Lemma 10.55.9. Let $R$ be a local Artinian ring. There is a commutative diagram

where the vertical maps are isomorphisms by Lemmas 10.55.7 and 10.55.8.

Lemma 10.55.9. Let $R$ be a local Artinian ring. There is a commutative diagram

\[ \xymatrix{ K_0(R) \ar[rr] \ar[d]_{\text{rank}_ R} & & K'_0(R) \ar[d]^{\text{length}_ R} \\ \mathbf{Z} \ar[rr]^{\text{length}_ R(R)} & & \mathbf{Z} } \]

where the vertical maps are isomorphisms by Lemmas 10.55.7 and 10.55.8.

**Proof.**
Let $P$ be a finite projective $R$-module. We have to show that $\text{length}_ R(P) = \text{rank}_ R(P) \text{length}_ R(R)$. By Lemma 10.55.8 the module $P$ is finite free. So $P \cong R^{\oplus n}$ for some $n \geq 0$. Then $\text{rank}_ R(P) = n$ and $\text{length}_ R(R^{\oplus n}) = n \text{length}_ R(R)$ by additivity of lengths (Lemma 10.52.3). Thus the result holds.
$\square$

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