A new family of algebras whose representation schemes are smooth

TL;DR

This paper establishes a precise criterion for the smoothness of schemes parameterizing n-dimensional representations of finitely generated associative algebras, extending classical results to a broader algebraic context.

Contribution

It provides a necessary and sufficient condition for smoothness of representation schemes, generalizing known finite-dimensional algebra results to finitely generated algebras.

Findings

Representation schemes are smooth at points where Ext^2 vanishes.

The smoothness criterion is both necessary and sufficient.

Points with Ext^2(M,M)=0 are regular.

Abstract

We give a necessary and sufficient smoothness condition for the scheme parameterizing the n-dimensional representations of a finitely generated associative algebra over an algebraically closed field of characteristic zero. In particular, our result implies that the points M of the above scheme, satisfying $Ext_A ^2(M,M)=0$, are regular. This generalizes well-known results on finite-dimensional algebras to finitely generated algebras.