The Nori-Hilbert scheme is not smooth for 2-Calabi Yau algebras

Abstract

Let $k$ be an algebraically closed field of characteristic zero and let $A$ be a finitely generated $k-$algebra. The Nori - Hilbert scheme of $A$, parameterizes left ideals of codimension $n$ in $A,$ and it is well known to be smooth when $A$ is formally smooth. In this paper we will study the Nori - Hilbert scheme for $2-$Calabi Yau algebras. The main examples of these are surface group algebras and preprojective algebras. For the former we show that the Nori-Hilbert scheme is smooth for $n=1$ only, while for the latter we show that the smooth components that contain simple representations are precisely those that only contain simple representation. Under certain conditions we can generalize this last statement to arbitrary $2-$Calabi Yau algebras.

30 pages, research paper. Accepted for publication in Journal of Noncommutative Geometry

References in corpus (3)

• Stable categories of Cohen-Macaulay modules and cluster categories
• Superpotential algebras and manifolds
• Noncommutative Tangent Cones and Calabi Yau Algebras