Lemma 15.118.7. Let $R$ be a ring. There is a map

$\det : K_0(R) \longrightarrow \mathop{\mathrm{Pic}}\nolimits (R)$

which maps $[M]$ to the class of the invertible module $\wedge ^ n(M)$ if $M$ is a finite locally free module of rank $n$.

Proof. This follows immediately from the constructions above and in particular Lemma 15.118.2 to see that the relations are mapped to $0$. $\square$

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