nLab
Grothendieck-Riemann-Roch theorem

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Integration theory

Contents

Idea

The Grothendieck-Riemann-Roch theorem describes (the failure of) the naturality of the behaviour of a Chern character under push forward along proper maps.

It says, in the formulation of (Atiyah-Hirzebruch 61), that for f:XYf \colon X \longrightarrow Y a flat morphism of schemes which are flat and regular quasi-projective varieties over the spectrum of a Dedekind domain, then the Chern character of the push-forward of some EE is the push-forward of the cup product of the Chern-character of EE with the Todd class. Hence it says that “Chern-cup-Todd is natural under pushforward” along proper maps, and generally along K-oriented maps.

If f:XYf \colon X \to Y is a proper map, then there is a commuting diagram

K 0(X) ch()Td(X) H (X,) f f K 0(Y) ch()Td(Y) H (Y,), \array{ K_0(X) &\stackrel{ch(-)\cup Td(X)}{\to}& H_\bullet(X, \mathbb{Q}) \\ \downarrow^{\mathrlap{\int_{f}}} && \downarrow^{\mathrlap{\int_f}} \\ K_0(Y) &\stackrel{ch(-)\cup Td(Y)}{\to}& H_\bullet(Y,\mathbb{Q}) } \,,

where…

If XX is an algebraic curve, then the Riemann-Roch theorem reduces to a statement about the Euler characteristic/curve. This generalizes to arithmetic geometry with the notion of genus of a number field.

References

The formulation in terms of topological K-theory is due to

For a general survey see

Discussion of Riemann-Roch over arithmetic curves is in

  • Jürgen Neukirch, chapter III, section 3 of Algebraische Zahlentheorie (1992), English translation Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften 322, 1999 (pdf)

The refinement to differential cohomology, hence differential K-theory, is discussed in section 6.2 of

Over algebraic stacks/Deligne-Mumford stacks the GRR theorem is discussed in

  • Roy Joshua, Riemann-Roch for algebraic stacks (pdf I, pdf II, pdf III)

  • Bertrand Toën, Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford, K-Theory 18 (1999), no. 1, 33–76. 1, 23

  • Dan Edidin, Riemann-Roch for Deligne-Mumford stacks (arXiv:1205.4742v1)

Revised on July 19, 2014 04:22:40 by Urs Schreiber (89.204.130.28)