Algebra bundles, projective flatness and rationally-deformed tori

We show that isomorphism classes $[\mathcal{A}]$ of flat $q\times q$ matrix bundles $\mathcal{A}$ (or projectively flat rank-$q$ complex vector bundles $\mathcal{E}$) on a pro-torus $\mathbb{T}$ are in bijective correspondence with the Čech cohomology group H^2(\mathbb{T},μ_q:=\text{q^{th}roots of unity}) (respectively $H^2(\mathbb{T},\mathbb{Z})$) via the image of $[A]\in H^1(\mathbb{T},PGL(q,{\mathcal{C}}_{\mathbb{T}}))$ through $H^1(\mathbb{T},PGL(q,{\mathcal{C}}_{\mathbb{T}}))\stackrel{}{\to }{H}^2(\mathbb{T},\mu (q,{\mathcal{C}}_{\mathbb{T}}))$ (respectively the first Chern class $c_1(\mathcal{E})$). 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 $B\mathrm{\Gamma }$ of compact Lie groups $\mathrm{\Gamma }$ (as opposed to ${\mathbb{T}}^n\cong B{\mathbb{Z}}^n$), on which flat non-trivial vector bundles abound. The discussion both recovers the Disney-Elliott-Kumjian-Raeburn classification of rational non-commutative tori ${\mathbb{T}}_{\theta }^n$ with a different, bundle-theoretic proof, and sheds some light on the connection between topological invariants associated to ${\mathbb{T}}_{\theta }^2$, $\theta \in \mathbb{Q}$ by Rieffel and respectively Høegh-Krohn-Skjelbred.