Let $R$ be a graded ring, finitely generated by $R_1$ as an $R_0$-algebra. Let $X=\mathop{\rm Proj}R$ and let $R':=\Gamma_*(\mathcal{O}_X)$ be the associated graded module of global sections of twists of the structure sheaf. There is a natural morphism $\alpha\colon R\to R'$. How do $X':=\mathop{\rm Proj}R'$ and $X$ relate? When are they isomorphic?

Maybe we add the assumption that $R_0$ is noetherian...

The only example I have comes from Hartshorne ex. II.5.14, for when $X$ is a normal, connected, projective variety, in which case $X$ and $X'$ are isomorphic... A related question, when is $\alpha$ an isomorphism in large degree?


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