  After introducing a noncommutative counterpart of commutative algebraic
geometry based on monoidal categories of quasi-coherent sheaves we show that
various constructions in noncommutative geometry (e.g. Morita equivalences,
Hopf-Galois extensions) can be given geometric meaning extending their
geometric interpretations in the commutative case. On the other hand, we show
that some constructions in commutative geometry (e.g. faithfully flat descent
theory, principal fibrations, equivariant and infinitesimal geometry) can be
interpreted as noncommutative geometric constructions applied to commutative
objects. For such generalized geometry we define global invariants constructing
cyclic objects from which we derive Hochschild, cyclic and periodic cyclic
homology (with coefficients) in the standard way.
