next up previous Next: 3 Divisors on varieties Up: Algebraic
Cycles Previous: 1 Model case of  2 The Grothendieck-Riemann-
Roch theorem  Let X be a non-singular variety over k. An
algebraic cycle of codimension p is an element of the free
Abelian group on irreducible subvarieties of X of codimension p;
the group of these cycles is denoted Zp(X). As in the case of
curves one can introduce the effective cycles Zp(X)$\scriptstyle
\geq$ 0 which is the sub-semigroup of Zp(X) consisting of non-
negative linear combinations. There is a subgroup Rp(X) $
\subset$ Zp(X), defined to be the subgroup generated by all the
cycles div(f )W where W ranges over irreducible subvarieties of
codimension p - 1 in X, and f $ \in$ k(W)*. The quotient $
\CH^{p}_{}$(X) = Zp(X)/Rp(X) is called the Chow group of
codimension p cycles on X modulo rational equivalence; if n =
dim X then we use the notation $ \CH_{p}^{}$(X) = $
\CH^{n-p}_{}$(X). For p = 1 and X a smooth projective curve the
Chow group $ \CH^{1}_{}$(X) is precisely the class g