The Brauer Group of a Commutative Ring

Abstract

Introduction. This paper contains the foundations of a general theory of separable algebras over arbitrary commutative rings. Of the various equiv-alent conditions for separability in the classical theory of algebras over a field, there is one which is most suitable for generalization; we say that an algebra A over a commutative ring £ is separable if A is a projective module over its enveloping algebra Ae = A®fiA°. The basic properties of separable algebras are developed in the first three sections. The results obtained show that a considerable portion of the classical theory is preserved in our generalization. For example, it is proved that separability is maintained under tensor products as well as under the forma-tion of factor rings. Furthermore, an £-algebra A is separable over £ if, and only if, A is separable over its center C and C is separable over £. This fact shows that the study of separability can be split into two parts: commutative algebras and central algebras. The purely commutative situation has been studied to some extent by Auslander and Buchsbaum in [l]. The present investigation is largely concerned with central algebras. In the classical case, an algebra which is separable over a field K, and has K for its center, is simple. One cann'ot expect this if the center is not a field; however, if A is central separable over £, then the two-sided ideals of A are all generated by ideals of £. In the fourth section we consider a different aspect of the subject, one which is more analogous to ramification theory. If A is an algebra over a ring £, the homological different (A/£) is an ideal in the center C of A which essentially describes the circumstances under which A®rRv is separable over £(>, when p is a prime ideal of £. (Suitable finiteness conditions must be im-posed in the statement of these theorems.) The general question of ramifica-tion in noncommutative algebras is only touched on in the present paper; various arithmetic applications will be treated in another publication. In the classical theory of central simple algebras, the full matrix algebras have a special significance. The proper analogue of the full matrix algebra in the present context is the endomorphism ring Homje(£, £) of a finitely gener-ated projective £-module £. It is easy to show for such a module £ that rlomR(E, E) is separable over £, and central if £ is faithful. By analogy with the classical theory, we introduce an equivalence relation between central separable algebras over a ring £, under which the equivalence classes form Received by the editors March 16, 1960.