Algebraic cycle

In mathematics, an algebraic cycle on an algebraic variety "V" is, roughly speaking, a homology class on "V" that is represented by a linear combination of subvarieties of "V". Therefore the algebraic cycles on "V" are the part of the algebraic topology of "V" that is directly accessible in algebraic geometry. With the formulation of some fundamental conjectures in the 1950s and 1960s, the study of algebraic cycles became one of the main objectives of the algebraic geometry of general varieties.

The nature of the difficulties is quite plain: the existence of algebraic cycles is easy to predict, but the methods of construction of them are currently deficient. The major conjectures on algebraic cycles include the Hodge conjecture and the Tate conjecture. In the search for a proof of the Weil conjectures, Alexander Grothendieck and Enrico Bombieri formulated what are now known as the standard conjectures of algebraic cycle theory.

Algebraic cycles have also been shown to be closely connected with algebraic K-theory.

For the purposes of a well-working intersection theory, one uses various equivalence relations on algebraic cycles. Particularly important is the so-called "rational equivalence". Cycles up to rational equivalence form a graded ring, the Chow ring, the multiplication is given by the intersection product. Further fundamental relations include "algebraic equivalence", "numerical equivalence", and "homological equivalence". They have (partly conjectural) applications in the theory of motives.

Definition

An "algebraic cycle" of an algebraic variety or scheme "X" is a formal linear combination "V" = ∑ "n_i·V_i" of irreducible reduced closed subschemes. A coefficient "n_i" is called multiplicity of "V_i" in "V". Ad hoc, the coefficients are integers, but rational coefficients are also widely used.

Under the correspondence :{"irreducible reduced closed subschemes V ⊂ X"} ↭ {"points of X"}("V" maps to its generic point (with respect to the Zariski topology), conversely a point maps to its closure (with the reduced subscheme structure))an algebraic cycle is thus just a formal linear combination of points of "X".

The group of cycles naturally forms a group "Z_*(X)" graded by the dimension of the cycles. The grading by codimension is also useful, then the group is usually written "Z^*(X)".

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' ⊂ X' ": : $f^*(\; [Y\text{'}]\; )\; =\; [f^\{-1\}(Y\text{'})]\; ,!$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:$f\_*(\; [Y]\; )\; =\; n\; [f(Y)]\; ,!$

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 and "0" otherwise.

By linearity, these definitions extend to homomorphisms of abelian groups:$f^*\; colon\; Z^k(X\text{'})\; o\; Z^k(X)\; ,!$ and $f\_*\; colon\; Z\_k(X)\; o\; Z\_k(X\text{'})\; ,!$

(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.

References

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