Algebra bundles, projective flatness and rationally-deformed tori

We show that isomorphism classes A of flat q q matrix bundles A (or projectively flat rank-q complex vector bundles E) on a pro-torus T are in bijective correspondence with the Čech cohomology group H² (T, μq: =q^th roots of unity) (respectively H² (T, Z) ) via the image of A H¹ (T, PGL (q, Cₓ) ) through H¹ (T, PGL (q, Cₓ) ) ² (T, μ (q, Cₓ) ) (respectively the first Chern class c₁ (E) ). This is in the spirit of Auslander-Szczarba's result identifying real flat bundles on the torus with their first two Stiefel-Whitney classes, and contrasts with classifying spaces BΓ of compact Lie groups Γ (as opposed to Tⁿ BZⁿ), on which flat non-trivial vector bundles abound. The discussion both recovers the Disney-Elliott-Kumjian-Raeburn classification of rational non-commutative tori Tⁿ_θ with a different, bundle-theoretic proof, and sheds some light on the connection between topological invariants associated to T²_θ, θ by Rieffel and respectively Høegh-Krohn-Skjelbred.