Noncommutative Localizations of Lie-Complete Rings

Abstract

In this paper we investigate the topological localizations of Lie-complete rings. It has been proved that a topological localization of a Lie-complete ring is commutative modulo its topological nilradical. Based on the topological localizations we define a noncommutative affine scheme X = Spf(A) for a Lie-complete ring A. The main result of the paper asserts that the topological localization A_(f) of A at f ∈ A is embedded into the ring 𝒪_A(X_f) of all sections of the structure sheaf 𝒪_A on the principal open set X_f as a dense subring with respect to the weak I₁-adic topology, where I₁ is the two-sided ideal generated by all commutators in A. The equality A_(f) = 𝒪_A(X_f) can only be achieved in the case of an NC-complete ring A.

Key Words:

• Lie-complete ring
• Lie-nilpotent ring
• Noncommutative affine scheme
• Noncommutative localization

1991 Mathematics Subject Classification:

• Primary: 16L30
• Secondary: 14A22
• 46H70

ACKNOWLEDGMENT

I wish to thank Yu. V. Turovskii, A. Yu. Pirkovskii, and X. Ma for useful discussions certain details of the present work. We also thank the referees for the proposal to include a technical appendix that made easier to read the main text of the paper.