Inertial subalgebras of algebras over commutative rings

Abstract

Introduction. The classical Wedderburn Principal Theorem states that if a finite-dimensional algebra A over a field has the property that A modulo its Jacobson radical N is separable, then A contains a subalgebra S with S + N A and S n N = (0). In [2], M. Auslander and 0. Goldman developed the theory of separable algebras over an arbitrary commutative ring. This concept of separability prompts one to ask to what degree the Wedderburn Principal Theorem can be generalized to algebras over a commutative ring. This paper is concerned with the following questions which bear directly on this problem. (1) Let A be a finitely generated R-algebra, where R is a commutative ring. Under what circumstances does A contain an R-separable subalgebra S stucl that S + N = A? We call such a subalgebra an inertial stubalgebr a of A. (2) Which commutative rings R have the property that every finitely generated R-algebra A such that A/rad(A) is separable over R contains an inertial subalgebra'? We call such a commutative ring an inertial coefficient ring. We say that the uniqueness statement holds for an inertial coefficient ring R if, whenever S and S' are inertial subalgebras of a finitely generated R-algebra A, there exists an element n in the radical of A such that (1 n) S(l n)= S'. In this setting the Wedderburn Principal Theorem along with Malcev's uniqueness assertion state that every field is an inertial coefficient ring for which the uniqueness statement holds. ?1 contains preliminaries. In ?2, we first investigate some properties of inertial subalgebras. Then necessary and sufficient conditions are obtained for the existence of an inertial subalgebra in a finitely generated, faithful, commutative R-algebra A having a finite group G of automorphisms such that AG = {a E A I (a) a, Va E G} = R, where R is a commutative ring containinig no idempotents but 0 and 1. ?3 is devoted to the second of the above questions, beginning with a discussion of some formal properties of inertial coefficient rings. Azumaya has proved in [3] that every Hensel ring is an inertial coefficient ring for which the uniqueness statement holds. We give the following partial