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Hochschild dimension and the prime radical of algebras

Author: Joseph A. Wehlen
Journal: Proc. Amer. Math. Soc. 83 (1981), 443-447
MSC: Primary 16A62; Secondary 13D05
MathSciNet review: 627665
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Abstract: Let $ R$ be a regular local ring and $ A$ an algebra over $ R$ which is finitely generated and free as an $ R$-module. Defining the Hochschild dimension of $ A$ as $ R - \dim A = {\text{left}}\;{\text{h}}{{\text{d}}_{{A^e}}}(A)$, we show the following: if A modulo its prime radical $ L(A)$ is $ R$-free and $ R - \dim A/L(A) = 0$, then $ R - \dim A = {\text{left}}\;{\text{h}}{{\text{d}}_A}(A/L(A))$. Using localization and sheaf theoretic techniques, the result is generalized to regular rings and to absolutely flat (von Neumann regular) rings. The relationship between the $ A$-homological dimension of the algebra $ A$ modulo its prime radical and the algebra modulo its Jacobson radical is explored in view of this result.

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Keywords: Hochschild dimension, regular ring, separable algebra, absolutely flat ring, prime radical
Article copyright: © Copyright 1981 American Mathematical Society

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