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| 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.
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| 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]].
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| == Definition ==
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| 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:
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| # (Linearity) The equivalence relation is compatible with addition of cycles.
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| # ([[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.
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| # (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.
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| The push-forward cycle in the last axiom is often denoted
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| :<math>\beta(\alpha) := (\pi_Y)_{*}(\beta \cdot (\alpha \times Y))</math>
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| 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.
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| == Examples of equivalence relations == | |
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| The most common equivalence relations, listed from strongest to weakest, are gathered below in a table.
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| {| class="wikitable" style="text-align:center"
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| ! !! definition !! remarks
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| ! rational equivalence
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| | ''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' ''].
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| || 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]]
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| ! algebraic equivalence
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| | ''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.
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| || strictly stronger than homological equivalence, see also [[Néron–Severi group]]
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| |-
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| ! smash-nilpotence equivalence
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| | ''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''.
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| || 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>
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| |-
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| ! homological equivalence
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| | 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
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| || depends a priori of the choice of ''H'', not assuming the [[standard conjectures on algebraic cycles|standard conjecture]] ''D''
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| |-
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| ! numerical equivalence
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| | ''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.)
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| || the coarsest equivalence relation
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| |}
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| == Notes ==
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| <references />
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| == References==
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| * {{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}}
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| * {{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}}
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| {{DEFAULTSORT:Adequate Equivalence Relation}}
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| [[Category:Algebraic geometry]] | |
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