# Local-global principles for Galois cohomology

### David Harbater

University of Pennsylvania, Philadelphia, United States### Julia Hartmann

RWTH Aachen, Germany### Daniel Krashen

University of Georgia, Athens, USA

## Abstract

This paper proves local-global principles for Galois cohomology groups over function fields $F$ of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for $H^n(F, \mathbb Z/m \mathbb Z(n-1))$, for all $n>1$. This is motivated by work of Kato and others, where such principles were shown in related cases for $n=3$. Using our results in combination with cohomological invariants, we obtain local-global principles for torsors and related algebraic structures over $F$. Our arguments rely on ideas from patching as well as the Bloch–Kato conjecture.

## Cite this article

David Harbater, Julia Hartmann, Daniel Krashen, Local-global principles for Galois cohomology. Comment. Math. Helv. 89 (2014), no. 1, pp. 215–253

DOI 10.4171/CMH/317