Tame Covers and Cohomology of Relative Curves over Complete Discrete Valuation Rings, with Applications to the Brauer Group

TL;DR

This paper explores the cohomology and covers of relative curves over complete discrete valuation rings, providing new constructions of division algebras over p-adic function fields, especially for curves lacking smooth models over Z_p.

Contribution

It introduces methods to lift cohomology classes in mixed characteristic settings, enabling the construction of noncrossed product and indecomposable division algebras over certain p-adic function fields.

Findings

Existence of noncrossed product division algebras over p-adic function fields.

Construction of indecomposable division algebras in new settings.

Generalization of previous results to curves without smooth models over Z_p.

Abstract

We prove the existence of noncrossed product and indecomposable division algebras over the function field of a smooth p-adic curve, especially when the curve does not admit a smooth model over Z_p. Thus we generalize arXiv 0907.0670. To make our constructions, we investigate the lifting of cohomology classes from the total fraction ring of the closed fiber to the function field of the curve, over an arbitrary discrete valuation ring of mixed characteristic.