Some classes of sequences of Linear Type

TL;DR

The paper investigates which homogeneous ideals $I$ in a graded ring are of linear type via $d$-sequences, and how $y$-regularity constrains their generators. It analyzes relations among $d$-sequences, $M$-sequences, and weak $d$-sequences via $\mathcal{R}(I)$ and $\operatorname{Sym}(I)$, and constructs explicit counterexamples where linear type holds but $d$-sequences do not. It demonstrates a class of linear-type ideals not generated by $d$-sequences (e.g., $I=P_{n-3}(C_n)$ for odd $n$) with increasing $y$-regularity, and provides gcd-based criteria for squarefree $M$-sequences to be $d$-sequences along with results on powers preserving Gröbner-type properties. Finally, it shows that in algebras with straightening laws, weak $d$-sequences become $d$-sequences under a total order, and establishes unconditioned $d$-sequence status for maximal Pfaffians of generic odd-order skew-symmetric matrices.

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 Gröbner 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.

Theorems & Definitions (7)

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