= Tensor compatibility of graded sheafification on degree-one-generated Proj
{title2=$\widetilde M\otimes_{\mathcal O_X}\widetilde N\cong\widetilde{M\otimes_SN}$}
Under the <degree-one generation condition for Proj>, cover $X=\operatorname{Proj}S$ by degree-one charts. The <degree-one localization of a graded module> identifies the two local modules in the displayed formula, using <localization commutes with tensor products>. The maps are natural and agree on overlaps, so they glue to an isomorphism of <sheaves of modules>. The analogous assertion can fail for general positively graded rings.
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