Artinian Gorenstein algebras that are free extensions over \sf k[t]/(tⁿ), and Macaulay duality

2018/07/08 by Iarrobino, Anthony, Marques, Pedro Macias, McDaniel, Chris
#13A50 #13D40 #13E10 (primary) #13H10 #14D06 (secondary) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.02881

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=\sf k[t]/(tⁿ) with fiber B. We give examples that explore 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.

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