Adequate equivalence relation

<h2 id="definition">Definition</h2>
Let Z*(X) := Z[X] be the <a href="/facts/Free_abelian_group/oaSRGmpY">free abelian group</a> on the algebraic cycles of X. Then an adequate equivalence relation is a family of <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relations</a>, ~X on Z*(X), one for each smooth projective variety X, satisfying the following three conditions:

<ol><li>(Linearity) The equivalence relation is compatible with addition of cycles.</li>
<li>(<a href="/facts/Chow%2527s_moving_lemma/3YE2DH7D">Moving lemma</a>) If 
 
 
 
 α
 ,
 β
 ∈
 
 Z
 
 ∗
 
 
 (
 X
 )
 
 
 {\displaystyle \alpha ,\beta \in Z^{*}(X)}
 
 are cycles on X, then there exists a cycle 
 
 
 
 
 α
 ′
 
 ∈
 
 Z
 
 ∗
 
 
 (
 X
 )
 
 
 {\displaystyle \alpha '\in Z^{*}(X)}
 
 such that 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 ~X 
 
 
 
 
 α
 ′
 
 
 
 {\displaystyle \alpha '}
 
 and 
 
 
 
 
 α
 ′
 
 
 
 {\displaystyle \alpha '}
 
 intersects 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 properly.</li>
<li>(Push-forwards) Let 
 
 
 
 α
 ∈
 
 Z
 
 ∗
 
 
 (
 X
 )
 
 
 {\displaystyle \alpha \in Z^{*}(X)}
 
 and 
 
 
 
 β
 ∈
 
 Z
 
 ∗
 
 
 (
 X
 ×
 Y
 )
 
 
 {\displaystyle \beta \in Z^{*}(X\times Y)}
 
 be cycles such that 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 intersects 
 
 
 
 α
 ×
 Y
 
 
 {\displaystyle \alpha \times Y}
 
 properly. If 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 ~X 0, then 
 
 
 
 (
 
 π
 
 Y
 
 
 
 )
 
 ∗
 
 
 (
 β
 ⋅
 (
 α
 ×
 Y
 )
 )
 
 
 {\displaystyle (\pi _{Y})_{*}(\beta \cdot (\alpha \times Y))}
 
 ~Y 0, where 
 
 
 
 
 π
 
 Y
 
 
 :
 X
 ×
 Y
 →
 Y
 
 
 {\displaystyle \pi _{Y}:X\times Y\to Y}
 
 is the projection.</li></ol>
The push-forward cycle in the last axiom is often denoted

β
        (
        α
        )
        :=
        (
        
          π
          
            Y
          
        
        
          )
          
            ∗
          
        
        (
        β
        ⋅
        (
        α
        ×
        Y
        )
        )
      
    
    {\displaystyle \beta (\alpha ):=(\pi _{Y})_{*}(\beta \cdot (\alpha \times Y))}

If 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 is the <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>, 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 <a href="/facts/Correspondence_(algebraic_geometry)/lplsPuTf">correspondences</a>. The last axiom allows us to push forward cycles by a correspondence.

<h2 id="examples-of-equivalence-relations">Examples of equivalence relations</h2>
The most common equivalence relations, listed from strongest to weakest, are gathered in the following table.

<table><tbody><tr><th></th><th>definition</th><th>remarks</th></tr><tr><th>rational equivalence</th><td>Z ~rat Z' if there is a cycle V on X × <a href="/facts/Projective_line/PhRgf87G">P1</a> <a href="/facts/Flat_morphism/CLM7e8d6">flat</a> over P1, such that [V ∩ X × {0}] − [V ∩ X × {∞}] = [Z] − [Z' ].</td><td>the finest adequate equivalence relation (Lemma 3.2.2.1 in Yves André's book<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>) "∩" denotes intersection in the cycle-theoretic sense (i.e. with multiplicities) and [.] denotes the cycle associated to a subscheme. see also <a href="/facts/Chow_ring/2gkmVppG">Chow ring</a></td></tr><tr><th>algebraic equivalence</th><td>Z ~alg Z′  if there is a <a href="/facts/Curve/xBQ0pTBW">curve</a> 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.</td><td>Strictly stronger than homological equivalence, as measured by the <a href="/facts/Griffiths_group/4AeqqVBP">Griffiths group</a>. See also <a href="/facts/N%25C3%25A9ron%25E2%2580%2593Severi_group/MKKDDgAW">Néron–Severi group</a>.</td></tr><tr><th>smash-nilpotence equivalence</th><td>Z ~sn Z′  if Z − Z′ is smash-nilpotent on X, that is, if ( Z − Z ′ ) ⊗ n {\displaystyle (Z-Z')^{\otimes n}} ~rat 0 on Xn for n >> 0.</td><td>introduced by Voevodsky in 1995.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></td></tr><tr><th>homological equivalence</th><td>for a given <a href="/facts/Weil_cohomology_theory/FSuTh4xq">Weil cohomology</a> H, Z ~hom Z′  if the image of the cycles under the cycle class map agrees</td><td>depends a priori of the choice of H, not assuming the <a href="/facts/Standard_conjectures_on_algebraic_cycles/54xlzvXF">standard conjecture</a> D</td></tr><tr><th>numerical equivalence</th><td>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.)</td><td>the coarsest equivalence relation (Exercise 3.2.7.2 in Yves André's book<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>)</td></tr></tbody></table>
<h2 id="notes">Notes</h2>

<ul><li>Kleiman, Steven L. (1972), "Motives", in Oort, F. (ed.), Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970), Groningen: Wolters-Noordhoff, pp. 53–82, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0382267">0382267</a></li>
<li>Jannsen, U. (2000), "Equivalence relations on algebraic cycles", The Arithmetic and Geometry of Algebraic Cycles, NATO, 200, Kluwer Ac. Publ. Co.: 225–260</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Samuel, Pierre (1958), "Relations d'équivalence en géométrie algébrique" (PDF), Proc. ICM, Cambridge Univ. Press: 470–487, archived from the original (PDF) on 2017-07-22, retrieved 2015-07-22 <a href="/wiki/Pierre_Samuel" target="_blank">/wiki/Pierre_Samuel</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">André, Yves (2004), Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, vol. 17, Paris: Société Mathématique de France, ISBN 978-2-85629-164-1, MR 2115000 <a href="978-2-85629-164-1" target="_blank">978-2-85629-164-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Voevodsky, V. (1995), "A nilpotence theorem for cycles algebraically equivalent to 0", Int. Math. Res. Notices, 4: 1–12 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">André, Yves (2004), Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, vol. 17, Paris: Société Mathématique de France, ISBN 978-2-85629-164-1, MR 2115000 <a href="978-2-85629-164-1" target="_blank">978-2-85629-164-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Adequate equivalence relation open-in-new

Adequate equivalence relation