In algebraic geometry, an adequate equivalence relation is used on algebraic cycles of smooth projective varieties to develop a well-defined theory, particularly for intersection products. Introduced by Pierre Samuel in 1958, this concept is central to the theory of motives, allowing the definition of the category of pure motives. Examples include rational, algebraic, homological, and numerical equivalence, all functorial with respect to push-forward and pull-back. Notably, codimension 1 cycles modulo rational equivalence form the classical group of divisors modulo linear equivalence, and cycles modulo rational equivalence constitute 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 relations, ~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.
- (Moving lemma) 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.
- (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.
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 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 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.
| definition | remarks | |
|---|---|---|
| rational equivalence | Z ~rat Z' if there is a cycle V on X × P1 flat over P1, such that [V ∩ X × {0}] − [V ∩ X × {∞}] = [Z] − [Z' ]. | the finest adequate equivalence relation (Lemma 3.2.2.1 in Yves André's book2) "∩" 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, as measured by the Griffiths group. See also Néron–Severi group. |
| smash-nilpotence equivalence | 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. | introduced by Voevodsky in 1995.3 |
| homological equivalence | for a given 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 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 (Exercise 3.2.7.2 in Yves André's book4) |
Notes
- 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, MR 0382267
- Jannsen, U. (2000), "Equivalence relations on algebraic cycles", The Arithmetic and Geometry of Algebraic Cycles, NATO, 200, Kluwer Ac. Publ. Co.: 225–260
References
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 /wiki/Pierre_Samuel ↩
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 978-2-85629-164-1 ↩
Voevodsky, V. (1995), "A nilpotence theorem for cycles algebraically equivalent to 0", Int. Math. Res. Notices, 4: 1–12 ↩
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 978-2-85629-164-1 ↩