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Algebraic cycle

While divisors on higher-dimensional varieties continue to play an important role in determining the structure of the variety, on varieties of dimension two or more there are also higher codimension cycles to consider. The behavior of these cycles is strikingly different from that of divisors. For example, every curve has a constant N such that every divisor of degree zero is linearly equivalent to a difference of two effective divisors of degree at most N. David Mumford proved that, on a smooth complete complex algebraic surface S with positive geometric genus, the analogous statement for the group ${\displaystyle \operatorname {CH} ^{2}($S)} of rational equivalence classes of codimension two cycles in S is false.^[1] The hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group ${\displaystyle H^{2}($S)} contains transcendental information, and in effect Mumford's theorem implies that, despite ${\displaystyle \operatorname {CH} ^{2}($S)} having a purely algebraic definition, it shares transcendental information with ${\displaystyle H^{2}($S)} . Mumford's theorem has since been greatly generalized.^[2]

The behavior of algebraic cycles ranks among the most important open questions in modern mathematics. The Hodge conjecture, one of the Clay Mathematics Institute's Millennium Prize Problems, predicts that the topology of a complex algebraic variety forces the existence of certain algebraic cycles. The Tate conjecture makes a similar prediction for étale cohomology. Alexander Grothendieck's standard conjectures on algebraic cycles yield enough cycles to construct his category of motives and would imply that algebraic cycles play a vital role in any cohomology theory of algebraic varieties. Conversely, Alexander Beilinson proved that the existence of a category of motives implies the standard conjectures. Additionally, cycles are connected to algebraic K-theory by Bloch's formula, which expresses groups of cycles modulo rational equivalence as the cohomology of K-theory sheaves.

Let X be a scheme which is finite type over a field k. An algebraic r-cycleonX is a formal linear combination

${\displaystyle \sum n_{i}[V_{i}]}$

ofr-dimensional closed integral k-subschemes of X. The coefficient n_i is the multiplicityofV_i. The set of all r-cycles is the free abelian group

${\displaystyle Z_{r}X=\bigoplus _{V\subseteq X}\mathbf {Z} \cdot [$V],}

where the sum is over closed integral subschemes VofX. The groups of cycles for varying r together form a group

${\displaystyle Z_{*}X=\bigoplus _{r}Z_{r}X.}$

This is called the group of algebraic cycles, and any element is called an algebraic cycle. A cycle is effectiveorpositive if all its coefficients are non-negative.

Closed integral subschemes of X are in one-to-one correspondence with the scheme-theoretic points of X under the map that, in one direction, takes each subscheme to its generic point, and in the other direction, takes each point to the unique reduced subscheme supported on the closure of the point. Consequently ${\displaystyle Z_{*}X}$ can also be described as the free abelian group on the points of X.

A cycle ${\displaystyle \alpha }$isrationally equivalent to zero, written ${\displaystyle \alpha \sim 0}$, if there are a finite number of ${\displaystyle (r+1)}$-dimensional subvarieties ${\displaystyle W_{i}}$of${\displaystyle X}$ and non-zero rational functions ${\displaystyle r_{i}\in k(W_{i})^{\times }}$ such that ${\displaystyle \alpha =\sum [\operatorname {div} _{W_{i}}(r_{i})]}$, where ${\displaystyle \operatorname {div} _{W_{i}}}$ denotes the divisor of a rational function on W_i. The cycles rationally equivalent to zero are a subgroup ${\displaystyle Z_{r}($X)_{\text{rat}}\subseteq Z_{r}(X)} , and the group of r-cycles modulo rational equivalence is the quotient

${\displaystyle A_{r}($X)=Z_{r}(X)/Z_{r}(X)_{\text{rat}}.}

This group is also denoted ${\displaystyle \operatorname {CH} _{r}($X)} . Elements of the group

${\displaystyle A_{*}($X)=\bigoplus _{r}A_{r}(X)}

are called cycle classesonX. Cycle classes are said to be effectiveorpositive if they can be represented by an effective cycle.

IfX is smooth, projective, and of pure dimension N, the above groups are sometimes reindexed cohomologically as

${\displaystyle Z^{N-r}X=Z_{r}X}$

and

${\displaystyle A^{N-r}X=A_{r}X.}$

In this case, ${\displaystyle A^{*}X}$ is called the Chow ringofX because it has a multiplication operation given by the intersection product.

There are several variants of the above definition. We may substitute another ring for integers as our coefficient ring. The case of rational coefficients is widely used. Working with families of cycles over a base, or using cycles in arithmetic situations, requires a relative setup. Let ${\displaystyle \phi \colon X\to S}$, where S is a regular Noetherian scheme. An r-cycle is a formal sum of closed integral subschemes of X whose relative dimension is r; here the relative dimension of ${\displaystyle Y\subseteq X}$ is the transcendence degree of ${\displaystyle k($Y)} over ${\displaystyle k({\overline {\phi ($Y)}})} minus the codimension of ${\displaystyle {\overline {\phi ($Y)}}} inS.

Rational equivalence can also be replaced by several other coarser equivalence relations on algebraic cycles. Other equivalence relations of interest include algebraic equivalence, homological equivalence for a fixed cohomology theory (such as singular cohomology or étale cohomology), numerical equivalence, as well as all of the above modulo torsion. These equivalence relations have (partially conjectural) applications to the theory of motives.

There is a covariant and a contravariant functoriality of the group of algebraic cycles. Let f : X → X' be a map of varieties.

Iffisflat of some constant relative dimension (i.e. all fibers have the same dimension), we can define for any subvariety Y' ⊂ X':

${\displaystyle f^{*}([Y'])=[f^{-1}(Y')]\,\!}$

which by assumption has the same codimension as Y′.

Conversely, if fisproper, for Y a subvariety of X the pushforward is defined to be

${\displaystyle f_{*}([$Y])=n[f(Y)]\,\!}

where n is the degree of the extension of function fields [k(Y) : k(f(Y))] if the restriction of ftoYisfinite and 0 otherwise.

By linearity, these definitions extend to homomorphisms of abelian groups

${\displaystyle f^{*}\colon Z^{k}(X')\to Z^{k}($X)\quad {\text{and}}\quad f_{*}\colon Z_{k}(X)\to Z_{k}(X')\,\!}

(the latter by virtue of the convention) are homomorphisms of abelian groups. See Chow ring for a discussion of the functoriality related to the ring structure.

• divisor (algebraic geometry)
• Relative cycle

• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. Third series. A Series of Modern Surveys in Mathematics, vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323
• Gordon, B. Brent; Lewis, James D.; Müller-Stach, Stefan; Saito, Shuji; Yui, Noriko, eds. (2000), The arithmetic and geometry of algebraic cycles: proceedings of the CRM summer school, June 7–19, 1998, Banff, Alberta, Canada, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1954-8