NONCOMMUTATIVE SCHEMES

The main purpose of this work is to introduce noncommutative relative schemes and establish some of basic properties of schemes and scheme m orphisms. In particular, we prove an analogue of the canonical bijecti on: Hom(schemes/k)((X, O), Spec(A)) similar or equal to Hom(k-alg)(A, Gamma(X, O)). We define a noncommutative version of the Cech cohomolog y of an affine cover and show that the Cech cohomology can be used to compute higher direct images. This fact is applied here to compute coh omology of invertible sheaves on skew projective spaces and in [LR3] t o study D-modules on quantum flag varieties.