Devissage Theorem

Quillen's devissage theorem is a powerful tool for computing K-groups of abelian categories. The name comes from the French "devissage" (unscrewing), reflecting the idea that K-groups are determined by the simple objects of the category, via successive Jordan--Holder-type filtrations.

Statement

Theorem2.2Quillen's Devissage

Let $\mathcal{A}$ be an abelian category and $\mathcal{B} \subseteq \mathcal{A}$ a full abelian subcategory such that:

$\mathcal{B}$ is closed under subobjects and quotients in $\mathcal{A}$ .
Every object $A \in \mathcal{A}$ has a finite filtration $0 = A_0 \subseteq A_1 \subseteq \cdots \subseteq A_n = A$ with successive quotients $A_i / A_{i-1} \in \mathcal{B}$ .

Then the inclusion $\mathcal{B} \hookrightarrow \mathcal{A}$ induces isomorphisms

$K_n(\mathcal{B}) \xrightarrow{\;\sim\;} K_n(\mathcal{A}) \quad \text{for all } n \geq 0.$

Classical applications

ExampleK-theory of a local ring via residue field

Let $(R, \mathfrak{m}, k)$ be a Noetherian local ring and let $\mathcal{A} = \mathcal{M}_{\mathfrak{m}}(R)$ be the category of finitely generated $R$ -modules supported at $\mathfrak{m}$ (i.e., annihilated by some power of $\mathfrak{m}$ ). Let $\mathcal{B}$ be the subcategory of $k$ -vector spaces (modules annihilated by $\mathfrak{m}$ ).

Every module in $\mathcal{A}$ has finite length and thus a composition series with quotients isomorphic to $k$ , so devissage applies:

$K_n(k) \xrightarrow{\;\sim\;} K_n(\mathcal{M}_\mathfrak{m}(R)) \quad \text{for all } n \geq 0.$

At the level of $K_0$ : $K_0(k) = \mathbb{Z}$ with generator $[k]$ , and $K_0(\mathcal{M}_\mathfrak{m}(R)) = \mathbb{Z}$ with generator $[k]$ as well. The map sends $[k^n] \mapsto n[k]$ , which is the length function.

ExampleReduction modulo nilpotents

Let $R$ be a Noetherian ring and $I \subseteq R$ a nilpotent ideal ( $I^N = 0$ ). Let $\bar{R} = R/I$ .

Consider $\mathcal{A} = \operatorname{mod}\text{-}R$ (modules annihilated by $I^N = 0$ , i.e., all modules) and $\mathcal{B} =$ modules annihilated by $I$ , which is equivalent to $\operatorname{mod}\text{-}\bar{R}$ .

A finitely generated $R$ -module $M$ has the filtration $M \supseteq IM \supseteq I^2M \supseteq \cdots \supseteq I^NM = 0$ , with quotients $I^kM/I^{k+1}M$ that are $\bar{R}$ -modules. Devissage gives:

$G_n(\bar{R}) = K_n(\operatorname{mod}\text{-}\bar{R}) \xrightarrow{\;\sim\;} K_n(\operatorname{mod}\text{-}R) = G_n(R).$

In other words, $G$ -theory is insensitive to nilpotent extensions: $G_n(R) \cong G_n(R_{\mathrm{red}})$ .

Proof sketch

Proof

The proof uses Quillen's Theorem A applied to the Q-construction.

Step 1: Reduction to Theorem A. We need to show $BQ\mathcal{B} \to BQ\mathcal{A}$ is a homotopy equivalence. By Theorem A, it suffices to show for each $A \in \mathcal{A}$ that the over-category $(\mathcal{B} \downarrow A)$ (objects of $\mathcal{B}$ mapping to $A$ in $Q\mathcal{A}$ ) has contractible classifying space.

Step 2: Analysis of the fiber category. The fiber category at $A$ consists of pairs $(B, \alpha)$ where $B \in \mathcal{B}$ and $\alpha: B \to A$ is a morphism in $Q\mathcal{A}$ , i.e., a diagram $B \twoheadleftarrow C \rightarrowtail A$ with $C$ a subquotient.

Step 3: Filtration argument. Choose a filtration $0 = A_0 \subset A_1 \subset \cdots \subset A_n = A$ with quotients in $\mathcal{B}$ . The subobjects $A_i$ provide a "tower" in the fiber category. The key observation is that any object in the fiber can be refined to factor through this filtration.

Step 4: Contractibility. The fiber category is shown to have an initial functor from a filtered category (essentially the poset of filtrations of $A$ with quotients in $\mathcal{B}$ ). Since any two such filtrations have a common refinement, this poset is filtered, and filtered categories have contractible classifying spaces.

Step 5: Technical verification. Condition (1) (closure under subobjects and quotients) ensures that the subcategory $\mathcal{B}$ behaves well with respect to the Q-construction morphisms, and condition (2) provides the necessary filtrations. $\square$

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Further applications

ExampleG-theory of Artinian schemes

For an Artinian local ring $(A, \mathfrak{m}, k)$ , every finitely generated $A$ -module has finite length, so devissage gives $G_n(A) \cong K_n(k)$ for all $n$ .

For an Artinian ring $A = \prod_{i=1}^r A_i$ (product of local Artinian rings with residue fields $k_i$ ):

$G_n(A) \cong \bigoplus_{i=1}^r K_n(k_i).$

This is used in the localization sequence: for a Noetherian scheme $X$ with closed point $x$ and open complement $U = X \setminus \{x\}$ :

$\cdots \to G_n(k(x)) \to G_n(X) \to G_n(U) \to G_{n-1}(k(x)) \to \cdots$

The devissage computation $G_n(\{x\}) = K_n(k(x))$ identifies the "fiber term."

RemarkLimitations of devissage

Devissage applies to G-theory (K-theory of all finitely generated modules) but typically not to K-theory of projective modules directly. The key distinction:

G-theory has devissage and is insensitive to nilpotents, but lacks good functoriality for non-flat maps.
K-theory of projectives has good functoriality (pullback along any ring map) but no devissage.

For regular rings these agree ( $K_n = G_n$ ), combining the best of both worlds. For singular rings, the difference $K_n(R)$ vs $G_n(R)$ measures the singularity.