Higher adeles and non-abelian Riemann–Roch

We show a Riemann–Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson–Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann–Roch formula as in Kapranov–Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P1P1. Our set-up provides an extension of several aspects of the classical Fröhlich theory of the Galois module structure of rings of integers of number fields to arithmetic surfaces.