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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