A special chain theorem for the embedding dimension

Abstract

This paper establishes an analogue of the special chain theorem for the embedding dimension of polynomial rings, with direct application on the (embedding) codimension. In particular, we recover a classic result on the transfer of regularity to polynomial rings (initially proved via a combination of Serre's result on finite global dimension and Hilbert theorem on syzygies). A second application characterizes regularity in general settings of localizations of polynomial rings, including Nagata rings and Serre's conjecture rings.