External degree-zero abelian dévissage

PDF p. 15: Supported degree-zero dévissage

Assume the Axiom of Choice. Let $\mathcal{B}\subseteq \mathcal{A}$ be essentially small abelian categories, with $\mathcal{B}$ full, inclusion exact, and $\mathcal{B}$ closed under subobjects and quotients taken in $\mathcal{A}$. Assume every object of $\mathcal{A}$ has a finite filtration whose quotients belong to $\mathcal{B}$. The inclusion induces an isomorphism of Grothendieck groups $G_0(\mathcal{B})\stackrel{\sim }{\to }{G}_0(\mathcal{A})\mathrm{.}$ Here $G_0(\mathcal{A})$ is generated by isomorphism classes of objects with $[M]=[M^{\mathrm{\prime }}]+[M^{\mathrm{\prime }\mathrm{\prime }}]$ for every short exact sequence $0\to M^{\mathrm{\prime }}\to M\to M^{\mathrm{\prime }\mathrm{\prime }}\to 0$. The inverse sends $[M]$ to the sum of the classes of the quotients of any permitted finite filtration. This sum is independent of filtration and is additive in short exact sequences. Extension-closure of $\mathcal{B}$ inside $\mathcal{A}$ is not a hypothesis.

Given: AC and the exact abelian inclusion and filtration hypotheses above, after choosing equivalent small presentations.

Goal: Import the established degree-zero filtration theorem and its inverse formula.

[F1] Weibel, The K-book, Dévissage Theorem II.6.3 with its proof supplies this exact assertion.

[A1] Choice is assumed in selecting equivalent small presentations and, on a small presentation, a permitted finite filtration for each object as in the source proof (The Axiom of Choice). Independence of these choices is part of [F1].

Import [F1] with its full source record and actual reading limits. The proof uses common refinements of finite filtrations for independence and concatenates a subobject filtration with the inverse image of a quotient filtration for additivity. Geometric applications must separately construct any required ideal-power filtrations and check the categorical hypotheses; no comparison of K-theory spectra is implicit.