A special chain theorem for the embedding dimension

TL;DR

This paper extends the special chain theorem to the embedding dimension of polynomial rings, providing new insights into regularity transfer and localizations, with applications to classical and modern algebraic structures.

Contribution

It introduces an analogue of the special chain theorem for embedding dimension, enhancing understanding of regularity transfer in polynomial ring extensions.

Findings

Established a new chain theorem for embedding dimension

Characterized regularity in various localizations of polynomial rings

Connected results to classical theorems on regularity and codimension

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.