This question is motivated by some issue raised by David Speyer in this question.

Let $R$ be a ring. Let $K_0(R)$ and $G_0(R)$ be the Grothendieck groups of f.g. projective modules and f.g. modules over $R$, respectively (you just kill all relations generated by short exact sequences). There is a natural map, called the *Cartan homomorphism* (see Serre's "Linear reps of finite groups", Chapter 15) $$c: K_0(R) \to G_0(R)$$ given by forgetting a module is projective.

In general, $c$ needs not be injective nor surjective. For non-surjectivity, take $R$ to be a local ring, then $K_0(R)=\mathbb Z$ but $G_0(R)$ can be huge (in particular, if $R$ is normal, $\mathbb Z\oplus \text{Cl}(R)$ is a quotient of $G_0(R)$). Examples of non-injectivity can be found by taking $R$ to be some group rings, as the Cartan matrix is not always invertible, see for example Section 4 of this paper by Martin Lorenz . But I don't know any commutative example of non-injectivity.

*Is $c$ always injective if $R$ is commutative? How about if $R$ is commutative and Noetherian?*

(If this is true, one can prove the original question quoted above with the assumption $G_0(R)=\mathbb Z$)

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