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Maybe I have not really understood what does it mean localization of a ring or a module at a prime ideal. I know the definition but i cannot really use it in the practice.

If I have an ideal, in my case $$I=\left(x^2+y^2-yz,xyz-z,y(y-z)(yz-1)\right),$$ what does it mean in the practice $I\mathbb{C}[x,y,z]_{(x,y)}$?

Thank you.

aleio1
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  • Possibly related: http://math.stackexchange.com/questions/72235 – Watson Jun 04 '16 at 21:54
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    At some point in math, you have to drop the idea of finding a convenient visualization for everything. That having been said, the canonical example to keep in mind is the localization of $\mathbb{Z}$ at a prime or $0$. – anomaly Jun 04 '16 at 22:09
  • @anomaly probably you are right. But I think that theese questions about localization need to be understood in the view of algebraic geometry (i think for instance to Zarisky Topology) and unlikely i miss this extended view of the subject. In this sense i ask for help. – aleio1 Jun 04 '16 at 22:21

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