Dimension theory of tensor products of AF-rings

TL;DR

This paper explores the dimension theory of tensor products of AF-rings, extending classical results and providing formulas for Krull and valuative dimensions, with applications to various algebraic constructions.

Contribution

It generalizes Wadsworth's results on AF-domains to AF-rings and derives formulas for their Krull and valuative dimensions in complex tensor product scenarios.

Findings

Extended Wadsworth's results to AF-rings

Provided formulas for Krull and valuative dimensions

Analyzed tensor products over zero-dimensional rings

Abstract

AF-rings are algebras over a field k which satisfy the Altitude Formula over k. This paper surveys a few works in the literature on the Krull and valuative dimensions of tensor products of AF-rings. The first section extends Wadsworth's classical results on the Krull dimension of AF-domains to the larger class of AF-rings. It also provides formulas for computing the valuative dimension with effect on the transfer of the (locally) Jaffard property. The second section studies tensor products of AF-rings over a zero-dimensional ring. Most results on algebras over a field are extended to these general constructions. The third section establishes formulas for the Krull and valuative dimensions of tensor products of pullbacks issued from AF-domains. Throughout, examples are provided to illustrate the scope and limits of the results.