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# 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) 0 which is the sub-semigroup of Zp(X) consisting of non-negative linear combinations. There is a subgroup Rp(X) 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 k(W)*. The quotient (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 (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) into an associative, commutative, graded ring, where (X) = Z, and (X) = 0 for p > dim X. The Chow ring is thus an algebraic analogue for the even cohomology ring (X,Z) in topology. A refined version of this analogy is examined in Section 6. In any case we note the following cohomology-like' properties.
1. X (X) is a contravariant functor from the category of smooth varieties over k to graded rings.
2. If X is projective and n = dim X, there is a well defined degree homomorphism deg : (X)Z given by deg(niPi) = ni. 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).
3. If f : XY 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').
4. f* : (X)(V) for any vector bundle f : VX.
5. If V is a vector bundle (i.e., locally free sheaf) of rank r, then there are Chern classes cp(V) (X), such that
1. c0(V) = 1,
2. cp(V) = 0 for p > r, and
3. for any exact sequence

0V1V2V3 0

we have c(V2) = c(V1)c(V3), where c(Ei) = cp(Vi) are the total Chern classes.
Moreover, we also have the following property.
6. 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 = c1(P(V)(1)), the first Chern class of the tautological line bundle, which is subject to the relation

- c1(V) + ... + (- 1)rcn(V) = 0

7. 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 cp(E) is the topological p-th Chern class of E.

In analogy with the case of curves we have that c1 : (X)(X) is an isomorphism. In fact more is true. If K0(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

ch : K0(X) Q(X) Q.

Identifying the group K0(X) with the Grothendieck group G0(X) of coherent sheaves, we may extend the definitions of Chern classes and Chern character to coherent sheaves; now the Grothendieck-Riemann-Roch theorem states that for any proper morphism f : XY, and any coherent sheaf on X, we have

f*(ch()td (X)) = ch(f!)td (Y),

where td (X) (X), td (Y) (Y) are the Todd classes of the tangent sheaves of X and Y respectively; here f! : G0(X)G0(Y) is f!() = (- 1)i[Rif*], and the Todd class of a coherent sheaf is a certain polynomail in its Chern classes. If X is proper over k (e.g., X is projective) of dimension n, and Y is a point, this gives a formula (the Grothendieck-Hirzebruch-Riemann-Roch formula)

(X,) = (- 1)idimk(X,) = degch()td (X),

where the subscript n means that we compute the degree of the component in (X). For further details, see [14], Chapter 15.

Next: 3 Divisors on varieties Up: Algebraic Cycles Previous: 1 Model case of
Kapil Hari Paranjape 2002-11-21