Rational Equivalence

Abstract

A cycle on an arbitrary algebraic variety (or scheme) X is a finite formal sum Σn _v [V] of (irreducible) subvarieties of X, with integer coefficients. A rational function r on any subvariety of X determines a cycle [div (r)]. Cycles differing by a sum of such cycles are defined to be rationally equivalent. Alternatively, rational equivalence is generated by cycles of the form [V(0)] — [V(∞)] for subvarieties V of $X \times {\mathbb{P}^1}$ which project dominantly to ${\mathbb{P}^1}$. The group of rational equivalence classes on X is denoted A _* X.

For a proper morphism f:X → Y, there is an induced push-forward of cycles. The fundamental theorem of this chapter states that rational equivalence pushes forward, so there is an induced homomorphism f _* from A _* X to A _* Y, making A _* a covariant functor for proper morphisms.

For flat morphisms f:X → Y (of constant relative dimension) there are contravariant pull-back homomorphisms f* from A _* Y to A _* X. There is a useful exact sequence

$$ {A_ * }Y \to {A_ * }X \to {A_ * }\left( {X - Y} \right) \to 0 $$

for a closed subscheme Y of X, and exterior products

$$ {A_ * }X \otimes {A_ * }Y\xrightarrow{X}{A_ * }\left( {X \times Y} \right) $$

The groups A _* X will play a role analogous to homology groups in topology. In succeeding chapters it will be shown how geometric objects (vector bundles, regularly imbedded subschemes,…) give rise to operations on these groups (Chern classes, intersection products,…). Eventually corresponding contravariant, ring-valued functors A* will be constructed, with cap-products from ${A^ * }X \otimes {A_ * }X$ To ${A_ * }X$ and other properties familiar from topology. When X is non-singular, ${A_ * }X \cong {A_ * }X$ in the non-singular case, but not in general, A _* X will have a ring structure. The actual relation of these groups to homology groups is discussed in Chapter 19.