Associated graded rings of one-dimensional analytically irreducible rings II

TL;DR

This paper addresses a flaw in previous work on associated graded rings of one-dimensional analytically irreducible rings, introduces the BF condition to fix it, and explores implications for semigroup rings and complete intersection properties.

Contribution

It introduces the BF condition to correct earlier results and establishes equivalences and applications for semigroup rings and their associated graded rings.

Findings

The BF condition ensures the correctness of associated graded ring properties.

Semigroup rings in embedding dimension at most three often have associated graded rings that are complete intersections.

Connections to Cortadella-Zarzuela's results extend the applicability of the findings.

Abstract

Lance Bryant noticed in his thesis that there was a flaw in our paper "Associated graded rings of one-dimensional analytically irreducible rings", J. Algebra 304 (2006), 349-358. It can be fixed by adding a condition, called the BF condition. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadella-Zarzuela. Finally we use our result to conclude when a semigroup ring in embedding dimension at most three has an associated graded which is a complete intersection.