June 17, 2010
Pseudo Geometry III
Copied from A.Rosenberg’s book
Geometry of noncommutative “spaces” and schemes
(a): A pretopology on is subcanonical iff for any presheaf of sets on (in other words,”descent” pretopologies on are precisely subcanonical pretopologies). In this case, ,where is the sheaf on associated with the presheaf and is natural full embedding.
(b). If a pretoplogy is of effective descent, then the above embedding becomes a categorical equivalence.
This theorem says that, roughly speaking, the category of quasi coherent presheaves knows itself which pretopologies to choose. It also indicates where one should look for a correct noncommutative version of the category (of sheaves of sets on the fpqc site of commutative affine schemes): this should be the category of sheaves of sets on the presite ,where is a pretopology of effective descent. From the minimalistic point of view, the best choice would be the finest pretopology of effective descent. But there is a more important consideration. The main role of a pretopology is that it is used for gluing new “spaces”. The pretopology that seems to be the most relevant for Grassmannians(in particular, for noncommutative projective space) and a number of other smooth noncommutative spaces constructed in [KR5] is the smooth topology introduced in [KR2].
The theorem is quite useful on a pragmatical level. Namely, if is a sheaf of sets on for an appropriate pretopology of effective descent and is a presheaf of sets on such that its associated sheaf is isomorphic to , and is an exact sequence of presheaves with and representable, then the category (hence the category ) is constructively described (unique up to equivalence) via pair of k-algebra representing . This consideration is used to describe the categories of quasi coherent sheaves on noncommutative “spaces”
3.8. Noncommutative stacks.
There is one more important observation in connection with this theorem: categories which appear in noncommutative algebraic geometry are categories of quasi coherent sheaves on noncommutative stacks.