Quasicoherent Sheaves on Complex Noncommutative Two-tori

We introduce the notion of a quasicoherent sheaf on a complex noncommutative two-torus T as an ind-object in the category of holomorphic vector bundles on T . We define the rank of a quasicoherent sheaf that can take arbitrary nonnegative real values. We study the category Qcoh(ηT ) obtained by taking the quotient of the category of quasicoherent sheaves by the subcategory of objects of rank zero (called torsion sheaves). In particular, we show that holomorphic vector bundles of the same rank become isomorphic in Qcoh(ηT ). We also prove that the subcategory of objects of finite rank in Qcoh(ηT ) is equivalent to the category of finitely presented modules over some semihereditary algebra.