The Brauer group and the center of generic matrices
Let k be a field and let R(k,n,r) be the ring generated by r \(n\times n\)- generic matrices. It is a domain with a central localization UD(k,n,r) which is a division algebra of degree n over its center Z(k,n,r). For all n, Z(k,n,r) is unirational and is known to be rational for \(n=2, 3\) and 4. In view of a construction, due to Artin and Mumford, of a field extension K/k, which is unirational but not rational, it is a natural question to ask if there exists a projective k-variety V such that k(V), the function field of V, is Z(k,n,r) and such that the Brauer group Br(V) is unequal to Br(k). In fact, the author shows that for any k,n,r the Brauer group of a V such that \(k(V)=Z(k,n,r)\) must be Br(k). As a step in the argument, he also shows that if W is the Brauer-Severi variety of UD(k,n,r) defined over \(K=Z(k,n,r)\), then K(W)/k is rational. In the last section, the author constructs a smooth subring T of Z(k,n,r) such that Br(T) viewed as in Br(Z(k,n,r)), is generated by the class of UD(k,n,r).
scientific article; zbMATH DE number 3939468
scientific article