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.