Structure Theory for a Class of Grade 3 Homogeneous Ideals Defining Type 2 Compressed Rings

TL;DR

This paper characterizes a class of grade 3 homogeneous ideals defining compressed rings with specific socle structures, providing explicit resolutions, bounds on generators, and analyzing their Tor-algebra classes, thus advancing understanding of their algebraic properties.

Contribution

It introduces a construction method for these ideals, derives minimal resolutions, establishes bounds on generators, and explores their Tor-algebra classes, partially answering realizability questions.

Findings

All such ideals are obtained by a specific trimming process.

Bounds on the minimal number of generators are sharp and explicitly constructed.

Rings have Tor algebra class G(r) for s ≤ r ≤ 2s-1.

Abstract

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s) \oplus k(-2s+1)$, where $s \geq3$ is some integer. We prove that all such ideals are obtained by a trimming process introduced by Christensen, Veliche, and Weyman. We also construct a general resolution for all such ideals which is minimal in sufficiently generic cases. Using this resolution, we can give bounds on the minimal number of generators $\mu(I)$ of $I$ depending only on $s$; moreover, we show these bounds are sharp by constructing ideals attaining the upper and lower bounds for all $s\geq 3$. Finally, we study the Tor-algebra structure of $R/I$. It is shown that these rings have Tor algebra class $G(r)$ for $s \leq r \leq 2s-1$. Furthermore, we produce ideals $I$ for all $s \geq 3$ and all $r$ with $s \leq r \leq 2s-1$ such that $\textrm{Soc} (R/I ) = k(-s) \oplus k(-2s+1)$ and $R/I$ has Tor-algebra class $G(r)$, partially answering a question of realizability posed by Avramov.