The main purpose of this work is to introduce noncommutative relative schemes and establish some of basic properties of schemes and scheme morphisms. In particular, we prove an analogue of the canonical bijection: Hom$_schemes/k$((X, O), Spec(A))[bsime ] Hom$_k-alg$ (A,Γ (X, O)). We define a noncommutative version of the Čech cohomology of an affine cover and show that the Čech cohomology can be used to compute higher direct images. This fact is applied here to compute cohomology of invertible sheaves on skew projective spaces and in [LR3] to study D-modules on quantum flag varieties.
