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Let $S_{\bullet}$ be a graded ring, generated in degree $1$ with $S_0 = k$ (a field). One can associate to $S_{\bullet}$ the twisted graded ring $$ \Gamma_{\bullet} = \left( \ \Gamma(\mathrm{Proj} S_{\bullet}, \mathcal{O}(n)) \ \right)_{n \geq 0}, $$ where $\mathcal{O}(n)$ is the standard twisting sheaf on $\mathrm{Proj} S_{\bullet}$. It is known that $S_{\bullet}$ and $\Gamma_{\bullet}$ must be isomorphic from some degree on as graded $S_{\bullet}$-modules.

I am looking for a simple example where $S_{\bullet}$ and $\Gamma_{\bullet}$ are not isomorphic in every degree.

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  • $\begingroup$ Take $S=k[x^4,x^3y,xy^3,y^4]$ (with the obvious grading). Then check $\Gamma= S[x^2y^2]$. $\endgroup$ – Mohan Feb 19 at 13:54

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