Initially regular sequences and depths of ideals

TL;DR

This paper introduces initially regular sequences in polynomial rings, linking their length to depth bounds of ideals, and demonstrates how combinatorial and polarization techniques can improve depth estimates for monomial ideals.

Contribution

It defines initially regular sequences, explores their properties, and connects them with regular sequences and polarization to enhance depth bounds for monomial ideals.

Findings

Initially regular sequences provide lower bounds for depth.

Constructed sequences of linear polynomials form initially regular sequences.

Results improve known depth bounds for monomial ideals using polarization.

Abstract

For an arbitrary ideal $I$ in a polynomial ring $R$ we define the notion of initially regular sequences on $R/I$. These sequences share properties with regular sequences. In particular, the length of an initially regular sequence provides a lower bound for the depth of $R/I$. Using combinatorial information from the initial ideal of $I$ we construct sequences of linear polynomials that form initially regular sequences on $R/I$. We identify situations where initially regular sequences are also regular sequences, and we show that our results can be combined with polarization to improve known depth bounds for general monomial ideals.