The lower central series invariants Mk of an associative algebra A are the two-sided ideals generated by k-fold iterated commutators; the Mk provide a filtration of A. We study the relationship between the geometry of X=Spec Aab and the associated graded components Nk of this filtration. We show that the Nk 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 Nk purely in terms of the geometry of X (and in particular, independent of A). Applying a construction of Kapranov, we exhibit the Nk as natural vector bundles on the category of smooth schemes.