Unramified Brauer groups of finite simple groups of Lie type Al

The cohomology theory of groups provides a way to obtain the Artin-Mumford birational invariant for a smooth compactification of a quotient space V by the action of a linear algebraic group G. This invariant distinguishes the quotient spaces V/G from rational varieties. There are numerous finite groups for which this invariant is non-zero and thus the corresponding quotient varieties V/ G are not rational nor even stably rational. As a consequence of our main result we prove that the first obstruction to stable rationality is trivial for quotient spaces V/G where G is a finite simple group of Lie type Al.;We present an algebraic framework for describing the subgroup B0(G) of the second cohomology group H2 (G, Q/Z ) of G consisting of the elements which have trivial restrictions to every Abelian subgroup of G. This group is identified with an unramified Brauer group coinciding with the Artin-Mumford group of the quotient variety V/G for faithful linear representations V of G. The initial step in this program is to define the set of equivalence classes of central extensions of G by Q/Z giving rise to the given action of G on Q/Z suitable for studying H2 ( G, ( Q/Z ). It is therefore desirable to understand the structure of such extensions with Abelian kernel corresponding to the elements of B 0(G) and to isolate the minimal data necessary to determine them. As the main tool in this analysis we use the notion of a Sylow subgroup. We demonstrate that for every finite group G, the unramified Brauer group B0(G) of G has a primary decomposition. Our knowledge of the structure of the subgroups of unramified Brauer groups of Sylow subgroups of finite simple groups allows us to state the conjecture which asserts that the unramified Brauer group of every finite simple group reduces to zero. Our main result establishes the validity of this conjecture for finite simple groups of Lie type Al.