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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]]. [[Pierre Samuel]] formalized the concept of an adequate equivalence relation in 1958.<ref>{{citation | last=Samuel | first=Pierre | title=Relations d'équivalence en géométrie algébrique | journal=Proc. ICM | year=1958 | publisher=Cambridge Univ. Press | pages=470–487 | url=http://www.mathunion.org/ICM/ICM1958/Main/icm1958.0470.0487.ocr.pdf | author-link=Pierre Samuel | access-date=2015-07-22 | archive-url=https://web.archive.org/web/20170722145343/http://www.mathunion.org/ICM/ICM1958/Main/icm1958.0470.0487.ocr.pdf | archive-date=2017-07-22 | url-status=dead }}</ref> Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the [[category (mathematics)|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 codimension) and pull-back of cycles is well-defined. Codimension 1 cycles modulo rational equivalence form the classical [[group (mathematics)|group]] of [[Divisor (algebraic geometry)|divisor]]s modulo linear equivalence. 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|graph]] of a [[function (mathematics)|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 (algebraic geometry)|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 in the following 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'' ×&thinsp;{0}] − [''V'' ∩ ''X'' ×&thinsp;{∞}] = [''Z''] − [''Z' ''].
|| the finest adequate equivalence relation (Lemma 3.2.2.1 in Yves André's book<ref>{{Citation | last1=André | first1=Yves | title=Une introduction aux motifs (motifs purs, motifs mixtes, périodes) | publisher=Société Mathématique de France | location=Paris | series=Panoramas et Synthèses | isbn=978-2-85629-164-1 |mr=2115000 | year=2004 | volume=17}}</ref>) "∩" 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> ''{{prime|Z}}''&thinsp; 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, as measured by the [[Griffiths group]]. See also [[Néron–Severi group]].
|-
! smash-nilpotence equivalence
| ''Z ~''<sub>sn</sub> ''{{prime|Z}}''&thinsp; if ''Z'' − ''{{prime|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> ''{{prime|Z}}''&thinsp; 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> ''{{prime|Z}}''&thinsp; if deg(''Z'' ∩ ''T'') = deg(''{{prime|Z}}'' ∩ ''T''), where ''T'' is any cycle such that dim&thinsp;''T'' = codim&thinsp;''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 (Exercise 3.2.7.2 in Yves André's book<ref>{{Citation | last1=André | first1=Yves | title=Une introduction aux motifs (motifs purs, motifs mixtes, périodes) | publisher=Société Mathématique de France | location=Paris | series=Panoramas et Synthèses | isbn=978-2-85629-164-1 |mr=2115000 | year=2004 | volume=17}}</ref>)
|}

== 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]]
[[Category:Equivalence (mathematics)]]