|analytic integration||cohomological integration|
|measurable space||Poincaré duality|
|measure||orientation in generalized cohomology|
|volume form||(virtual) fundamental class|
|Riemann/Lebesgue integration of differential forms||push-forward in generalized cohomology/in differential cohomology|
It says, in the formulation of (Atiyah-Hirzebruch 61), that for 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 is the push-forward of the cup product of the Chern-character of 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 is a proper map, then there is a commuting diagram
If 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.
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
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)