Higher adeles and non-abelian Riemann-Roch

2012/04/20 by Ted Chinburg, Chinburg, T., G. Pappas +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1204.4520

openalex publication_date 2012/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 P¹. Our set-up provides an extension of several aspects of the classical Fr"ohlich theory of the Galois module structure of rings of integers of number fields to arithmetic surfaces.

Citations

• Infinite-dimensional vector bundles in algebraic geometry (an introduction)
• An introduction to higher dimensional local fields and adeles
• K₁ of a p-adic group ring II. The determinantal kernel SK₁

Related

• Grothendieck's proof of Hirzebruch-Riemann-Roch theorem
• Adelic openness without the Mumford-Tate conjecture
• Cohomological arithmetic Chow rings
• K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems
• Riemann-Roch for Spec\mathbb Z