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Tag 08XC

Lemma 34.4.24. 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 34.4.10, $$0 \to C(Q \otimes_R M) \to C(P \otimes_R M) \to C(N \otimes_R M) \to 0$$ is exact. By (34.4.11.1), this last sequence can be rewritten as $$0 \to \mathop{\rm Hom}\nolimits_R(Q, C(M)) \to \mathop{\rm Hom}\nolimits_R(P, C(M)) \to \mathop{\rm Hom}\nolimits_R(N, C(M)) \to 0.$$ Hence $C(M)$ is an injective object of $\text{Mod}_R$. $\square$

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 1323–1326 (see updates for more information).

\begin{lemma}
\label{lemma-flat-to-injective}
If $M \in \text{Mod}_R$ is flat, then $C(M)$ is an injective $R$-module.
\end{lemma}

\begin{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 \ref{lemma-C-is-faithful},
$$0 \to C(Q \otimes_R M) \to C(P \otimes_R M) \to C(N \otimes_R M) \to 0$$
is exact. By (\ref{equation-adjunction}), this last sequence can be rewritten
as
$$0 \to \Hom_R(Q, C(M)) \to \Hom_R(P, C(M)) \to \Hom_R(N, C(M)) \to 0.$$
Hence $C(M)$ is an injective object of $\text{Mod}_R$.
\end{proof}

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