Example 15.62.2. Let $R \to S$ be a ring map. Let $M$ be an $R$-module and let $N$ be an $S$-module. Then there is a spectral sequence

\[ \text{Tor}^ S_ n(\text{Tor}^ R_ m(M, S), N) \Rightarrow \text{Tor}^ R_{n + m}(M, N). \]

To construct it choose a $R$-free resolution $P_\bullet $ of $M$. Then we have

\[ M \otimes _ R^{\mathbf{L}} N = P^\bullet \otimes _ R N = (P^\bullet \otimes _ R S) \otimes _ S N \]

and then apply the first spectral sequence of Example 15.62.1.

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