Grothendieck-Riemann-Roch theorem

In mathematics, specifically in algebraic geometry, the Grothendieck-Riemann-Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch-Riemann-Roch theorem , about complex manifolds, which is itself a generalisation of the classical Riemann-Roch theorem for line bundles on compact Riemann surfaces .

Riemann-Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees , or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann-Roch theorem does this for curves and line bundles, whereas the Hirzebruch-Riemann-Roch theorem generalises this to vector bundles over manifolds. The Grothendieck-Riemann-Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves.

The theorem has been very influential, not least for the development of the Atiyah-Singer index theorem. Conversely, complex analytic analogues of the Grothendieck-Riemann-Roch theorem can be proved using the families index theorem . Alexander Grothendieck, its author, did not publish his theorem because he was not satisfied with his 1957 proof. Instead Armand Borel and Jean-Pierre Serre, wrote up and published Grothendieck's preliminary (as he saw it) proof.

Let X be a smooth quasi-projective scheme over a field. Under these assumptions, the Grothendieck group

K₀(X)

of bounded complexes of coherent sheaves is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles. Using this isomorphism, consider the Chern character

(a rational combination of Chern classes) as a functorial transformation

where

is the Chow group of cycles on X of dimension d modulo rational equivalence , tensored with the rational numbers. In case X is defined over the complex numbers, the latter group maps to the topological cohomology group

.

Now consider a proper morphism

between smooth quasi-projective schemes and a bounded complex of sheaves

.

The Grothendieck-Riemann-Roch theorem relates the push forward maps

and the pushforward

by the formula

Here td(X) is the Todd genus of (the tangent bundle of) X. Thus the theorem gives a precise measure for the lack of commutativity of taking the push forwards in th eabove senses and the chern character and shows that the needed correction factors depends on X and Y only. In fact, since the Todd genus is functorial and multiplicative in exact sequences, we can rewrite the Grothendieck Riemann Roch formula to

where T_f is the relative tangent sheaf of f. This is often useful in applications, for example if f is a locally trivial fibration.

Generalisations of the theorem can be made to the non-smooth case by considering a proper generalisation of the combination ch( - )td(X) and to the non-proper case by considering cohomology with supports .

References

• A. Borel and J-P Serre, Le Théorème de Riemann-Roch, Bull. Soc. Math France 36 (1958), 97 - 136.
• W. Fulton Intersection theory. Springer 1987.