cohomology

integration

# 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 $f \colon X \longrightarrow Y$ 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 $E$ is the push-forward of the cup product of the Chern-character of $E$ 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 $f \colon X \to Y$ is a proper map, then there is a commuting diagram

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

where…

If $X$ 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)