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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