Arithm\'etique des sch\'emas en groupes de Bruhat-Tits sur un anneau de Dedekind semi-local

TL;DR

This paper develops foundational cohomological methods for Bruhat-Tits group schemes over semi-local Dedekind rings, providing simplified proofs of key conjectures and extending results to semisimple simply connected groups.

Contribution

It introduces a cohomological framework for Bruhat-Tits schemes over Dedekind rings and proves the Grothendieck-Serre conjecture in this setting, including for semisimple simply connected groups.

Findings

Simplified proof of the Grothendieck-Serre conjecture over semi-local Dedekind rings.

Extension of analogous results to Bruhat-Tits group schemes of semisimple simply connected groups.

Foundational work for future cohomological studies of Bruhat-Tits group schemes.

Abstract

The aim of this paper is to lay the foundations for the cohomological study of Bruhat-Tits group schemes over a semi-local Dedekind ring. In particular, we obtain a simplified proof of the Grothendieck-Serre conjecture in this case and also an analogous result for Bruhat-Tits group schemes of a semisimple simply connected group.