Chow Ring - Details of The Construction

Details of The Construction

The definition of Ak(X) given above requires some clarification regarding "modulo rational equivalence". The relevant technical detail is that, as in the computation of the Chow ring of projective space, it is sometimes (in fact, usually) the case that two cycles which are not the cycles associated to a variety may be rationally equivalent, yet rational equivalence as stated appears to take notice only of the set structure. The solution is via scheme theory, namely, that a subvariety Y defined by a sheaf of ideals can be considered to have a multiplicity d if we replace with . Then the classical statement of rational equivalence is inadequate, and we must pay close attention to the details of flat families. Finally, a formal sum of classes, such as aY + bZ, should be considered as the disjoint union of the varieties-with-degrees aY and bZ. Once these conventions are established, we may impose rational equivalence as a relation on the free abelian group of cycles to get the Chow ring.

The definition of the intersection product is somewhat more complex. The main problem is that of maintaining the correct dimension in the intersection. If Y and Z are two subvarieties of codimensions k and l, it is not always the case that their intersection has codimension k + l; for a trivial example, they could be equal. To handle this difficulty, the "moving lemma" is proved, which states that in any two rational equivalence classes we can always find representatives that intersect "generically transversely", in which case their intersection behaves well. Transversality of subvarieties is defined similarly as for manifolds: one defines the Zariski tangent spaces to the subvarieties, which are naturally subspaces of that of X, and if these subspaces span, then the intersection is transverse. It is generically transverse if transversality holds on an open, dense subset of the intersection.

In a sense it is disingenuous to claim that the Chow ring yields simpler proofs for facts that can be proved for cohomology as well. The machinery of scheme theory, flat families and flat limits in particular, and the moving lemma all furnish a great deal of technical difficulty underlying the Chow ring. However, these technical details for the most part underlie the theory, and once they are established the geometric advantage becomes clear.