Modules de cycles et classes non ramifi\'ees sur un espace classifiant

TL;DR

This paper develops a new method using exact sequences and residue morphisms to analyze unramified elements in Serre's invariants of finite groups, extending previous results in unramified cohomology.

Contribution

It introduces a novel construction of exact sequences for detecting unramified elements in Serre's invariants using residue morphisms related to subgroup and homomorphism pairs.

Findings

Reproduces Bogomolov and Peyre's results on unramified cohomology.

Generalizes unramified cohomology results to broader contexts.

Provides explicit residue morphisms for analyzing invariants.

Abstract

Let G be a finite group of exponent m and let k be a field of characteristic prime to m, containing the m-th roots of unity. For any Rost cycle module M over k, we construct exact sequences detecting the unramified elements in Serre's group of invariants of G with values in M in terms of "residue" morphisms associated to pairs (D,g), where D runs through the subgroups of G and g runs through the homomorphisms \mu_m \to G whose image centralises D. This allows us to recover results of Bogomolov and Peyre on the unramified cohomology of fields of invariants of G, and to generalise them.