Splitting families in Galois cohomology

TL;DR

This paper constructs specific varieties over a field that characterize the vanishing of cohomology classes in Galois cohomology, providing a geometric criterion for triviality of classes in terms of rational points.

Contribution

It establishes a method to represent the triviality of Galois cohomology classes via rational points on a family of varieties, generalizing previous results and introducing an ind-variety structure.

Findings

For any cohomology class, there exists a family of varieties detecting its triviality.

In the case n=2, a single variety suffices to detect triviality.

The varieties can be organized into an ind-variety, enhancing the geometric framework.

Abstract

Let $k$ be a field, with absolute Galois group $\Gamma$. Let $A/k$ be a finite \'etale group scheme of multiplicative type, i.e. a discrete $\Gamma$-module. Let $n \geq 2$ be an integer, and let $x \in H^n(k,A)$ be a cohomology class. We show that there exists a countable set $I$, and a familiy $(X_i)_{i \in I}$ of (smooth, geometrically integral) $k$-varieties, such that the following holds. For any field extension $l/k$, the restriction of $x$ vanishes in $H^n(l,A)$ if and only if (at least) one of the $X_i$'s has an $l$-point. We moreover show that the $X_i$'s can be made into an ind-variety. In the case $n=2$, we note that one variety is enough.