Resolution Theorem

Quillen's resolution theorem is a fundamental comparison result in algebraic K-theory. It states that for a regular ring $R$ , the K-theory of projective modules agrees with the G-theory (K-theory of all finitely generated modules). This bridges the "geometric" K-groups $K_n$ with the more algebraic $G_n$ groups.

Statement

Theorem2.1Quillen's Resolution Theorem

Let $\mathcal{C} \subseteq \mathcal{D}$ be an exact inclusion of exact categories such that:

Every object of $\mathcal{D}$ has a finite resolution by objects of $\mathcal{C}$ : for every $D \in \mathcal{D}$ , there exists an exact sequence $0 \to C_n \to C_{n-1} \to \cdots \to C_1 \to C_0 \to D \to 0$ with all $C_i \in \mathcal{C}$ .
If $0 \to C \to D \to D' \to 0$ is exact in $\mathcal{D}$ with $C \in \mathcal{C}$ and $D' \in \mathcal{D}$ , then $D \in \mathcal{C}$ implies $D' \in \mathcal{C}$ , and vice versa.

Then the inclusion $\mathcal{C} \hookrightarrow \mathcal{D}$ induces isomorphisms

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

Key application: K-theory equals G-theory for regular rings

Definition2.7G-theory

For a Noetherian ring $R$ , define the G-theory groups

$G_n(R) = K_n(\operatorname{mod}\text{-}R)$

where $\operatorname{mod}\text{-}R$ is the exact category of finitely generated $R$ -modules with all short exact sequences. The natural inclusion $\mathcal{P}(R) \hookrightarrow \operatorname{mod}\text{-}R$ induces a map $K_n(R) \to G_n(R)$ called the Cartan homomorphism.

ExampleRegular rings

When $R$ is a regular Noetherian ring (every finitely generated module has finite projective dimension), the resolution theorem applies with $\mathcal{C} = \mathcal{P}(R)$ and $\mathcal{D} = \operatorname{mod}\text{-}R$ :

Condition (1): Every finitely generated $R$ -module has a finite projective resolution (by regularity).
Condition (2): In a short exact sequence $0 \to P \to M \to N \to 0$ with $P$ projective, $M$ is projective iff $N$ is projective (by the horseshoe lemma and dimension shifting).

Therefore $K_n(R) \cong G_n(R)$ for all $n \geq 0$ when $R$ is regular. This includes:

Fields, PIDs, Dedekind domains
$k[x_1, \ldots, x_n]$ for a field $k$ (polynomial rings are regular)
Regular local rings (by Auslander--Buchsbaum--Serre)

ExampleFailure for singular rings

For the singular ring $R = k[x]/(x^2)$ :

$K_0(R) \cong \mathbb{Z}$ (every projective is free since $R$ is local).
$G_0(R) \cong \mathbb{Z}$ with generator $[k]$ , where $k = R/(x)$ .
The Cartan map $K_0(R) \to G_0(R)$ sends $[R] \mapsto 2[k]$ since $R$ has a composition series $0 \subset (x) \subset R$ of length 2.

For higher groups, $K_n(k[x]/(x^2))$ contains copies of $\Omega^{n-1}_{k/\mathbb{Z}}$ (Kahler differentials) via the Dennis trace, while $G_n(k[x]/(x^2)) \cong K_n(k)$ by devissage. So the Cartan map is far from an isomorphism.

Proof sketch

Proof

The proof uses Quillen's "Theorem A" and "Theorem B" from the theory of categories and homotopy.

Step 1: Setup. We must show $BQ\mathcal{C} \to BQ\mathcal{D}$ is a homotopy equivalence. By Quillen's Theorem A, it suffices to show that for each $D \in Q\mathcal{D}$ , the fiber category $(D \downarrow Q\mathcal{C})$ (objects of $\mathcal{C}$ under $D$ ) has contractible classifying space.

Step 2: Filtration by resolution length. Let $\mathcal{D}_n$ be the full subcategory of objects in $\mathcal{D}$ having a resolution of length $\leq n$ by objects of $\mathcal{C}$ . We have $\mathcal{D}_0 = \mathcal{C}$ and $\mathcal{D} = \bigcup_n \mathcal{D}_n$ .

Step 3: Induction on resolution length. Show that $K_*(\mathcal{D}_n) \cong K_*(\mathcal{D}_{n-1})$ for each $n \geq 1$ . This uses condition (2) and the observation that any object $D \in \mathcal{D}_n$ fits into a short exact sequence $0 \to D' \to C \to D \to 0$ with $C \in \mathcal{C}$ and $D' \in \mathcal{D}_{n-1}$ .

Step 4: Contractibility of fibers. For a fixed $D \in \mathcal{D}$ , the fiber category has a final object coming from the chosen resolution of $D$ . More precisely, the category of resolutions of $D$ by objects of $\mathcal{C}$ is filtered (any two resolutions can be compared), and filtered categories have contractible classifying spaces.

Step 5: Passage to the limit. Since $\mathcal{D} = \bigcup_n \mathcal{D}_n$ and each inclusion $\mathcal{D}_n \hookrightarrow \mathcal{D}_{n+1}$ induces an equivalence on K-groups, we conclude $K_*(\mathcal{C}) \cong K_*(\mathcal{D})$ . $\square$

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Related comparison results

RemarkDevissage and localization

The resolution theorem is one of Quillen's three fundamental comparison tools:

Resolution theorem (above): Compares $K(\mathcal{C})$ and $K(\mathcal{D})$ when $\mathcal{C}$ provides resolutions in $\mathcal{D}$ .
Devissage: If $\mathcal{A}$ is an abelian category and $\mathcal{B} \subseteq \mathcal{A}$ is a Serre subcategory such that every object of $\mathcal{A}$ has a finite filtration with subquotients in $\mathcal{B}$ , then $K_n(\mathcal{B}) \cong K_n(\mathcal{A})$ .
Localization: For a Serre subcategory $\mathcal{B} \subseteq \mathcal{A}$ with quotient $\mathcal{A}/\mathcal{B}$ , there is a long exact sequence: $\cdots \to K_{n+1}(\mathcal{A}/\mathcal{B}) \to K_n(\mathcal{B}) \to K_n(\mathcal{A}) \to K_n(\mathcal{A}/\mathcal{B}) \to \cdots$

Together, these tools form the calculational backbone of algebraic K-theory.