Some classes of sequences of Linear Type

TL;DR

This paper investigates classes of ideals in graded rings, focusing on those of linear type, and explores the properties of $d$-sequences and their relation to Gr"obner linear type, revealing new subclasses and algebraic conditions.

Contribution

It identifies a smaller subset of $d$-sequence ideals within ideals of linear type and characterizes classes where weak $d$-sequences are actual $d$-sequences, advancing understanding of their algebraic structure.

Findings

$d$-sequence ideals form a smaller subset of linear type ideals.

Certain $d$-sequences have powers generating Gr"obner linear type ideals.

A class of algebras where weak $d$-sequences are genuine $d$-sequences.

Abstract

Given a graded ring $A$ and a homogeneous ideal $I$, the ideal is said to be of linear type if the Rees algebra of $I$ is isomorphic to the symmetric algebra of $I$. In general, $y$-regularity of Rees algebra of $I$ is $0 \Rightarrow$ $I$ is generated by a $d$-sequence $\Rightarrow I$ is of linear type. We show that $d$-sequence ideals represent a significantly smaller subset of ideals of linear type in terms of $y$-regularity. Moreover, we identify a class of $d$-sequences whose arbitrary powers generate ideals of Gr\"obner linear type. Notably, while $d$-sequences are inherently weak $d$-sequences, we highlight a specific class of algebras where weak $d$-sequences are indeed $d$-sequences.