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arXiv · 1302.3992

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

David Jordan·Hendrik Orem
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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.

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David Jordan, Hendrik Orem. 2013-02-16. An algebro-geometric construction of lower central series of associative algebras. https://doi.org/10.1093/imrn%2Frnu125

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