Some classes of sequences of Linear Type
Neeraj Kumar, Chitra Venugopal

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 and a homogeneous ideal , the ideal is said to be of linear type if the Rees algebra of is isomorphic to the symmetric algebra of . In general, -regularity of Rees algebra of is is generated by a -sequence is of linear type. We show that -sequence ideals represent a significantly smaller subset of ideals of linear type in terms of -regularity. Moreover, we identify a class of -sequences whose arbitrary powers generate ideals of Gr\"obner linear type. Notably, while -sequences are inherently weak -sequences, we highlight a specific class of algebras where weak -sequences are indeed -sequences.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Rings, Modules, and Algebras · Algebraic structures and combinatorial models
