Lemma 15.15.7. Let $R$ be a ring. Suppose that $\varphi : R^ n \to R^ n$ be an injective map of finite free modules of the same rank. Then $\mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Coker}}(\varphi ), R) = 0$.

Proof. Let $\varphi ^ t : R^ n \to R^ n$ be the transpose of $\varphi$. The lemma claims that $\varphi ^ t$ is injective. With notation as in Lemma 15.15.6 we see that the rank of $\varphi ^ t$ is $n$ and that $I(\varphi ) = I(\varphi ^ t)$. Thus we conclude by the equivalence of (1) and (2) of the lemma. $\square$

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