Hochschild dimension and the prime radical of algebras

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.