Resolution and Tor algebra structures of grade 3 ideals defining compressed rings (Q2049370)

Let \(R=k[x,y,z]\) be a standard graded polynomial ring over a field \(k\). In this article, the author studies grade \(3\) homogeneous ideals \(I\subseteq R\) defining compressed rings with socle \(\mathrm{Soc}(R/I) = k(-s)^l \oplus k(-2s + 1)\), where \(s\geqslant 3\) and \(l \geqslant 1\) are integers. The case `\(l = 1\)' was previously studied by the author in [``Structure theory for a class of grade 3 homogeneous ideals defining type 2 compressed rings'', J. Commut. Algebra, in press]; a generically minimal resolution was constructed for all such ideals. In another paper [\textit{K. VandeBogert}, Commun. Algebra 49, No. 3, 1017--1036 (2021; Zbl 1470.13026)], he generalizes this resolution in the guise of (iterated) trimming complexes. In the present paper, he shows that all ideals of the above form are resolved by an iterated trimming complex. Moreover, he applies this machinery to construct ideals \(I\) such that \(R/I\) is a ring of Tor algebra class \(G(r)\) for some fixed \(r \geqslant 2\), 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 \textit{L. Winther Christensen} et al. [J. Commut. Algebra 11, No. 3, 325--339 (2019; Zbl 1441.13059)].