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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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this is answered here, although only with a reference to an exercise… – 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

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