Mayer-Vietoris sequences and Brauer groups of nonnormal domains

Abstract

Let R be a Noetherian domain with finite integral closure R ¯ \bar R . We study the map from the Brauer group of R , B ( R ) R,B(R) , to B ( R ¯ ) B(\bar R) : first, by embedding B ( R ) B(R) into the Čech etale cohomology group H 2 ( R , U ) {H^2}(R,U) and using a Mayer-Vietoris sequence for Čech cohomology of commutative rings; second, via Milnor’s theorem from algebraic K-theory. We apply our results to show, i.e., that if R is a domain with quotient field K a global field, then the map from B ( R ) B(R) to B ( K ) B(K) is 1-1.