Lemma 10.12.16. Let $M, N$ be $R$-modules, then there is a canonical $S^{-1}R$-module isomorphism $f : S^{-1}M \otimes _{S^{-1}R}S^{-1}N \to S^{-1}(M \otimes _ R N)$, given by

$f((m/s)\otimes (n/t)) = (m \otimes n)/st$

Proof. We may use Lemma 10.12.7 and Lemma 10.12.15 repeatedly to see that these two $S^{-1}R$-modules are isomorphic, noting that $S^{-1}R$ is an $(R, S^{-1}R)$-bimodule:

\begin{align*} S^{-1}(M \otimes _ R N) & \cong S^{-1}R \otimes _ R (M \otimes _ R N)\\ & \cong S^{-1}M \otimes _ R N\\ & \cong (S^{-1}M \otimes _{S^{-1}R}S^{-1}R)\otimes _ R N\\ & \cong S^{-1}M \otimes _{S^{-1}R}(S^{-1}R \otimes _ R N)\\ & \cong S^{-1}M \otimes _{S^{-1}R}S^{-1}N \end{align*}

This isomorphism is easily seen to be the one stated in the lemma. $\square$

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