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 $k{(-s)}^{\ell }\oplus k(-2s+1)$, where $s⩾3$ and $\ell ⩾1$ are integers. The case for $\ell =1$ was previously studied in [8]; a generically minimal resolution was constructed for all such ideals. The paper [7] generalizes this resolution in the guise of (iterated) trimming complexes. In this paper, we show that all ideals of the above form are resolved by an iterated trimming complex. Moreover, we apply this machinery to construct ideals I such that $R/I$ is a ring of Tor algebra class $G(r)$ for some fixed $r⩾2$, and $R/I$ may be chosen to have arbitrarily large type. In particular, this provides a new class of counterexamples to a conjecture of Avramov not already constructed by Christensen, Veliche, and Weyman in [5].

让 $\mathrm{电阻}=克[X,是,z]$是一个标准的分级 3 变量多项式环，其中k表示任何域。我们研究 3 年级的齐次理想$\mathrm{一世}\subseteq \mathrm{电阻}$ 用 socle 定义压缩环 $克{(-秒)}^{\ell }\oplus 克(-2秒+1)$， 在哪里 $秒⩾3$ 和 $\ell ⩾1$是整数。的情况$\ell =1$以前在[8]中研究过；为所有这些理想构建了一个一般最小的分辨率。论文 [7] 以（迭代的）修剪复合体的名义概括了该解决方案。在本文中，我们证明了上述形式的所有理想都可以通过迭代修整复数来解决。此外，我们应用这种机制来构建理想I，使得$\mathrm{电阻}/\mathrm{一世}$ 是 Tor 代数类的环 $G(r)$ 对于一些固定 $r⩾2$， 和 $\mathrm{电阻}/\mathrm{一世}$可以选择具有任意大的类型。特别是，这为 Avramov 猜想提供了一类新的反例，这些猜想尚未由 Christensen、Veliche 和 Weyman 在 [5] 中构建。