Local-global principles for Galois cohomology

TL;DR

This paper establishes local-global principles for Galois cohomology groups over function fields of curves over complete discretely valued fields, extending known results and applying patching and Bloch-Kato techniques.

Contribution

It proves new local-global principles for higher Galois cohomology groups over such function fields, generalizing previous cases for n=3.

Findings

Local-global principles hold for $H^n(F, Z/mZ(n-1))$ for all $n>1$

Results enable local-global analysis of torsors over these fields

Uses patching and Bloch-Kato conjecture in proofs

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 $H^n(F, Z/mZ(n-1))$, 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.