  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
$T$. Extending the results of math.QA/0211262 and math.QA/0308136 we prove that
the derived category of quasicoherent sheaves on $T$ is equivalent to the
derived category of usual quasicoherent sheaves on the corresponding elliptic
curve. We define the rank of a quasicoherent sheaf that can take arbitrary
nonnegative real values. We study the category $\Qcoh(\eta_T)$ obtained by
taking the quotient of the category of quasicoherent sheaves by the subcategory
of objects of rank zero (called torsion sheaves). We show that projective
objects of finite rank in $\Qcoh(\eta_T)$ are classified up to an isomorphism
by their rank. We also prove that the subcategory of objects of finite rank in
$\Qcoh(\eta_T)$ is equivalent to the category of finitely presented modules
over a semihereditary algebra.
