2019/12/14 by Keller VandeBogert, VandeBogert, Keller · 1 citation
Computer Science · Mathematics · #Commutative Algebra and Its Applications #Polynomial and algebraic computation #Rings, Modules, and Algebras #math.AC
paper · pdf · doi:10.48550/arxiv.1912.06949
24 pages
arxiv created 2020/02/19 · arxiv updated 2020/02/21
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. We also construct a general resolution for all such ideals which is minimal in sufficiently generic cases. Using this resolution, we can 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 \textrmSoc (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.