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 *Z*^{p}(*X*). As in the case of curves one can
introduce the *effective cycles*
*Z*^{p}(*X*)^{ 0} which is the
sub-semigroup of *Z*^{p}(*X*) consisting of non-negative linear
combinations. There is a subgroup
*R*^{p}(*X*) *Z*^{p}(*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* *k*(*W*)^{*}. The quotient
(*X*) = *Z*^{p}(*X*)/*R*^{p}(*X*) is
called the *Chow group* of codimension *p* cycles on *X* modulo
rational equivalence; if *n* = dim *X* then we use the notation
(*X*) = (*X*). For *p* = 1 and *X* a smooth projective curve the Chow
group (*X*) is precisely the class group (*X*) introduced above.

The generalisation of Schubert calculus on the Grassmannians
is the *intersection product*

(*X*) (*X*)(*X*)

making
(-
*X*(*X*) is a contravariant functor from the category of smooth varieties over*k*to graded rings. - If
*X*is projective and*n*= dim*X*, there is a well defined*degree homomorphism*deg : (*X*)*Z*given by deg(*n*_{i}*P*_{i}) =*n*_{i}. This allows one to define intersection numbers of cycles of complementary dimension, in a purely algebraic way, which agree with those defined via topology when*k*=*C*(see item 7 below). - If
*f*:*X**Y*is a proper morphism of smooth varieties, there are `Gysin' maps*f*_{*}: (*X*)(*Y*) for all*p*, where*d*= dim*Y*- dim*X*; here if*p*+*d*< 0, we define*f*_{*}to be 0; the induced map (*X*) (*Y*) is (*Y*)-linear (`projection formula'). -
*f*^{*}: (*X*)(*V*) for any vector bundle*f*:*V**X*. - If
*V*is a vector bundle (*i.e.*, locally free sheaf) of rank*r*, then there are Chern classes*c*_{p}(*V*) (*X*), such that*c*_{0}(*V*) = 1,*c*_{p}(*V*) = 0 for*p*>*r*, and- for any exact sequence
0we have
*V*_{1}*V*_{2}*V*_{3}0*c*(*V*_{2}) =*c*(*V*_{1})*c*(*V*_{3}), where*c*(*E*_{i}) =*c*_{p}(*V*_{i}) are the total Chern classes.

- If
*f*:*P*(*V*)*X*is the projective bundle associated to a vector bundle of rank*r*, (*P*(*V*)) is a (*X*)-algebra generated by =*c*_{1}(_{P(V)}(1)), the first Chern class of the tautological line bundle, which is subject to the relation-*c*_{1}(*V*) +^{ ... }+ (- 1)^{r}*c*_{n}(*V*) = 0 - If
*k*=*C*, there are cycle class homomorphisms (*X*)(*X*,*Z*) such that the intersection product corresponds to the cup product in cohomology, and for a vector bundle*E*, the cycle class of*c*_{p}(*E*) is the topological*p*-th Chern class of*E*.

In analogy with the case of curves we have that
*c*_{1} : (*X*)(*X*) is an isomorphism. In fact more is true.
If *K*_{0}(*X*) is the Grothendieck ring of vector bundles on *X*, the
*Chern character* (defined using Chern classes by the same formula as
in topology) gives a ring isomorphism

(*X*,) = (- 1)^{i}dim_{k}(*X*,) = deg*ch*()*td* (*X*),

where the
subscript