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Noncommutative schemes

Published online by Cambridge University Press:  04 December 2007

ALEXANDER L. ROSENBERG
ALEXANDER L. ROSENBERG
Affiliation:
Le Bois-Marie, 35 Route de Chartres, 91440, Bures-sur-Yvette, France. e-mail: rosenber@ihes.fr
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Abstract

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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.

Keywords

Quasi-schemeZariski covercontinuous morphismflat localizationskew projective spacequantum flag varietycontinuous monadresolutions of endofunctors.
Type
Research Article
Information
Compositio Mathematica , Volume 112 , Issue 1 , May 1998 , pp. 93 - 125
Copyright
© 1998 Kluwer Academic Publishers