Hochschild dimension of a separably generated field

Abstract

Let K be an Nk-generated field extension of the field F with transcendence degree n. Set bidim(K)=the projective dimension of K as a K OF K-module. Then K9ocally separably generated implies bidim(K)<k+n+l, and K separably generated implies bidim(K)=k+n+ 1. Let F be a commutative ring, K an F-algebra. The Hochschild or bidimension of K over F, written bidim(K), is the projective dimension of K as a module over the ring R=K ?F K0P. In [3], Hochschild showed that if K is finite over a field F, then bidim(K)=O if and only if K is separable over F, i.e., for all field extensions L of F, K ?F L is semisimple. Noether [8] had done this for commutative K earlier. Rosenberg and Zelinsky [12] showed bidim(K)=O implies [K:F]<ox. They also obtained results in the case K is a field relating bidim(K) and the transcendence degree of K over F. MacRae [6] took their work and characterized all fields K of bidimension 1 over the field F by showing they are countably generated and so, by Rosenberg and Zelinsky, either separable algebraic extensions of dimension No or finite separable extensions of rational function fields in one variable. The purpose of this note is to compute the bidimension of any separably generated extension field K of the field F. If K is not locally separably generated, Rosenberg and Zelinsky showed bidim(K) is infinite, but only an upper bound will be obtained in this paper on the locally separably generated case. Let &-1 denote any finite cardinal and X0O any cardinal ?g. #B will denote the cardinality of the set B, and [K: F] the dimension of the vector space K over the field F. We show the THEOREM. Let K be afield extension of thefield F with separating transcendence basis B. Let [K: F(B)] = Hm. Then bidim(K)=#B+m + I. Received by the editors March 7, 1973. AMS (MOS) subject classifications (1970). Primary 18H20, 13D05; Secondary 16A62.