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Example 15.62.3. Consider a commutative diagram

\[ \xymatrix{ B \ar[r] & B' = B \otimes _ A A' \\ A \ar[r] \ar[u] & A' \ar[u] } \]

and $B$-modules $M, N$. Set $M' = M \otimes _ A A' = M \otimes _ B B'$ and $N' = N \otimes _ A A' = N \otimes _ B B'$. Assume that $A \to B$ is flat and that $M$ and $N$ are $A$-flat. Then there is a spectral sequence

\[ \text{Tor}^ A_ i(\text{Tor}_ j^ B(M, N), A') \Rightarrow \text{Tor}^{B'}_{i + j}(M', N') \]

The reason is as follows. Choose free resolution $F_\bullet \to M$ as a $B$-module. As $B$ and $M$ are $A$-flat we see that $F_\bullet \otimes _ A A'$ is a free $B'$-resolution of $M'$. Hence we see that the groups $\text{Tor}^{B'}_ n(M', N')$ are computed by the complex

\[ (F_\bullet \otimes _ A A') \otimes _{B'} N' = (F_\bullet \otimes _ B N) \otimes _ A A' = (F_\bullet \otimes _ B N) \otimes ^{\mathbf{L}}_ A A' \]

the last equality because $F_\bullet \otimes _ B N$ is a complex of flat $A$-modules as $N$ is flat over $A$. Hence we obtain the spectral sequence by applying the spectral sequence of Example 15.62.1.

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