Mayer - Vietoris Equences and Brauer Groups of Nonnormal Domains

Let R be a Noetherian domain with finite integral closure R. We study the map from the Brauer group of R, B(R), to B(R): first, by embedding B(R) into the Cech etale cohomology group H (R, U) and using a Mayer-Vietoris sequence for Cech 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) to B(K) is 1-1. Let R be a Noetherian integral domain with finite integral closure R, conductor c and quotient field K. The object of this paper is to try to describe relationships between the Brauer group of R, B(R), and B(R), B(R/c), and B(K). Questions of this kind were considered by M. Auslander and 0. Goldman, who showed that if 7? = R is regular, the map from B(R) to B(K) is 1-1. Our approach in the first three sections is to glean information from a long exact Mayer-Vietoris sequence of Cech cohomology. This sequence extends a six-term Mayer-Vietoris K-theory sequence for the category Pic of Milnor and Bass, and when B(R) is isomorphic to the second etale cohomology group with coefficients in the sheaf of units (multiplicative group) the extended sequence describes the kernel and image of the map from B(R) to B(R) © B(R/c). In particular, for R of dimension 1 the kernel is trivial. When 7? has dimension 1, R is regular, so the only candidates for elements in the kernel of the map from B(R) to B(K) are elements of B(R) which become trivial in B(R) but not in B(R/c). Auslander and Goldman's counterexample to B(R) —» B(K) being 1-1 is of this kind. As a consequence it follows that if R is any ring with quotient field K a global field, the map B(R) —» B(K) is 1-1. We get the following splitting result: If A is any Azumaya R-algebra, R a ring with quotient field a global field K, and A ®R K is split by a finite extension field L, then every order over R in L splits A. When B(R) cannot be identified cohomologically, cohomological methods do not give precise information on the kernel of the map B(R) —» B(R) © B(R/c). In Received by the editors January 31, 1972 and, in revised form, October 15, 1973. AMS (.MOS) subject classifications (1970). Primary 13A20; Secondary 13B20, 14F20. (') Partially supported by the National Science Foundation Grants GP 29652 XI and SD GU 3171. Copyright O 1974, American Mathematical Society 51 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use