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 divisors.
The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves.
Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space.
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
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
contains transcendental information, and in effect Mumford's theorem implies that, despite
having a purely algebraic definition, it shares transcendental information with
[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-cycle on X is a formal linear combination of r-dimensional closed integral k-subschemes of X.
The set of all r-cycles is the free abelian group where the sum is over closed integral subschemes V of X.
A cycle is effective or positive if all its coefficients are non-negative.
denotes the divisor of a rational function on Wi.
Elements of the group are called cycle classes on X.
If X is smooth, projective, and of pure dimension N, the above groups are sometimes reindexed cohomologically as and In this case,
is called the Chow ring of X because it has a multiplication operation given by the intersection product.
An r-cycle is a formal sum of closed integral subschemes of X whose relative dimension is r; here the relative dimension of
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.
Conversely, if f is proper, for Y a subvariety of X the pushforward is defined to be 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 and 0 otherwise.
By linearity, these definitions extend to homomorphisms of abelian groups (the latter by virtue of the convention) are homomorphisms of abelian groups.