Quillen's dévissage theorem

Remarks

Quillen proved it with his Theorem A, applied to $Q\mathcal{B} \to Q\mathcal{A}$ for the Q-construction: the filtrations show that the relevant comma categories are contractible. In degree $0$ the statement is elementary: the filtration writes $[M] = \sum_i [M_i/M_{i-1}]$ in $K_0(\mathcal{A})$. This degree-$0$ case is the classical dévissage theorem for $K_0$.

Example: let $R$ be a left Noetherian ring and $I$ a nilpotent two-sided ideal, $I^m = 0$. Take for $\mathcal{A}$ the finitely generated left $R$-modules and for $\mathcal{B}$ those killed by $I$, which are the finitely generated $R/I$-modules. Each $M$ has the filtration $0 = I^m M \subseteq I^{m-1} M \subseteq \cdots \subseteq I M \subseteq M$, whose quotients are killed by $I$. So $G_n(R/I) \cong G_n(R)$ in G-theory. In the same way, the finite abelian $p$-groups have the same K-groups as the finite field $\mathbb{F}_p$.

The filtration hypothesis cannot be dropped. For $\mathcal{A}$ the finite abelian groups and $\mathcal{B}$ the finite abelian $2$-groups, $K_0(\mathcal{B}) \cong \Z$, but $K_0(\mathcal{A})$ is free abelian on the classes $[\Z/p]$, one for each prime $p$.

Going deeper

For a student

Dévissage is one of the three structural theorems Quillen derived from the Q-construction, with the resolution theorem and the localization theorem. A typical use concerns a Noetherian scheme $X$ with a closed subscheme $Z$. The coherent sheaves on $X$ supported on $Z$ have the same K-theory as the coherent sheaves on $Z$. This is how the first term of the localization sequence in G-theory is identified with $G_n(Z)$.