Local dimension theory of tensor products of algebras over a ring

Abstract

Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring $R$. Actually, we translate the theory initiated by A. Grothendieck and R. Sharp and subsequently developed by A. Wadsworth on Krull dimension of tensor products of algebras over a field $k$ into the general setting of algebras over an arbitrary ring $R$. For this sake, we introduce and study the notion of a fibred AF-ring over a ring $R$. This concept extends naturally the notion of AF-ring over a field introduced by A. Wadsworth in \cite{W} to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fibre rings of tensor products of algebras over a ring. Also, given a triplet of rings $(R,A,B)$ consisting of two $R$-algebras $A$ and $B$ such that $A{\otimes }_{R}B\ne \{0\}$, we introduce the inherent notion to $(R,A,B)$ of a $B$-fibred AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product $A{\otimes }_{R}B$. As an application, we provide a formula for the Krull dimension of $A{\otimes }_{R}B$ when $A$ and $B$ are $R$-algebras with $A$ is zero-dimensional as well as for the Krull dimension of $A{\otimes }_{\mathbb{Z}}B$ when $A$ is a fibred AF-ring over the ring of integers $\mathbb{Z}$ with nonzero characteristic and $B$ is an arbitrary ring. This enables us to answer a question of Jorge Matinez on evaluating the Krull dimension of $A{\otimes }_{\mathbb{Z}}B$ when $A$ is a Boolean ring. Actually, we prove that if $A$ and $B$ are rings such that $A{\otimes }_{\mathbb{Z}}B$ is not trivial and $A$ is a Boolean ring, then dim$(A\otimes_{\mathbb{Z}}B)=\mbox {dim}\Big (\displaystyle {\frac B{2B}}\Big )$.