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Higher adeles and non-abelian Riemann-Roch. (English) Zbl 1349.14035

Summary: 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 \(\mathbb{P}^1\). 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.

MSC:

14C40 Riemann-Roch theorems
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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