Invertible $R$-module, $R$ local ring, $L \cong R$

Question

In the last 5 lines of Lemma 17.22.4, the author seems to claim:

If $L$ is an invertible $R$-module, and $R$ is a local ring, then $L \cong R$.

The algebra section doesn't address this case. How is this true?

Answer 1

The last statement of Lord Shark the Unknown can be found also in stacksproject, 10.84.4. Let me elaborate on the first part. I recall this from a lecture notes of Vivek Shende.

As $M$ is invertible then exists $N$, $M\otimes_R N \cong R$, hence some $m_i \in M, n_i \in N$, $\sum m_i \otimes n_i =1$. There is a surjective map

$$ R^k \otimes N =N^k \rightarrow M \otimes_R N \cong R, \quad (s_i) \mapsto \sum m_i \otimes s_i$$ As $R$ is free, $R^k \otimes N = R \oplus Q$, tensoring by $M$, yields $R^{k} = M \oplus Q ' $ so $M$ is projective.