Dévissage

Remarks

The point is that $\sum_i [A_{i-1}/A_i] \in K_0(\mathcal{B})$ does not depend on the filtration. Any two such filtrations have refinements with isomorphic successive quotients, up to order, by the Schreier refinement theorem. Refining a filtration does not change the sum, because the quotients of a refinement are subquotients of quotients in $\mathcal{B}$. Additivity on short exact sequences comes from splicing a filtration of a subobject with a filtration of the quotient. Quillen extended the theorem to all higher K-groups.

Example: let $R$ be a commutative Noetherian ring, $\mathcal{A}$ the category of $R$-modules of finite length, and $\mathcal{B}$ the semisimple ones. A composition series is a filtration with simple quotients $R/\mathfrak{m}$, $\mathfrak{m}$ maximal, and $K_0(\mathcal{B})$ is free abelian on these simple modules. So $K_0(\mathcal{A}) \cong \bigoplus_{\mathfrak{m}} \Z$, and the class of a module records how often each $R/\mathfrak{m}$ occurs in a composition series. For a local Artinian ring $R$ with residue field $k$, this gives $G_0(R) \cong \Z$, generated by $[k]$, and $[M]$ is the length of $M$.

The filtration hypothesis is necessary. For $\mathcal{A}$ the finitely generated abelian groups and $\mathcal{B}$ the finite ones, $\Z$ has no such filtration. The map $K_0(\mathcal{B}) \to K_0(\mathcal{A})$ is zero, but $K_0(\mathcal{B}) \cong \bigoplus_p \Z$ over the primes $p$ is not.

Going deeper

For a student

Dévissage is a basic tool for computing $G$-theory. For a Noetherian ring $R$ and an ideal $I$, every finitely generated module killed by a power of $I$ has a filtration with quotients killed by $I$, which are $R/I$-modules. So the $K_0$ of finitely generated modules supported on $V(I)$ is $G_0(R/I)$. Combined with the localization sequence $G_0(R/I) \to G_0(R) \to G_0(R[s^{-1}]) \to 0$ for $I = (s)$, it lets one compute $G$-groups by cutting a ring into simpler pieces.