Lemma 10.10.1. Exactness and $\mathop{\mathrm{Hom}}\nolimits _ R$. Let $R$ be a ring.

1. Let $M_1 \to M_2 \to M_3 \to 0$ be a complex of $R$-modules. Then $M_1 \to M_2 \to M_3 \to 0$ is exact if and only if $0 \to \mathop{\mathrm{Hom}}\nolimits _ R(M_3, N) \to \mathop{\mathrm{Hom}}\nolimits _ R(M_2, N) \to \mathop{\mathrm{Hom}}\nolimits _ R(M_1, N)$ is exact for all $R$-modules $N$.

2. Let $0 \to M_1 \to M_2 \to M_3$ be a complex of $R$-modules. Then $0 \to M_1 \to M_2 \to M_3$ is exact if and only if $0 \to \mathop{\mathrm{Hom}}\nolimits _ R(N, M_1) \to \mathop{\mathrm{Hom}}\nolimits _ R(N, M_2) \to \mathop{\mathrm{Hom}}\nolimits _ R(N, M_3)$ is exact for all $R$-modules $N$.

Proof. Omitted. $\square$

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