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Let $S$ be a graded ring ($S_n=0$ for $n<0$), $f\in S$ a homogeneous element, and $M, N$ two graded $S$-modules. I'm trying to prove that $$(M\otimes_S N)_{(f)}\simeq M_{(f)}\otimes_{S_{(f)}}N_{(f)},$$ where $M_{(f)}$ is the degree zero component of the localized module $M_f$.

I got the map $M_{(f)}\otimes_{S_{(f)}}N_{(f)} \to (M\otimes_S N)_{(f)}$ but don't know how to define its inverse. I'd appreciate suggestions...

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