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The object of this paper is to study the kernel of the map of the Brauer group of an integrally closed noetherian domain A into the direct product of the Brauer groups of the localizations of A at prime ideals. It is shown that this kernel is isomorphically contained in the torsion subgroup of the first cohomology group of the sheaf of Cartier divisors over Spec A. As a consequence, the author describes several new sets of conditions on A which guaran- tee that the kernel is trivial. We recall that M. Auslander and 0. Goldman (2) define the Brauer group of a commutative ring A as follows: let %(A) be the isomor- phism classes of central separable A -algebras, and SI0(^1) the subset of %(A) consisting of endomorphism rings of finitely generated projec- tive faithful ^4-modules. Let Si and 22 in 11(A) be equivalent if there are algebras 9ii and 9i2 in 3to(^4) such that Si ®A 9ii is isomorphic to ?2 ®a %. Then 93(^4), the set of equivalence classes of 21(^4) under this relation, is a group, the Brauer group of A. If S is a commutative yl-algebra, then there is a homomorphism from %$(A) to 33(5) reduced by the operation 2—>S ®a ?• In particular, for each prime ideal p of A there is a homomorphism from 33(^4) to $&(A»). Thus we have a homomorphism 0 from %5(A) to IlV33(ylp), where p ranges over the prime ideals of A, defined by mapping the class of 8 (which we will also denote 2 without fear of confusion), to {&>}. We would like to examine the kernel of this map in the case where A is an integrally closed noetherian domain. We are particularly interested in knowing when this kernel is trivial. In other words, we would like to answer the following ques- tion: given a central separable A -algebra L such that for every prime ideal p of A the localization Ln is isomorphic to Hoiru^F), F«) for some free finitely generated ^4p-module Fp, need there exist a finitely gener- ated projective A -module P such that L is isomorphic to Hom^(P, P)? According to results known at this time, it may be that the answer to

Central separable algebras which are locally endomorphism rings of free modules