# A note on Gersten's conjecture for \'etale cohomology over two-dimensional henselian regular local rings

## TL;DR

This paper proves Gersten's conjecture for étale cohomology over two-dimensional henselian regular local rings without equi-characteristic assumptions, and applies it to establish a local-global principle for Galois cohomology in mixed characteristic cases.

## Contribution

It extends the validity of Gersten's conjecture to mixed characteristic two-dimensional henselian regular local rings and derives a local-global principle for Galois cohomology.

## Key findings

- Gersten's conjecture holds for étale cohomology in the specified setting.
- Established a local-global principle for Galois cohomology in mixed characteristic.
- No equi-characteristic assumption needed for the main result.

## Abstract

We show the Gersten's conjecture for \'etale cohomology over two dimensional henselian regular local rings without assuming equi-characteristic. As application, we obtain the local-global principle for Galois cohomology over mixed characteristic two-dimensional henselian local rings.