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The lower central series invariants M_k of an associative algebra A are thetwo-sided ideals generated by k-fold iterated commutators; the M_k provide afiltration of A. We study the relationship between the geometry of X = SpecA_ab and the associated graded components N_k of this filtration. We show thatthe N_k form coherent sheaves on a certain nilpotent thickening of X, and thatZariski localization on X coincides with noncommutative localization of A.Under certain freeness assumptions on A, we give an alternative construction ofN_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 vectorbundles on the category of smooth schemes.

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