= Degree-one localization of a graded module
{title2=$M_f\cong S_f\otimes_{(S_f)_0}(M_f)_0$}
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>.
Back to article page
