February 18, 2010

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}).

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