Separable Jordan algebras over commutative rings. I

Abstract

This paper presents a theory of separable Jordan algebras over commutative rings. We define a Jordan algebra over a commutative ring with & to be separable if its unital universal multiplication algebra is a separable associative algebra. In Section 1 we develop the basic properties of separable Jordan algebras over commutative rings. In Section 2 we prove that a central separable R-algebra J is an R-progenerator and that there is a one-to-one correspondence between the ideals of J and the ideals of R. The rest of the paper centers around the decomposition theorem of Section 6, that a separable Jordan algebra is a direct sum of homogenous components corresponding to the isomorphism classes of finite-dimensional simple Jordan algebras over an algebraically closed field. In Section 3, we obtain analogous decompositions for separable associative algebras and separable associative algebras with involution. In Sections 7 and 8 we apply the decomposition theorems to study the structure of central separable Jordan algebras and their universal envelopes. In particular, we relate the decompositions of separable Jordan algebras and separable associative algebras with involution. More precisely, the decomposition theorem states that a separable Jordan R-algebra J can be written J = Ji @ ... @ Js for distinct ordered pairs (pi , qi),..., (ps , qs) such that, if m is a maximal ideal of R and F is the algebraic closure of R/m, then ( Ji/m Ji) aRlln F is a direct sum of simple F-algebras of degree pi and dimension qi . We note that the isomorphism class of a finitedimensional simple algebra over an algebraically closed field is determined by its degree and dimension. The key fact needed to prove the decomposition theorem is that, if J is separable with center Z(J), its special universal envelope SzcJ)( J) is finitely spanned Z( J)-projective along with J. This implies that Z(J) is the direct sum of ideals Ci such that Ci J and S,,(Ci J) have constant rank over C, . On the other hand, * * Portions of the results presented here are contained in the author’s doctoral dissertation, written at Yale University under the direction of Professor N. Jacobson. The author would like to express his gratitude to Professor Jacobson for his guidance and encouragement.