G-theory

Remarks

For $n = 0$, clause 2 gives back clause 1. The same construction applied to coherent sheaves on a Noetherian scheme $X$ gives $G_n(X)$. The inclusion of finitely generated projective modules into $\mathcal{M}(R)$ induces the Cartan map $K_n(R) \to G_n(R)$, an isomorphism when $R$ is regular by the resolution theorem.

Example: $G_0(\Z) \cong \Z$ through the rank. The rank is additive on short exact sequences, so it defines a homomorphism $G_0(\Z) \to \Z$. It is an isomorphism, because every finitely generated abelian group is $\Z^r \oplus T$ with $T$ finite, and $[\Z/m\Z] = [\Z] - [\Z] = 0$ from the sequence $0 \to \Z \to \Z \to \Z/m\Z \to 0$, with first map multiplication by $m$.

Non-example: for a non-Noetherian ring the finitely generated modules need not form an abelian category. Over $R = k[x_1, x_2, \ldots]$, polynomials in infinitely many variables over a field $k$, the map $R \to k$ sending each $x_i$ to $0$ has kernel $(x_1, x_2, \ldots)$, which is not finitely generated.