# 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.