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

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.