On the smoothness of the scheme of linear representations and the Nori-Hilbert scheme of an associative algebra

TL;DR

This paper proves that the scheme of n-dimensional representations of a hereditary and coherent algebra is smooth, leading to the smoothness of the associated Nori-Hilbert scheme, advancing understanding of algebraic representation schemes.

Contribution

It establishes the smoothness of the scheme of representations for hereditary and coherent algebras and extends this to the Nori-Hilbert scheme, a novel result in algebraic geometry.

Findings

Scheme of n-dimensional representations is smooth for hereditary and coherent algebras.

Nori-Hilbert scheme associated to such algebras is also smooth.

Provides conditions under which these schemes are smooth.

Abstract

Let $k$ be an algebraically closed field and let $A$ be a finitely generated $k-$algebra. We show that the scheme of n-dimensional representations of $A$ is smooth when $A$ is hereditary and coherent. We deduce from this the smoothness of the Nori-Hilbert scheme associated to $A.$