A bound to kill the ramification over function fields

TL;DR

This paper establishes a bound on ramification in function fields over characteristic zero fields, showing that certain cohomology classes become unramified after adjoining r-th roots of specific functions.

Contribution

It provides a new bound on ramification for elements in cohomology groups over function fields, extending understanding of ramification behavior in algebraic geometry.

Findings

Cohomology classes become unramified after adjoining r-th roots of n^2 functions.

The result applies to fields containing all r-th roots of unity.

The bound depends on the dimension n of the variety.

Abstract

Let k be a field of characteristic zero, let X be a geometrically integral k-variety of dimension n and let K be its field of fractions. Under the assumption that K contains all r-th roots of unity for an integer r, we prove that, given an element in H^m(K, Z/r), it becomes unramified in the extension of K obtained by adding r-th roots of some n^2 functions in K.