil cohomology theory|Weil cohomology]] ''H'', ''Z ∼hom Z' '' if the image of the cycles under the cycle class map agrees
|| depends a priori of the choice of ''H'', not assuming the [[standard conjectures on algebraic cycles|standard conjecture]] ''D''
|-
! numerical equivalence
| ''Z ∼num Z' '' if ''deg(Z ∩ T) = deg(Z' ∩ T)'', where ''T'' is any cycle such that ''dim T = codim Z'' (The intersection is a linear combination of points and we add the intersection multiplicities at each point to get the degree.)
|| the coarsest equivalence relation
|}

== Notes ==
<references />

== References==
* {{Citation | last1=Kleiman | first1=Steven L. | editor1-last=Oort | editor1-first=F. | title=Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970) | publisher=Wolters-Noordhoff | location=Groningen | mr=0382267 | year=1972 | chapter=Motives | pages=53–82}}
* {{Citation | last=Jannsen | first=U. | title=Equivalence relations on algebraic cycles | journal=The Arithmetic and Geometry of Algebraic Cycles, NATO, 200 | publisher=Kluwer Ac. Publ. Co. | year=2000 | pages=225–260}}

{{DEFAULTSORT:Adequate Equivalence Relation}}
[[Category:Algebraic geometry]]