Lemma 10.75.1. Let $R$ be a ring. Let $M_1, M_2, N$ be $R$-modules. Suppose that $F_\bullet$ is a free resolution of the module $M_1$ and that $G_\bullet$ is a free resolution of the module $M_2$. Let $\varphi : M_1 \to M_2$ be a module map. Let $\alpha : F_\bullet \to G_\bullet$ be a map of complexes inducing $\varphi$ on $M_1 = \mathop{\mathrm{Coker}}(d_{F, 1}) \to M_2 = \mathop{\mathrm{Coker}}(d_{G, 1})$, see Lemma 10.71.4. Then the induced maps

$H_ i(\alpha ) : H_ i(F_\bullet \otimes _ R N) \longrightarrow H_ i(G_\bullet \otimes _ R N)$

are independent of the choice of $\alpha$. If $\varphi$ is an isomorphism, so are all the maps $H_ i(\alpha )$. If $M_1 = M_2$, $F_\bullet = G_\bullet$, and $\varphi$ is the identity, so are all the maps $H_ i(\alpha )$.

Proof. The proof of this lemma is identical to the proof of Lemma 10.71.5. $\square$

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