Degree-one localization of a graded module ($M_f\cong S_f{\otimes }_{(S_f{)}_0}(M_f{)}_0$)

If $f$ is homogeneous of degree one, multiplication by powers of the invertible element $f$ identifies every homogeneous component of $M_f$ with $(M_f{)}_0$. A homogeneous element $m$ of degree $a$ is written $f^a(f^{-a}m)$. Thus the displayed map is an isomorphism of graded modules. In particular, the degree-zero part of a tensor product of localized graded modules is the tensor product of their degree-zero parts over $(S_f{)}_0$. This proves tensor compatibility of graded sheafification on degree-one-generated Proj.