# Gerbe patching and a Mayer-Vietoris sequence over arithmetic curves

## TL;DR

This paper extends patching techniques and local-global principles to gerbes over arithmetic curves, establishing a Mayer-Vietoris sequence for non-abelian hypercohomology and applying these results to points on homogeneous spaces.

## Contribution

It introduces a 2-categorical analogue of patching for gerbes, develops a Mayer-Vietoris sequence for non-abelian hypercohomology, and proves local-global principles for points on certain homogeneous spaces.

## Key findings

- Established a Mayer-Vietoris sequence for non-abelian hypercohomology.
- Proved local-global principles for points on homogeneous spaces under linear algebraic groups.
- Extended patching techniques to gerbes and bitorsors over arithmetic curves.

## Abstract

We discuss patching techniques and local-global principles for gerbes over arithmetic curves. Our patching setup is that introduced by Harbater, Hartmann and Krashen. Our results for gerbes can be viewed as a 2-categorical analogue on their results for torsors. Along the way, we also discuss bitorsor patching and local-global principles for bitorsors. As an application of these results, we obtain a Mayer-Vietoris sequence with respect to patches for non-abelian hypercohomology sets with values in the crossed module G->Aut(G) for G a linear algebraic group. Using local-global principles for gerbes, we also prove local-global principles for points on homogeneous spaces under linear algebraic groups H that are special (e.g. SL_n and Sp_2n) for certain kind of stabilizers.