The Stacks project

Remark 27.8.5. The map from $M_0$ to the global sections of $\widetilde M$ is generally far from being an isomorphism. A trivial example is to take $S = k[x, y, z]$ with $1 = \deg (x) = \deg (y) = \deg (z)$ (or any number of variables) and to take $M = S/(x^{100}, y^{100}, z^{100})$. It is easy to see that $\widetilde M = 0$, but $M_0 = k$.