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:=qᵗʰroots of unity) (respectively H²(T,Z)) via the image of [A]∈ H¹(T,PGL(q,CT)) through H¹(T,PGL(q,CT))H²(T,μ(q,CT)) (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.