# Why do we have $\operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x\right>^i)\cong \operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x^2\right>^i)$?

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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this is answered here, although only with a reference to an exercise mathoverflow.net/questions/46202/… –  user29743 Jul 1 '12 at 18:50
Thanks countinghaus! I will definitely take a look at that link. –  math-visitor Jul 1 '12 at 18:58