Example 15.60.3. Consider a commutative diagram

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

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

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 15.60.1.

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