Structure theory for a class of grade 3 homogeneous ideals defining type 2 compressed rings

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), s≥3 is some integer. prove that all such are obtained by trimming process introduced Christensen, Veliche, and Weyman (J. Commut. Algebra 11:3 (2019), 325–339). also construct general resolution for which minimal in sufficiently generic cases. Using this resolution, we give bounds on the number of generators μ(I) I depending only s; moreover, show these sharp constructing attaining upper lower s≥3. Finally, Tor-algebra structure R∕I. It shown have Tor algebra class G(r) s≤r≤2s−1. Furthermore, produce r s≤r≤2s−1 Soc(R∕I)=k(−s)⊕k(−2s+1) R∕I has G(r), partially answering question realizability posed Avramov Pure Appl. 216:11 (2012), 2489–2506).