The Stacks project

Lemma 10.10.1. Exactness and $\mathop{\mathrm{Hom}}\nolimits _ R$. Let $R$ be a ring. Let $M_1$, $M_2$, $M_3$ be $R$-modules. Let $M_1 \to M_2$ and $M_2 \to M_3$ be $R$-module maps.

  1. $M_1 \to M_2 \to M_3 \to 0$ 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. $0 \to M_1 \to M_2 \to M_3$ 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. $\square$

Comments (2)

Comment #10135 by on

It would be nice to weaken the hypotheses by replacing "complex" with "sequence" in each part (that is, not to assume a priori that the composition of every two consecutive morphisms is 0).

There are also:

  • 3 comment(s) on Section 10.10: Internal Hom

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View Lemma 10.10.1 as pdf