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

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 $s\ge 3$ 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 $s\ge 3$. Finally, we study the Tor-algebra structure of $R/I$. It is shown that these rings have Tor algebra class $G(r)$ for $s\le r\le 2s-1$. Furthermore, we produce ideals $I$ for all $s\ge 3$ and all $r$ with $s\le r\le 2s-1$ such that $\text{Soc}(R/I)=k(-s)\oplus k(-2s+1)$ and $R/I$ has Tor-algebra class $G(r)$, partially answering a question of realizability posed by Avramov.

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