T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras over a base with fiber . The free extensions are deformations of the usual tensor product; when is also Gorenstein, so are and , and it is natural to ask for the relation among the Macaulay dual generators for the algebras. Writing a dual generator for as a homogeneous “polynomial” in and the dual variables for , and given the dual generator for , we give sufficient conditions on that ensure that is a free extension of with fiber . We give examples exploring the sharpness of the statements. We also consider a special set of coinvariant algebras which are free extensions of , but which do not satisfy the sufficient conditions of our main result.
Anthony Iarrobino. Pedro Macias Marques. Chris McDaniel. "Artinian Gorenstein algebras that are free extensions over , and Macaulay duality." J. Commut. Algebra 14 (4) 553 - 569, Winter 2022. https://doi.org/10.1216/jca.2022.14.553