February 18, 2010

Noncommutative Zariski topology?

Posted in Uncategorized at 11:58 am by noncommutativeag

There are a lot of discussions on mathoverflow on Zariski topology in commutative algebraic geometry. What I want to do is to see whether we can formulate noncommutative version Zariski topology which should play roles to cover noncommutative schemes.

First,we took a look at the usual Zariski topology. If R is a commutative ring(k-algebra), then the open set of Zariski topology on Spec(R) is U(\alpha)=(p\in SpecR|\alpha\nsubseteq p). Actually, if we make more precise description: we take \alpha as radical ideal. Then following the fact that:

Qcoh_{(U(\alpha),O_{U(\alpha)})}=Qcoh_{(U(rad\alpha),O_{U(rad\alpha)})}.

Radical ideal determines the (open)schemes uniquely.

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Why spectrum of category?

Posted in algebraic geometry at 1:20 am by noncommutativeag

In commutative algebraic geometry, spectrum of a commutative ring R plays basically important roles. Once we have Spec(R), naturally we have Zariski topology on this spectrum.  We can define sheaf of rings O_{Spec(R)} on the open sets(in particular, basis D_{f},f\in R). Then we can recover R by so called global section functor \Gamma as R=\Gamma(SpecR,O_{Spec(R)}). Geometrically, we glued bunches of pieces together to reconstruct affine schemes. What we did can be described by the following diagrams:

(Spec(R),O_{Spec(R)})\rightarrow (sheaf of rings)\rightarrow \Gamma(SpecR,O_{Spec(R)})=R, then if we take prime spectrum, we got the underlying space of original affine schemes.

Then,if we switch to consider category of modules(in particular, quasi-coherent modules). Then the picture should become:

(Spec(R),O_{Spec(R)}) \rightarrow Qcoh_{(Spec(R),O_{Spec(R)})} \rightarrow R-mod

The reason to consider this picture is that the first reconstruction process works only for affine case. Another less important reason is that sheaves of modules is usually easy to deal with than sheaves of rings.

For the second picture, one need to define a “spectrum” for R-mod to get the underlying space of schemes. So this is why we can not avoid the notion Spec(R-mod), or more naturally, Spec(C_{X}),where C_{X} is category of quasi coherent sheaves on scheme (X,O_{X}). In fact, not only we can reconstruct the underlying space of the schemes but only the structure sheaf. Before going to the reconstruction theorem, I will explain the definition of Spec(C_{X}).

February 9, 2010

Plan

Posted in algebraic geometry at 8:49 am by noncommutativeag

I plan to write something on three topics:

1. Several basic spectrum for noncommutative ring and how do them come into the story of representation theory of reductive Lie algebra.

I will do some explicit computations. The motivation for me is for the research topic course which is now given by A.Rosenberg.I want to understand all of them in detail.

2. I will start with Zariski topology in commutative algebraic geometry and try to do direct imitation for noncommutative ring.

Unfortunately, one will find direct imitation failed. Therefore, another appropriate topology for noncommutative algebraic geometry will be formulated.After finish it, I will go to the purely functorial point of view to algebraic geometry. I might restrict myself to commutative algebraic geometry.

3. D-module theory and Lie algebra and their quantum analogue

I will talk about how noncommutative algebraic geometry naturally comes into the life of D-modules. I will push myself to write localization theory for category of D-modules in almost full details.