Cohomological dimension and global dimension of algebras

Abstract

Let R be a regular local ring and A an algebra over R which is an R-progenerator. Defining the cohomological dimension of A as R-dim A=l hdAe(A), one obtains the Hochschild cohomological dimension of A as an R-algebra. We show the following under the additional hypothesis that R-dim A is finite: (1) R-dim A =n iff A/N is R-separable and l hdA(A/N)=n+gl R; (2) A= R-dim A +gl R; (3) A is R-separable iff Ae=gl R. The purpose of this note is to extend a result of Eilenberg on the Hochschild cohomological dimension of associative algebras. The results obtained will relate the global dimension of an algebra A which is a progenerator as an R-module, the global dimension of the ground ring R, and the cohomological dimension of the algebra. Throughout we will assume that all rings have one. We shall say that an R-algebra A is an R-progenerator in case A is finitely generated, projective, and faithful as a module over the commutative ring R. N will denote the Jacobson radical of the algebra A. We will always mean by hd and gl dim the left homological dimension and the left global dimension. Recall that if a ring is commutative or noetherian, the left and right global dimensions coincide. We define R-dimA=hdA (A), where Ae=A ORA* and A* is the algebra antiisomorphic to A. Since A*=A, its left and right global dimensions also coincide. Hence for the main theorems, global dimension is well defined. We require two well-known results, the first of Eilenberg, Rosenberg, and Zelinsky [4, Proposition 2] and the second due to Kaplansky [5, p. 172]: RESULT 1. If A and B are R-algebras and A is R-flat, then (a) B?A <R-dim A +gl B. If further, A is also R-projective and contains R as an R-direct summand, then (b) B<gl BOA. Received by the editors December 21, 1970. AMS 1970 subject classifications. Primary 18H15, 16A46, 13H05; Secondary 13D05, 16A62.