Artinian Gorenstein algebras that are free extensions over ${\sf k}[t]/(t^n)$, and Macaulay duality

Abstract

T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras C over a base A with fibre 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]/(t n ) with fiber B. We give examples exploring

Cohomological Blow Ups of Graded Artinian Gorenstein Algebras Along Surjective Maps

“…The following elementary observation will be useful and we record it as a lemma. We offer our own proof here for completeness, but it can also be deduced from [27,Lemma 1.8] and [43,Lemma 2.1]. Recall that for graded Artinian algebras A, B, and C, we say that C is a free extension of A with fiber B if there are maps ι : A → C making C into a free A-module and π : C → B with kernel ker(π) = m A • C where m A is the maximal ideal of A.…”