Chow Ring - Rational Equivalence

Rational Equivalence

Further information: Equivalence relations on algebraic cycles

Before defining the Chow ring, we must define the notion of "rational equivalence", which as the name indicates, is an equivalence relation on a certain set. If X is an algebraic variety and Y, Z are two subvarieties, we say that Y and Z are rationally equivalent if there is a flat family parameterized by P1, contained in the product family P1 × X, two of whose fibers are Y and Z. In more classical language, we want a subvariety V of the product family two of whose fibers are Y and Z, and all of whose fibers are subvarieties of X with the same Hilbert polynomial. If we think of P1 as a line, then this notion is an algebraic analogue of cobordism.