Algebras and varieties

TL;DR

This paper introduces new algebraic varieties parametrizing associative algebras, revealing shared properties and deformation behaviors, especially for finite-dimensional and graded algebras, with implications for algebras of global dimension two.

Contribution

It defines new affine algebraic varieties for associative algebras and studies their properties, including deformation and homological invariants, extending the understanding of algebra classifications.

Findings

Algebras in the same variety share homological properties.

Finite dimensional and graded algebras are subvarieties of the new varieties.

Algebras of global dimension two deform to monomial algebras.

Abstract

In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same variety have the same dimension. The case of finite dimensional algebras as well as that of graded algebras arise as subvarieties of the varieties we define. As an application we show that for algebras of global dimension two over the complex numbers, any algebra in the variety continuously deforms to a monomial algebra.