Groupe de Brauer non ramifi\'e de quotients par un groupe fini

TL;DR

This paper extends Bogomolov's algebraic formula for the unramified Brauer group of quotients SL(n)/G over algebraically closed fields to any characteristic zero field, generalizing previous number field results.

Contribution

It develops a purely algebraic method to compute the unramified Brauer group for quotients by finite groups over any characteristic zero field, broadening the scope of Bogomolov's original formula.

Findings

Generalization of Bogomolov's formula to arbitrary characteristic zero fields.

Recovery of Harari and Demarche's results over number fields.

Proving triviality of the unramified Brauer group for Q and G of odd order.

Abstract

Let k be a field, G a finite group embedded in the k-group SL(n). For k an algebraically closed field, Bogomolov gave a formula for the unramified Brauer group of the quotient SL(n)/G. We develop his method over any characteristic zero field. This purely algebraic method enables us to recover and generalize results of Harari and of Demarche over number fields, such as the triviality of the unramified Brauer group for k=Q and G of odd order. --- Soient k un corps et G un groupe fini plong\'e dans le k-groupe SL(n).Pour k alg\'ebriquement clos, Bogomolov a donn\'e une formule pour le groupe de Brauer non ramifi\'e du quotient SL(n)/G. On examine ce que donne sa m\'ethode sur un corps k quelconque (de caract\'eristique nulle). Par cette m\'ethode purement alg\'ebrique, on retrouve et \'etend des r\'esultats obtenus par Harari et par Demarche au moyen de m\'ethodes arithm\'etiques, comme la trivialit\'e du groupe de Brauer non ramifi\'e pour k= Q et G d'ordre impair.