  We construct a canonical noncommutative spectral triple for every oriented
closed Riemannian manifold, which represents the fundamental class in the
twisted K-homology of the manifold. This so-called "projective spectral triple"
is Morita equivalent to the well-known commutative spin spectral triple
provided that the manifold is spin-c. We give an explicit local formula for the
twisted Chern character for K-theories twisted with torsion classes, and with
this formula we show that the twisted Chern character of the projective
spectral triple is identical to the Poincar\'e dual of the A-hat genus of the
manifold.
