Chow Ring - Definition of The Chow Ring

Definition of The Chow Ring

It is part of the definition of rational equivalence that it only holds between subvarieties of equal dimension. For the purposes of constructing the Chow ring, we are interested in the codimension of the subvariety (that is, the difference between its dimension and that of X) since it makes the product work properly, so we define the groups Ak(X), for integers k satisfying, to be the abelian group of formal sums of subvarieties of X of codimension k modulo rational equivalence. The Chow ring itself is the direct sum of these, namely,

The ring structure is given by intersection of varieties: that is, if we have two classes in Ak(X) and Al(X) respectively, we define their product to be

This definition has a number of technicalities that will be discussed below; here it suffices to say that in the best case, which can be shown always to hold up to rational equivalence, this intersection has codimension k + l, hence lies in Ak + l(X). This makes the Chow ring into a graded ring. As a matter of notation, an element of the Chow ring is often called a "cycle".