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In mathematics, an algebraic cycle on an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
''V'' is a formal linear combination of
subvarieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
of ''V''. These are the part of the
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...
of ''V'' that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety. The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
, and to extrinsic properties, such as embeddings of the curve into projective space. 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 David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded ...
proved that, on a smooth complete complex algebraic surface ''S'' with positive
geometric genus In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for comp ...
, the analogous statement for the group \operatorname^2(S) of rational equivalence classes of codimension two cycles in ''S'' is false. The hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group H^2(S) contains transcendental information, and in effect Mumford's theorem implies that, despite \operatorname^2(S) having a purely algebraic definition, it shares transcendental information with H^2(S). Mumford's theorem has since been greatly generalized.Voisin, Claire, ''Chow Rings, Decomposition of the Diagonal, and the Topology of Families'', Annals of Mathematics Studies 187, February 2014, . The behavior of algebraic cycles ranks among the most important open questions in modern mathematics. The Hodge conjecture, one of the
Clay Mathematics Institute The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address is now in Denver, Colorado. CMI's sc ...
's
Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem. According ...
, 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 In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conject ...
. 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 Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1 ...
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.


Definition

Let ''X'' be a scheme which is finite type over a field ''k''. An algebraic ''r''-cycle on ''X'' is a formal linear combination :\sum n_i _i/math> of ''r''-dimensional closed integral ''k''-subschemes of ''X''. The coefficient ''n''''i'' is the ''multiplicity'' of ''V''''i''. The set of all ''r''-cycles is the free abelian group :Z_r X = \bigoplus_ \mathbf \cdot where the sum is over closed integral subschemes ''V'' of ''X''. The groups of cycles for varying ''r'' together form a group :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 effective or positive 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 Z_* X can also be described as the free abelian group on the points of ''X''. A cycle \alpha is rationally equivalent to zero, written \alpha \sim 0, if there are a finite number of (r + 1)-dimensional subvarieties W_i of X and non-zero rational functions r_i \in k(W_i)^\times such that \alpha = \sum operatorname_(r_i)/math>, where \operatorname_ denotes the divisor of a rational function on ''W''''i''. The cycles rationally equivalent to zero are a subgroup Z_r(X)_ \subseteq Z_r(X), and the group of ''r''-cycles modulo rational equivalence is the quotient :A_r(X) = Z_r(X) / Z_r(X)_. This group is also denoted \operatorname_r(X). Elements of the group :A_*(X) = \bigoplus_r A_r(X) are called cycle classes on ''X''. Cycle classes are said to be effective or positive if they can be represented by an effective cycle. If ''X'' is smooth, projective, and of pure dimension ''N'', the above groups are sometimes reindexed cohomologically as :Z^ X = Z_r X and :A^ X = A_r X. In this case, A^* X is called the Chow ring of ''X'' because it has a multiplication operation given by the
intersection product In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
. 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 \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 Y \subseteq X is the transcendence degree of k(Y) over k(\overline) minus the codimension of \overline in ''S''. 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.


Flat pullback and proper pushforward

There is a covariant and a contravariant functoriality of the group of algebraic cycles. Let ''f'' : ''X'' → ''X''' be a map of varieties. If ''f'' is flat of some constant relative dimension (i.e. all fibers have the same dimension), we can define for any subvariety ''Y''' âŠ‚ ''X''': : f^*( ' =
^(Y') Caret is the name used familiarly for the character , provided on most QWERTY keyboards by typing . The symbol has a variety of uses in programming and mathematics. The name "caret" arose from its visual similarity to the original proofrea ...
,\! which by assumption has the same codimension as ''Y′''. Conversely, if ''f'' is proper, for ''Y'' a subvariety of ''X'' the pushforward is defined to be :f_*( = n (Y),\! where ''n'' is the degree of the extension of function fields 'k''(''Y'') : ''k''(''f''(''Y''))if the restriction of ''f'' to ''Y'' is
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...
and 0 otherwise. By linearity, these definitions extend to homomorphisms of abelian groups :f^* \colon Z^k(X') \to Z^k(X) \quad\text\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.


See also

*
divisor (algebraic geometry) In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mum ...
* Relative cycle


References

* * {{Citation , editor1-last=Gordon , editor1-first=B. Brent , editor2-last=Lewis , editor2-first=James D. , editor3-last=Müller-Stach , editor3-first=Stefan , editor4-last=Saito , editor4-first=Shuji , editor5-first=Noriko , editor5-last=Yui, title=The arithmetic and geometry of algebraic cycles: proceedings of the CRM summer school, June 7–19, 1998, Banff, Alberta, Canada , publisher=American Mathematical Society , location=Providence, R.I. , isbn=978-0-8218-1954-8 , year=2000 Algebraic geometry