Let O q ( G ) be the algebra of quantized functions on an algebraic group G and O q ( B ) its quotient algebra corresponding to a Borel subgroup B of G. We define the category of sheaves on the “quantum flag variety of G” to be the O q ( B ) -equivariant O q ( G ) -modules and prove that this is a proj-category. We construct a category of equivaria...

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P 2 . The generic noncommutative plane corre...

This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild co...