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I'm just curious but why is it that $ \operatorname{Proj}\left(\oplus_{i=0}^\infty t^i \left< x\right>^i\right) $ isomorphic to $ \operatorname{Proj} \left(\oplus_{i=0}^\infty t^i \left< x^2\right>^i\right), $ where $\left< x\right>$ is an ideal in $\mathbb{C}[x]$?

$ $ Note: if the above is true, then isn't it reasonable to also have $ \operatorname{Proj}\left(\oplus_{i=0}^\infty t^i \left< x\right>^i\right) \cong \operatorname{Proj}\left(\oplus_{i=0}^\infty t^i \left< x^k\right>^i\right) $ where $k\geq 1$?

Edit: In fact, is it true that $\operatorname{Proj}\left(\oplus_{i=0}^\infty t^i I^i\right) \cong \operatorname{Proj}\left(\oplus_{i=0}^\infty t^i I^{ki}\right)$ for any $I \subseteq R$ and $k\geq 1$?

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    Thanks countinghaus! I will definitely take a look at that link.2012-07-01

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