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⊆R defining compressed rings with socle k(−s)⊕k(−2s+1), where s≥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 μ(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≥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≤r≤2s−1. Furthermore, we produce ideals I for all s≥3 and all r with s≤r≤2s−1 such that Soc(R∕I)=k(−s)⊕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).