Existence of a Nonsmoothable Local Gorenstein Algebra with Smoothable Q(0)

Abstract

We prove that there exists a local Artinian Gorenstein algebra \(A\) which is not smoothable, although the first symmetric quotient \(Q_A(0)\) in the symmetric decomposition of the associated graded algebra is smoothable. The proof uses divided-power inverse systems and gives such algebras of length \(31\) and embedding dimension \(14\) over every algebraically closed field.