On the representability of ${\Cal Hilb}^nk[x]_{(x)}$

Abstract

Let $k[x]_{(x)}$ be the polynomial ring $k[x]$ localized in the maximal ideal $(x)\subseteq k[x]$. We study the Hilbert functor parameterizing ideals of colength $n$ in this ring {\it having support at the origin}. The main result of this article is that this functor is not representable. We also give a complete description of the functor as a limit of representable functors.