Quillen's Q-Construction

Quillen's Q-construction provides an alternative definition of higher algebraic K-groups using exact categories, avoiding the choice of a classifying space. For an exact category $\mathcal{C}$ , it produces a topological space (the classifying space of a certain category $Q\mathcal{C}$ ) whose homotopy groups are the K-groups.

Exact categories

Definition2.3Exact category

An exact category is an additive category $\mathcal{C}$ equipped with a class $\mathcal{E}$ of sequences $A \rightarrowtail B \twoheadrightarrow C$ (called conflations or short exact sequences) where $A \rightarrowtail B$ is an admissible monomorphism (inflation) and $B \twoheadrightarrow C$ is an admissible epimorphism (deflation), satisfying:

The identity $\operatorname{id}_0: 0 \to 0$ is a deflation.
Deflations are closed under composition.
Deflations are stable under pullback along arbitrary morphisms.
Inflations are closed under composition.
Inflations are stable under pushout along arbitrary morphisms.

The prototypical example is the category of finitely generated projective $R$ -modules with split exact sequences, or the category of all finitely generated $R$ -modules (when $R$ is Noetherian) with all short exact sequences.

ExampleImportant exact categories

$\mathcal{P}(R)$ : Finitely generated projective $R$ -modules with split exact sequences. $K_n(\mathcal{P}(R)) = K_n(R)$ .
$\mathcal{M}(R)$ : Finitely generated $R$ -modules (for Noetherian $R$ ) with all short exact sequences. $K_n(\mathcal{M}(R)) = G_n(R)$ , the $G$ -theory groups.
$\mathcal{V}(X)$ : Vector bundles on a scheme $X$ , with short exact sequences that are locally split. $K_n(\mathcal{V}(X)) = K_n(X)$ .
$\operatorname{Coh}(X)$ : Coherent sheaves on a Noetherian scheme $X$ . $K_n(\operatorname{Coh}(X)) = G_n(X)$ .
$\mathcal{C}^b(\mathcal{P}(R))$ : Bounded chain complexes of finitely generated projectives with degreewise split exact sequences.

The Q-construction

Definition2.4The category QC

For an exact category $\mathcal{C}$ , define the category $Q\mathcal{C}$ as follows:

Objects: The same as objects of $\mathcal{C}$ .
Morphisms from $A$ to $B$ : Isomorphism classes of diagrams $A \twoheadleftarrow C \rightarrowtail B$ , where $C \twoheadrightarrow A$ is an admissible epimorphism and $C \rightarrowtail B$ is an admissible monomorphism.
Composition: Given $A \twoheadleftarrow C \rightarrowtail B$ and $B \twoheadleftarrow D \rightarrowtail E$ , form the pullback $C \times_B D$ and compose to get $A \twoheadleftarrow C \times_B D \rightarrowtail E$ .

The K-groups of $\mathcal{C}$ are defined as

$K_n(\mathcal{C}) = \pi_{n+1}(BQ\mathcal{C}, 0) \quad \text{for } n \geq 0$

where $BQ\mathcal{C}$ is the classifying space (geometric realization of the nerve) and $0$ is the zero object.

RemarkWhy the shift in indexing?

The shift $K_n = \pi_{n+1}$ is because $Q\mathcal{C}$ has a "built-in delooping." The fundamental group $\pi_1(BQ\mathcal{C})$ already gives $K_0$ :

$K_0(\mathcal{C}) = \pi_1(BQ\mathcal{C}, 0)$

which is the usual Grothendieck group. The loop $A \twoheadleftarrow 0 \rightarrowtail A$ composed with $A \twoheadleftarrow A \rightarrowtail 0$ gives the class $[A] \in K_0$ . The relation $[B] = [A] + [C]$ for a conflation $A \rightarrowtail B \twoheadrightarrow C$ follows from the homotopy theory of $BQ\mathcal{C}$ .

Agreement with the plus construction

RemarkQuillen's equivalence theorem

For the exact category $\mathcal{P}(R)$ of finitely generated projective $R$ -modules, Quillen proved

$\Omega BQ\mathcal{P}(R) \simeq K_0(R) \times BGL(R)^+$

establishing that the Q-construction and plus construction give the same K-groups:

$\pi_{n+1}(BQ\mathcal{P}(R)) \cong \pi_n(K_0(R) \times BGL(R)^+) = K_n(R) \quad \text{for all } n \geq 0.$

The proof uses the "group completion" properties of the $H$ -space structure on $\coprod_n BGL_n(R)$ induced by direct sum. This is a non-trivial result that validates both approaches to higher K-theory.

ExampleQ-construction for abelian categories

When $\mathcal{C} = \mathcal{A}$ is an abelian category (e.g., $\operatorname{Coh}(X)$ ), every monomorphism is admissible and every epimorphism is admissible, so the Q-construction morphisms from $A$ to $B$ are equivalence classes of subquotients: subobjects $C \hookrightarrow B$ together with epimorphisms $C \twoheadrightarrow A$ .

For $\mathcal{A} = \operatorname{Coh}(\operatorname{Spec} R)$ with $R$ Noetherian:

$K_0(\operatorname{Coh}(\operatorname{Spec} R)) = G_0(R) = K_0(\text{f.g. } R\text{-modules}) / \langle \text{exact sequences} \rangle.$

When $R$ is regular, the resolution theorem gives $K_n(R) \cong G_n(R)$ for all $n$ .