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Bruhat-Tits group schemes over higher dimensional base-II

Vikraman Balaji, Yashonidhi Pandey

Abstract

We prove that split reductive BT group schemes over a higher dimensional base are {\em affine}. Our method also gives a new construction of higher BT-group schemes more general than parahoric ones. The new ingredients are an extension of J.-K.Yu's construction in \cite{yu} to higher dimensional bases, Néron-Raynaud dilatations of subgroup schemes on divisors, combined with techniques from \cite{bt2} and the structure theory developed in \cite{bp}.

Bruhat-Tits group schemes over higher dimensional base-II

Abstract

We prove that split reductive BT group schemes over a higher dimensional base are {\em affine}. Our method also gives a new construction of higher BT-group schemes more general than parahoric ones. The new ingredients are an extension of J.-K.Yu's construction in \cite{yu} to higher dimensional bases, Néron-Raynaud dilatations of subgroup schemes on divisors, combined with techniques from \cite{bt2} and the structure theory developed in \cite{bp}.
Paper Structure (23 sections, 13 theorems, 136 equations)

This paper contains 23 sections, 13 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Layout
  3. The recursive step of J.K. Yu yu
  4. Over arbitrary DVRs
  5. On a result of McNinch on Levi decompositions
  6. Parahoric Higher BT group scheme in the split reductive case are affine
  7. When $G$ is not simply-connected
  8. We now consider the case $G$ is split reductive
  9. The recursive step over higher dimensional bases
  10. Reduction to the case $D$ is a single divisor assuming Theorem \ref{['first']}
  11. Relation with $\mathfrak{G}_{(f_0,\cdots,f_n)} \rightarrow X$ of bp
  12. Proof of \ref{['first']} when $\text{dim}(X) = 2$
  13. When $\text{dim}(X) \geq 3$
  14. Towards the completion of Theorem \ref{['first']}
  15. Step 1: The prolongation of the unipotent radical and a Levi-like decomposition of $\mathfrak{G}_{_{Y}}$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $G$ be a split reductive connected Chevalley group scheme over $\mathbb{Z}$ with a split maximal torus $T$. Let $\boldsymbol{\boldsymbol{\Phi}}$ denote the root system of $G$ relative to $T$. For $0 \leq i \leq n$, let $f_{_i}: \boldsymbol{\Phi} \cup \{0 \} \rightarrow \mathbb{R}$ be concave fun be a reduced normal crossing divisor. For $0 \leq i \leq n$, we denote the generic point of the com

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Theorem 4.1
  • proof
  • proof
  • Theorem A.1
  • Theorem A.2
  • Definition A.3
  • Theorem A.4
  • ...and 9 more