Projective Modules over Higher-dimensional Non-commutative Tori

The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n-torus rn with its algebra, C(rn), of continuous complex-valued functions under pointwise multiplication. But C(rn) is the universal C*-algebra generated by n commuting unitary operators. By definition, [15, 16, 50], a non-commutative n-torus is the universal C*-algebra generated by n unitary operators which, while they need not commute, have as multiplicative commutators various fixed scalar multiples of the identity operator. As Connes has shown [8, 10], these algebras have a natural differentiable structure, defined by a natural ergodic action of rn as a group of automorphisms. The non-commutative tori behave in many ways like ordinary tori. For instance, it is an almost immediate consequence of the work of Pimsner and Voiculescu [37] that the K-groups of a non-commutative torus are the same as those of an ordinary torus of the same dimension. (In particular, non-commutative tori are KK-equivalent to ordinary tori by Corollary 7.5 of [52].) Furthermore, the structure constants of non-commutative tori can be continuously deformed into those for ordinary tori. (This is exploited in [17].) In this paper we study the non-stable behavior of (finitely generated) projective modules over non-commutative tori. These are the appropriate generalization of complex vector bundles over ordinary tori, according to a theorem of Swan [54, 45]. It is well known that for higher-dimensional ordinary tori the non-stable behavior of vector bundles is quite complicated. Our main theme is that, in contrast, as soon as there is any irrationality present, then the non-stable behavior of projective modules over non-commutative tori is quite regular. To make this more precise, let us introduce some notation. A non-commutative torus is specified by giving the multiplicative commutators for its generators. For our purposes this is most conveniently done by giving a skew bicharacter on Zn, or better (as first exploited by Elliott [17]) by giving a real skew bilinear form, say B, on Zn. To each x E Zn we can associate a product, say ux' of the unitary generators, in