Chow group

In algebraic geometry, a branch of mathematics, Chow groups are an important invariant of varieties.

Definition

Let be a smooth, irreducible, projective variety over an algebraically closed field.

The group of algebraic Cycles of codimension i

Is defined as the free abelian group generated by the irreducible (not necessarily smooth ) subvarieties of codimension. An element is thus a finite sum

With irreducible subvariety and.

Two subvarieties

Hot rationally equivalent if there is a subvariety

As well as with

There. Rational equivalence defines an equivalence relation on the Zykelgruppe.

The Chow group is defined as the quotient of modulo rational equivalence Cycles Group:

Chow ring

The intersection product defines a map

For everyone. The Chow ring is the direct sum of the Chow groups

With the plane defined by the average product multiplication.

Examples

• For any smooth, irreducible variety

• Is the Picardgruppe