Non-trivial stably free modules over crossed products

TL;DR

This paper investigates the existence of non-free projective modules over certain noncommutative algebras, extending known results to crossed products of noetherian domains with universal enveloping algebras, including Weyl algebras.

Contribution

It introduces a sufficient condition for the existence of non-trivial stably free modules over crossed products, broadening the understanding of noncommutative Serre's problem.

Findings

Identifies conditions for non-free projective modules over crossed products.

Applies lifting methods from Ore extensions to crossed products.

Includes examples from RIT algebras in physics.

Abstract

We consider the class of crossed products of noetherian domains with universal enveloping algebras of Lie algebras. For algebras from this class we give a sufficient condition for the existence of projective non-free modules. This class includes Weyl algebras and universal envelopings of Lie algebras, for which this question, known as noncommutative Serre's problem, was extensively studied before. It turns out that the method of lifting of non-trivial stably free modules from simple Ore extensions can be applied to crossed products after an appropriate choice of filtration. The motivating examples of crossed products are provided by the class of RIT algebras, originating in non-equilibrium physics.