# difference between graded ring and its twisted global sections

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.

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