Bimodules, the Brauer group, Morita equivalence, and cohomolgy

Abstract

The purpose of this note is to focus attention on an equivalence relation which we call ‘congruence of k-algebras’ and define in terms of the categories of bimodules. (Here k is a commutative unital ring.) Many familiar Morita invariant properties are trivially seen to be congruence invariant. Examples include the center, the lattice of two-sided ideals, and the Hochschild and cyclic cohomologies. In Section 3 we prove that Morita equivalence implies congruence, thereby obtaining new, uniform, and more efficient proofs of the Morita invariance of these properties. (Actually, the invariance of cohomology seems to have been previously noticed for only one type of Morita equivalence: that between an algebra A and the r X r matrix algebra hlr(A).) We also give new cochain maps inducing the cohomology isomorphism. In Section 2 we use congruence to answer Zelinsky’s concluding question in [ll] and to give an economical description of the Brauer group as consisting of the Morita equivalence classes of congruence-trivial algebras (under tensor product). This definition provides a categorical interpretation of the Brauer group and is available, without introducing separability, immediately after proving the Eilenberg-Watts theorem characterizing functors between module categories. (The Morita theorems, which are invoked, at least tacitly, in most treatments of the Brauer group, follow easily