Expanding the socle of a codimension 3 complete intersection (Q2374282)

The list of known structure theorems for the minimal free resolution of a quotient of a commutative Noetherian local ring \(R\) is fairly short. The Hilbert-Burch Theorem describes the resolution of \(R/I\) when \(I\) is a grade two perfect ideal of \(R\) and the Buchsbaum-Eisenbud Theorem describes the resolution of \(R/I\) when \(I\) is a grade three Gorenstein ideal. The list of such theorems can be extended if one is satisfied with less information about the \(R\)-module structure of \(R/I\); in particular, if one is satisfied with a classification of the algebra structure of \(\text{Tor}_\bullet^R(R/I,k)\), where \(k\) is the residue field of \(R\). For example, there appears to be little hope for a classification of the resolutions of all perfect ideals of grade three; but the corresponding \(\text{Tor}\)-algebras were classified in the 1980's by Weyman and Avramov-Kustin-Miller. Amazingly, there is a very small collection of families of such \(\text{Tor}\)-algebras: \(\mathbf B\), \(\mathbf{C}\), \(\mathbf{G}(r)\), \(\mathbf{H}(p,q)\), and \(\mathbf T\). The names conjured up particular examples in the minds of the classifiers: \(\mathbf{C}\) stands for complete intersection, \(\mathbf T\) for truncated exterior algebra, \(\mathbf{G}(r)\) includes the family of Gorenstein ideals, and \(\mathbf{H}(p,q)\) includes the family of hypersurfaces. At the time of the classification, only one example was known for an ideal whose \(\text{Tor}\)-algebra had the form \(\mathbf B\); this example was introduced in Anne Brown's thesis. The present paper gives many more ideals with \(\text{Tor}\)-algebra of form \(\mathbf B\).