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The Stacks project

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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