Krull dimension of tensor products of pullbacks

TL;DR

This paper investigates the Krull dimension of tensor products of algebras over a field, providing explicit dimension formulas for certain pullback constructions and characterizing these pullbacks as generalized AF-domains.

Contribution

It answers an open problem by computing the dimension of tensor products involving pullback algebras and establishes that such pullbacks are generalized AF-domains.

Findings

Computed dim$(A\otimes_k B)$ for specific pullback algebras.

Resolved an open problem in the dimension theory of tensor products.

Showed that certain pullback algebras are generalized AF-domains.

Abstract

This paper is concerned with the study of the dimension theory of tensor products of algebras over a field $k$. We answer an open problem set in [6] and compute dim$(A\otimes_kB)$ when $A$ is a $k$-algebra arising from a specific pullback construction involving AF-domains and $B$ is an arbitrary $k$-algebra. On the other hand, we deal with the question (Q) set in [5] and show, in particular, that such a pullback $A$ is in fact a generalized AF-domain.