Unramified Brauer groups of finite simple groups of Lie type Aℓ

We study the subgroup B₀(G) of H²(G, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]) consisting of all elements which have trivial restrictions to every Abelian subgroup of G. The group B₀(G) serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties V/G where V is a faithful complex linear representation of the group G. We prove that B₀(G) is trivial for finite simple groups of Lie type A_ℓ.