Finitely-Generated Projective Modules over the Theta-deformed 4-sphere

TL;DR

This paper studies the structure of finitely-generated projective modules over theta-deformed 3- and 4-spheres, revealing their classification, cancellation properties, and the existence of nontrivial modules when theta is irrational.

Contribution

It classifies all finitely-generated projective modules over C(S^{4}_{theta}) and shows that all over C(S^{3}_{theta}) are free, highlighting differences based on rationality of theta.

Findings

All modules over C(S^{3}_{theta}) are free.

C(S^{4}_{theta}) has the cancellation property.

Existence of nontrivial rank-1 modules when theta is irrational.

Abstract

We investigate the "theta-deformed spheres" C(S^{3}_{theta}) and C(S^{4}_{theta}), where theta is any real number. We show that all finitely-generated projective modules over C(S^{3}_{theta}) are free, and that C(S^{4}_{theta}) has the cancellation property. We classify and construct all finitely-generated projective modules over C(S^{4}_{\theta}) up to isomorphism. An interesting feature is that if theta is irrational then there are nontrivial "rank-1" modules over C(S^{4}_{\theta}). In that case, every finitely-generated projective module over C(S^{4}_{\theta}) is a sum of a rank-1 module and a free module. If theta is rational, the situation mirrors that for the commutative case theta=0.