ARTINIAN GORENSTEIN ALGEBRAS THAT ARE FREE EXTENSIONS OVER k[t]/(tn), AND MACAULAY DUALITY

T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras C over a base A with fiber B. The free extensions are deformations of the usual tensor product; when C is also Gorenstein, so are A and B, and it is natural to ask for the relation among the Macaulay dual generators for the algebras. Writing a dual generator F for C as a homogeneous "polynomial" in T and the dual variables for B, and given the dual generator for B, we give sufficient conditions on F that ensure that C is a free extension of A = k[t]/(tn) with fiber B. We give examples exploring the sharpness of the statements. We also consider a special set of coinvariant algebras C which are free extensions of A, but which do not satisfy the sufficient conditions of our main result.