Lemma 15.96.6. Let $A \to B$ be a ring map. Let $f \in A$ be a nonzerodivisor. Let $M^\bullet $ be a bounded complex of finite free $A$-modules. Assume $f$ maps to a nonzerodivisor $g$ in $B$ and $I_ i(M^\bullet , f)$ is a principal ideal for all $i \in \mathbf{Z}$. Then there is a canonical isomorphism $\eta _ fM^\bullet \otimes _ A B = \eta _ g(M^\bullet \otimes _ A B)$.
Proof. Set $N^ i = M^ i \otimes _ A B$. Observe that $f^ iM^ i \otimes _ A B = g^ iN^ i$ as submodules of $(N^ i)_ g$. The maps
\[ (\eta _ fM)^ i \otimes _ A B \to g^ iN^ i \otimes g^{i + 1}N^{i + 1} \quad \text{and}\quad (\eta _ gN)^ i \to g^ iN^ i \otimes g^{i + 1}N^{i + 1} \]
are inclusions of direct summands by Lemma 15.96.5. Since their images agree after localizing at $g$ we conclude. $\square$
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