Lemma 15.96.3. 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 $B$. Then $I_ i(M^\bullet , f)B = I_ i(M^\bullet \otimes _ A B, g)$.

Proof. The minors of $(f, d^ i) : M^ i \to M^ i \oplus M^{i + 1}$ map to the corresponding minors of $(g, d^ i) : M^ i \otimes _ A B \to M^ i \otimes _ A B \oplus M^{i + 1} \otimes _ A B$. $\square$

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