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Quasicoherent sheaves in commutative and noncommutative geometry. (English. Russian original) Zbl 1077.14021

Izv. Math. 67, No. 3, 535-554 (2003); translation from Izv. Ross. Akad. Nauk Ser. Mat. 67, No. 3, 119-138 (2003).
This paper gives an unified definition of quasi-coherent sheaves on commutative and non-commutative schemes. The idea is to consider both objects as presheaves of sets on the category of affine commutative or non-commutative schemes. In the commutative case, the category of affine schemes is equivalent to the opposite of the category of commutative algebras; in the non-commutative case it is defined as the opposite of the category of non-commutative algebras. Then to each presheaf of this kind the author associates a category of quasi-coherent sheaves on it. The definition is very natural and people familiar with Grothendieck topologies and sites will easily see that it generalises the usual one. Moreover, the definition is shown to be well behaved with respect to sheafification in the commutative case, in the sense that if one considers the topology whose covering sieves are the sieves generated by effective descent families, then the categories of quasi-coherent sheaves on a presheaf and on its associated sheaf with respect to that topology are equivalent. This topology, called effective descent topology by the author, is coarser than the flat topology. Finally, using also presheaves of groupoids the author describes an embedding of commutative geometry in non-commutative geometry which has the property that the coherent sheaves on a (commutative) presheaf are the same that the coherent sheaves on its image presheave in the non-commutative setting. The paper is very well written and will become a foundational reference for non-commutative algebraic geometry.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14A22 Noncommutative algebraic geometry
18F10 Grothendieck topologies and Grothendieck topoi
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