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 (J. Commut. Algebra 11:3 (2019), 325–339). We also construct a general resolution for all such ideals which is minimal in sufficiently generic cases. Using this resolution, we 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 (J. Pure Appl. Algebra 216:11 (2012), 2489–2506).