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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)$\scriptstyle \geq$ 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 group $ \Cl$(X) introduced above.

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

$\displaystyle \CH^{p}_{}$(X) $\displaystyle \otimes_{{\Bbb Z}}^{}$ $\displaystyle \CH^{q}_{}$(X)$\displaystyle \to$$\displaystyle \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$$ \Bbb$Z given by deg($ \sum_{i}^{}$niPi) = $ \sum_{i}^{}$ni. This allows one to define intersection numbers of cycles of complementary dimension, in a purely algebraic way, which agree with those defined via topology when k = $ \Bbb$C (see item 7 below).
  3. If f : X$ \to$Y is a proper morphism of smooth varieties, there are `Gysin' maps f* : $ \CH^{p}_{}$(X)$ \to$$ \CH^{p+d}_{}$(Y) for all p, where d = dim Y - dim X; here if p + d < 0, we define f* 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 cp(V) $ \in$ $ \CH^{p}_{}$(X), such that
    1. c0(V) = 1,
    2. cp(V) = 0 for p > r, and
    3. for any exact sequence

      0$\displaystyle \to$V1$\displaystyle \to$V2$\displaystyle \to$V3$\displaystyle \to$ 0

      we have c(V2) = c(V1)c(V3), where c(Ei) = $ \sum_{p}^{}$cp(Vi) 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}$$\scriptstyle \Bbb$P(V)(1)), the first Chern class of the tautological line bundle, which is subject to the relation

    $\displaystyle \xi^{r}_{}$ - c1(V)$\displaystyle \xi^{r-1}_{}$ + ... + (- 1)rcn(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 cp(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) $\displaystyle \otimes$ $\displaystyle \Bbb$Q$\displaystyle \;\stackrel{\cong}{\to}\;$$\displaystyle \CH^{*}_{}$(X) $\displaystyle \otimes$ $\displaystyle \Bbb$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($\displaystyle \cal {F}$)td (X)) = ch(f!$\displaystyle \cal {F}$)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)

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

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

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Kapil Hari Paranjape 2002-11-21