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 Z^p(X). As in the case of curves one can introduce the effective cycles Z^p(X)^{$\geq$ 0} which is the sub-semigroup of Z^p(X) consisting of non-negative linear combinations. There is a subgroup R^p(X) $\subset$ 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 $\in$ k(W)^*. The quotient $\CH^{p}_{}$(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 $\CH_{p}^{}$(X) = $\CH^{n-p}_{}$(X). For p = 1 and X a smooth projective curve the Chow group $\CH^{1}_{}$(X) is precisely the class group $\Cl$(X) introduced above.

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

$$\CH^{p}_{}$$(X) $$\otimes_{{\Bbb Z}}^{}$$ $$\CH^{q}_{}$$(X)$$\to$$$$\CH^{p+q}_{}$$(X)

making $\CH^{*}_{}$(X) = $\oplus_{p}^{}$ $\CH^{p}_{}$(X) into an associative, commutative, graded ring, where $\CH^{0}_{}$(X) = $\Bbb$Z, and $\CH^{p}_{}$(X) = 0 for p > dim X. The Chow ring is thus an algebraic analogue for the even cohomology ring $\oplus_{i=0}^{n}$ $\HH^{2i}_{}$(X,$\Bbb$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 $\mapsto$ $\oplus_{p}^{}$ $\CH^{p}_{}$(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 : $\CH^{n}_{}$(X)$\to$$\Bbb$Z given by deg($\sum_{i}^{}$n_iP_i) = $\sum_{i}^{}$n_i. This allows one to define intersection numbers of cycles of complementary dimension, in a purely algebraic way, which agree with those defined via topology when k = $\Bbb$C (see item 7 below).
3. If f : X$\to$Y is a proper morphism of smooth varieties, there are `Gysin' maps f<sub>*</sub> : $\CH^{p}_{}$(X)$\to$$\CH^{p+d}_{}$(Y) for all p, where d = dim Y - dim X; here if p + d < 0, we define f<sub>*</sub> to be 0; the induced map $\oplus_{p}^{}$ $\CH^{p}_{}$(X)$\to$ $\oplus_{p}^{}$ $\CH^{p}_{}$(Y) is $\oplus_{p}^{}$ $\CH^{p}_{}$(Y)-linear (`projection formula').
4. f^* : $\CH^{*}_{}$(X)$\;\stackrel{\cong}{\to}\;$$\CH^{*}_{}$(V) for any vector bundle f : V$\to$X.
5. If V is a vector bundle (i.e., locally free sheaf) of rank r, then there are Chern classes c_p(V) $\in$ $\CH^{p}_{}$(X), such that
   1. c₀(V) = 1,
   2. c_p(V) = 0 for p > r, and
   3. for any exact sequence
      0$$\to$$V₁$$\to$$V₂$$\to$$V₃$$\to$$ 0
      we have c(V₂) = c(V₁)c(V₃), where c(E_i) = $\sum_{p}^{}$c_p(V_i) are the total Chern classes.
   Moreover, we also have the following property.
6. If f : $\Bbb$P(V)$\to$X is the projective bundle associated to a vector bundle of rank r, $\CH^{*}_{}$($\Bbb$P(V)) is a $\CH^{*}_{}$(X)-algebra generated by $\xi$ = c₁($\cal {O}$_$\Bbb$P(V)(1)), the first Chern class of the tautological line bundle, which is subject to the relation
   $$\xi^{r}_{}$$ - c₁(V)$$\xi^{r-1}_{}$$ + ^... + (- 1)^rc_n(V) = 0

7. If k = $\Bbb$C, there are cycle class homomorphisms $\CH^{p}_{}$(X)$\to$$\HH^{2p}_{}$(X,$\Bbb$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 c₁ : $\Pic$(X)$\to$$\CH^{1}_{}$(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) $$\otimes$$ $$\Bbb$$Q$$\;\stackrel{\cong}{\to}\;$$$$\CH^{*}_{}$$(X) $$\otimes$$ $$\Bbb$$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 : X$\to$Y, and any coherent sheaf $\cal {F}$ on X, we have

f_*(ch($$\cal {F}$$)td (X)) = ch(f_!$$\cal {F}$$)td (Y),

where td (X) $\in$ $\CH^{*}_{}$(X), td (Y) $\in$ $\CH^{*}_{}$(Y) are the Todd classes of the tangent sheaves of X and Y respectively; here f_! : G₀(X)$\to$G₀(Y) is f_!($\cal {F}$) = $\sum_{i\geq 0}^{}$(- 1)ⁱ[Rⁱf_*$\cal {F}$], 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)

$$\chi$$(X,$$\cal {F}$$) = $$\sum_{i\geq 0}^{}$$(- 1)ⁱdim_k$$\HH^{i}_{}$$(X,$$\cal {F}$$) = deg$$\left(\vphantom{ch({\cal F})td(X)}\right.$$ch($$\cal {F}$$)td (X)$$\left.\vphantom{ch({\cal F})td(X)}\right)_{n}^{}$$,

where the subscript n means that we compute the degree of the component in $\CH^{n}_{}$(X). For further details, see [14], Chapter 15.