($\tilde M$ and $\Gamma_*$)
For \(S\) a graded ring and \(M\) a graded \(S\)-module, \(\tilde M\) is the sheaf on \(\Proj S\) with \(\tilde M(D_+(f)) = (M_f)_0\), the degree-zero part of the localization. Conversely, for \(\mcf \in \mods{\OO_X}\) with \(X = \Proj S\), \[ \Gamma_*(\mcf) \da \bigoplus_{n \in \ZZ} \Gamma(X, \mcf(n)) , \] a graded \(S\)-module, where \(\mcf(n) \da \mcf \tensor_{\OO_X} \OO_X(n)\).
For \(S\) generated in degree \(1\) over a Noetherian ring \(S_0\), every quasicoherent \(\mcf\) on \(\Proj S\) satisfies \(\widetilde{\Gamma_*(\mcf)} \cong \mcf\), and \(\OO_X(n) = \widetilde{S(n)}\).
Under the hypotheses of the theorem, the functor \(M\mapsto\tilde M\) is essentially surjective onto quasicoherent sheaves, with \(\mcf\cong\widetilde{\Gamma_*(\mcf)}\), and it is not an equivalence. A graded homomorphism \(M\to M'\) that is an isomorphism in all sufficiently large degrees induces an isomorphism \(\tilde M\cong\tilde M'\), and a graded module with \(M_n=0\) for all \(n\gg0\) has \(\tilde M=0\); for example, \(\widetilde{S/S_+}=0\) while \(S/S_+\ne0\). The category \(\QCoh(\Proj S)\) is the quotient of the category of graded \(S\)-modules by the modules in which every element is annihilated by a power of \(S_+\).