rational equivalence

Let XX be a noetherian scheme. One defines a notion of rational equivalence on XX which, roughly speaking, identifies algebraic cycles that are connected by a family of cycles parametrized by the projective line.

A cycle CZ k(X)C \in Z_k(X) is rationally equivalent to zero if there are closed integral subschemes Z αZ_\alpha of dimension k+1k + 1 and invertible rational functions r αR(Z α) *r_\alpha \in R(Z_\alpha)^* such that

C= αj α,*(div(r α)), C = \sum_\alpha j_{\alpha,*}(div(r_\alpha)) \,,


  • divdiv denote the associate Weil divisors;

  • j α,*j_{\alpha,*} denotes the direct image along the inclusion Z αXZ_\alpha \hookrightarrow X.

Rational equivalence generalizes linear equivalence? of Weil divisors. It is an example of an adequate equivalence relation.


Last revised on May 30, 2013 at 16:09:50. See the history of this page for a list of all contributions to it.