June 17, 2010

Pseudo Geometry III

Posted in Uncategorized at 9:45 pm by noncommutativeag

Copied from A.Rosenberg’s book

Geometry of noncommutative “spaces” and schemes

3.7. Theorem

(a): A pretopology \tau on Aff_k is subcanonical iff Qcoh(X,\tau)=Qcoh(X) for any presheaf of sets on Aff_k(in other words,”descent” pretopologies on Aff_k are precisely subcanonical pretopologies). In this case, Qcoh(X)=Qcoh(X,\tau)\rightarrow Qcoh(X^{\tau})=Qcoh(X^{\tau},\tau),where X^{\tau} is the sheaf on (Aff_k,\tau) associated with the presheaf X and \rightarrow is natural full embedding.

(b). If a pretoplogy \tau is of effective descent, then the above embedding becomes a categorical equivalence.

This theorem says that, roughly speaking, the category Qcoh(X) 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 Esp (of sheaves of sets on the fpqc site of commutative affine schemes): this should be the category NEsp_{\tau} of sheaves of sets on the presite (Aff_k,\tau),where \tau 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  \mathfrak{X} is a sheaf of sets on (Aff_k, \tau) for an appropriate pretopology of effective descent and X is a presheaf of sets on Aff_k such that its associated sheaf is isomorphic to \mathfrak{X}, and \mathfrak{R}\Rightarrow \mathfrak{U}\rightarrow X is an exact sequence of presheaves with \mathfrak{R} and \mathfrak{U} representable, then the category Qcoh(X)(hence the category Qcoh(\mathfrak{X})) is constructively described (unique up to equivalence) via pair of k-algebra A\Rightarrow R representing \mathfrak{R}\Rightarrow \mathfrak{U}. 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.

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