Lemma 35.4.24. If $M \in \text{Mod}_ R$ is flat, then $C(M)$ is an injective $R$-module.

Proof. Let $0 \to N \to P \to Q \to 0$ be an exact sequence in $\text{Mod}_ R$. Since $M$ is flat,

$0 \to N \otimes _ R M \to P \otimes _ R M \to Q \otimes _ R M \to 0$

is exact. By Lemma 35.4.10,

$0 \to C(Q \otimes _ R M) \to C(P \otimes _ R M) \to C(N \otimes _ R M) \to 0$

is exact. By (35.4.11.1), this last sequence can be rewritten as

$0 \to \mathop{\mathrm{Hom}}\nolimits _ R(Q, C(M)) \to \mathop{\mathrm{Hom}}\nolimits _ R(P, C(M)) \to \mathop{\mathrm{Hom}}\nolimits _ R(N, C(M)) \to 0.$

Hence $C(M)$ is an injective object of $\text{Mod}_ R$. $\square$

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