An algebro-geometric construction of lower central series of associative algebras

TL;DR

This paper explores the geometric structure of the lower central series of associative algebras, linking algebraic invariants to coherent sheaves and vector bundles on associated schemes, providing a new geometric perspective.

Contribution

It introduces a geometric construction of the lower central series invariants of associative algebras using algebraic geometry and sheaf theory, independent of the algebra's specific structure.

Findings

N_k form coherent sheaves on nilpotent thickenings of Spec A_ab

Zariski localization matches noncommutative localization of A

N_k can be realized as vector bundles on smooth schemes

Abstract

The lower central series invariants M_k of an associative algebra A are the two-sided ideals generated by k-fold iterated commutators; the M_k provide a filtration of A. We study the relationship between the geometry of X = Spec A_ab and the associated graded components N_k of this filtration. We show that the N_k form coherent sheaves on a certain nilpotent thickening of X, and that Zariski localization on X coincides with noncommutative localization of A. Under certain freeness assumptions on A, we give an alternative construction of N_k purely in terms of the geometry of X (and in particular, independent of A). Applying a construction of Kapranov, we exhibit the N_k as natural vector bundles on the category of smooth schemes.