# Local-global principles for tori over arithmetic curves

## TL;DR

This paper investigates local-global principles for algebraic tori over semi-global fields, providing criteria for when these principles hold or fail, using patching methods and flasque resolutions.

## Contribution

It introduces new methods to compute obstructions to local-global principles for tori over arithmetic curves, including conditions for their vanishing and explicit examples.

## Key findings

- Obstructions can be computed via patching techniques.
- Sufficient conditions for the vanishing of obstructions are provided.
- Examples demonstrate both trivial and nontrivial obstructions.

## Abstract

In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the underlying discrete valuation ring, the obstruction to a local-global principle with respect to discrete valuations can be computed using methods coming from patching. We give a sufficient condition for the vanishing of the obstruction, as well as examples were the obstruction is nontrivial or even infinite. A major tool is the notion of a flasque resolution of a torus.