Bloch’s lecture 10.27.03, Beilinson’s office hours 10.30.03. Subject: Quillen’s two constructions of K-groups.

References: articles by Quillen and Quillen-Grayson in Springer Lecture Notes 341, 551.

A. The +-construction. Let X be a connected topological space (which means, say, CW complex), N ⊂ π_{1}(X) a normal subgroup (one need not specify a base point due to normality). Suppose N is perfect, i.e., coincides with its commutant.

Claim. There is a morphism ϕ : X → X^{+} which satisfies properties (o)–(iii) below. Here X̃^{+} is a universal covering of X^{+} and X̃ its pull-back to X.

(o) X^{+} is connected,

(i) π_{1}(X)/N →^{∼} π_{1}(X^{+}),

(ii) X̃^{+}/X̃ := Cone(ϕ̃ : X̃ → X̃^{+}) is contractible. Here X̃^{+} is a universal covering of X^{+} and X̃ its pull-back to X.

One can replace (ii) by either of the equivalent properties (ii)’, (ii)”. Below H_{·}(?) := H_{·}(?, ℤ).

(ii)’ H_{·}(X̃) →^{∼} H_{·}(X̃^{+}).

(ii)” H_{·}(X, ϕ^{∗}L) →^{∼} H_{·}(X^{+}, L) for every local system L of abelian groups on X^{+}.

Construction. We define X ↪ Y ↪ X^{+} where Y is obtained from X by gluing 2-cells, and X^{+} from Y by gluing 3-cells. The sets of 2- and 3-cells are in a canonical bijection.

Take any set {γ_{α}} of loops that generate N (as a normal subgroup). Get Y by gluing 2-cells B_{α}^{2} by γ_{α}.

The composition π_{2}(Y) → H_{2}(Y) → H_{2}(Y/X) is surjective. Proof of surjectivity: Let Ỹ be a universal covering of Y, X̃ its pull-back to X. Then our arrow is the composition π_{2}(Y) ←^{∼} π_{2}(Ỹ) →^{∼} H_{2}(Ỹ) ↠ H_{2}(Ỹ /X̃) ↠ H_{2}(Y/X). Here the second arrow is an isomorphism since π_{1}(Ỹ) = 0, the third one is surjective since H_{1}(X̃) = 0 (for π_{1}(X̃) = N is perfect).

Choose any section ⊕ℤ_{α} = H_{2}(Y/X) → π_{2}(Y). The images of the generators form a family of maps ξ_{α} : S^{2} → Y. Glue 3-cells B_{α}^{3} to Y by means of ξ_{α}. This is X^{+}. Properties (0)–(ii) are evident. □

Universality property. Let ψ : X → Z be a map which kills N. Then for any X^{+} as above, there is a unique morphism ψ^{+} : X^{+} → Z in the homotopy category such that ψ = ψ^{+}ϕ (the equality should hold in the homotopy category).

Proof. It suffices to consider the X^{+} built by the above construction (for a morphism from this X^{+} to any other version of X^{+}, which exists by universality, is a homotopy equivalence).

The set P_{1} of homotopy classes of extensions ψ_{Y} of ψ to Y is non-empty, and it is naturally a ∏ π_{2}(Z, z_{α})-torsor. Here z_{α} ∈ Z is a point on the loop ψ(γ_{α}) (which is homotopic to 0). The obstruction c(ψ_{Y}) to the existence of an extension of a given ψ_{Y} to X^{+} lies in ∏ π_{2}(Z, z_{α}). The map ψ_{Y} ↦ c(ψ_{Y}) is affine with respect to the ∏ π_{2}(Z, z_{α})-translations. Thus there is a unique ψ_{Y} that can be extended to X^{+}.

Take such ψ_{Y} (as a map). The homotopy classes of its extensions to X^{+} form a ∏ π_{3}(Z, z_{α})-torsor P_{2}. The group of (homotopy classes of) self-homotopies of ψ_{Y} leaving ψ fixed acts on P_{2}. One easily checks that this action is transitive, so all possible ψ^{+} are homotopic (with the restriction to X fixed). □

Definition. For an associative algebra A one defines K_{0}(A) as the K-group of the category of finitely generated projective A-modules. For i ≥ 1 we set K_{i}(A) := π_{i}(BGL_{∞}(A)^{+}). Here GL_{∞}(A) := ∪GL_{n}(A), BGL_{∞}(A) is the corresponding Eilenberg-MacLane space, and the +-construction is taken with respect to the commutant E(A) of GL_{∞}(A) (it is known that E(A) is the subgroup generated by elementary matrices, so it is perfect).

Exercise. Choose some BGL_{∞}(ℤ) ↪ BGL_{∞}(ℤ)^{+}. Then for any A the space BGL_{∞}(A) ⊔_{BGL_{∞}(ℤ)} BGL_{∞}(ℤ)^{+} is BGL_{∞}(A)^{+}. (The condition to check is that

E(A) ⊂ GL_{∞}(A) is generated as a normal subgroup by elementary matrices with integral coefficients). By the way, one can get BGL_{∞}(ℤ)^{+} by adding to BGL_{∞}(ℤ) only one 2-cell and one 3-cell.

Denote by P = P(A) the groupoid of finitely-generated projective A-modules (the morphisms are isomorphisms). Notice that P is a symmetric monoidal category with respect to the direct sum operation. Thus the classifying space BP is an Hspace. It is highly non-connected: π_{0}P is the semi-group of isomorphism classes of finitely-generated projective A-modules. Its group completion is K_{0}(A).

Suppose we have an H-space T such that π_{0}T is a commutative group and a morphism of H-spaces ν : BP → T.

Lemma. Such ν yields ν_{i} : K_{i}(A) → π_{i}T.

Proof. The group completion of the semi-group π_{0}BP equals K_{0}(A), hence we get ϕ_{0}. To get ϕ_{>0} we will define a (canonical up to homotopy) morphism ν̄ : BGL_{∞}(A) → T^{0} where T^{0} is the neutral component of T. By the universality property of +-construction (and commutativity of π_{1}(T^{0})) it extends then to a morphism BGL_{∞}(A)^{+} → T^{0}. Our ν_{i} are the corresponding morphisms of the homotopy groups.

Since π_{0}(T) is a group, our T is a group in the homotopy category; denote by − the “substraction” morphism T × T → T. For P ∈ P we have a map ν̄_{P} : BAutP → T^{0}, ν̄_{P} := ν|_{BAutP} − ν(P). Here ν(P) ∈ T is the ν-image of P ∈ BP. In particular, we have maps ν̄_{A^{n}} : BGL_{n}(A) → T^{0}. It is clear that the restriction of ν̄_{A^{n}} to BGL_{n−1}(A) ↪ BGL_{n}(A) (the standard embedding) is canonically homotopic to ν̄_{A^{n−1}}. Applying the telescope construction, we get the promised ν̄ : BGL_{∞}(A) → T^{0}. □

B. The Q-construction. Let M be an exact category. Thus it is an additive category equipped with a class of “exact sequences” satisfying some axioms. They say, essentially, that M can be realized as a full subcategory in an abelian category closed under extensions; the exact sequences are those coming from exact sequences in the abelian category. The essential example is the category of finitely-generated projective A-modules. Below “injections” and “surjection” in M mean morphisms that can be included (appropriately) in the “exact sequences” (Quillen’s terms are “admissible monomorphism” and admissible epimorphism”). Notation: ↪, ↠.

Quillen’s category Q = QM has the same objects as M, and a morphism X → Y in Q is an identification of X with a subquotient of Y. Thus we have a filtration Y_{0} ↪ Y_{1} ↪ Y and an isomorphism X →^{∼} Y_{1}/Y_{0}. The composition is evident (the notion of subquotient is transitive).

The direct sum in M defines a symmetric monoidal structure on Q. Since BQ is connected (the set of morphisms from 0 to any object is non-empty), in the next definition we need not refer to the base point for π_{i}’s, and π_{1} is commutative:

Definition. K_{i}(M) := π_{i+1}(BQ).

Special subcategories. Below P ⊂ Q is the groupoid whose objects are the same as those of Q and morphisms are all isomorphisms in Q.

Remark. An isomorphism X →^{∼} Y in M can be considered as an identification of X with a subobject of Y, or of Y with a quotient object of X. Thus, depending on your fancy, P identifies canonically either with the groupoid M^{iso} of isomorphisms in M, or with its inverse.

Among subquotients there are subobjects and quotient objects, These define two subcategories Q^{+}, Q^{−} ⊂ Q. They have the same objects as Q. Clearly Q^{+} is the subcategory of injections in M, and Q^{−} is opposite to the subcategory of surjections in M. One has P = Q^{+} ∩ Q^{−}.

Two important facts:

(a) The subcategories Q^{+}, Q^{−} are contractible (since 0 is the initial object in either of them).

(b) Any subquotient can be considered either as a quotient of a subobject, or a subobject of a quotient object. Thus any morphism f in Q can be uniquely represented as the composition of a morphism in Q^{+} and a morphism in Q^{−} in either order, i.e., written as the diagonal in a commutative square with columns in Q^{+} and rows in Q^{−}. We refer to it as the canonical commutative square of f.

Adding to P ⊂ Q^{+} all morphisms from the initial object 0 ∈ Q^{+}, we get a “cone” subcategory C^{+}P ⊂ Q^{+}. Similarly, we get a cone subcategory C^{−}P ⊂ Q^{−}. Set SP := C^{+}P ∪ C^{−}P ⊂ Q. This is a subcategory in Q whose classifying space is the suspension of BP, i.e., BSP = SBP. By adjunction, the embedding SP ⊂ Q yields a map

ν : BP → ΩBQ.

The monoidal structure on Q makes BQ is an H-space. So ΩBQ carries two natural H-structures, which coincide by a standard general result from algebraic topology. The same monoidal structure makes BP an H-space, and ν is a morphism of Hspaces (clear for one of the H-structures).

Proposition. The map π_{0}ν : {isomophism classes of objects in M} → π_{0}ΩBQ = K_{0}(M) identifies K_{0}(M) with the quotient of the free abelian group generated by the isoorphism classes modulo the usual relation coming from exact sequences.

Proof. Contracting BQ^{−} ⊂ BQ to a point, we get a space BQ/BQ^{−} homotopically equivalent to Q. It has a single vertex, thus its fundamental group is generated by 1-simplices with relations coming from 2-simplices. Each 1-simplex comes from a morphism in Q. Looking at its canonical square (see (b) above) we see that our 1-simplex is (canonically) homotopic to one Z ↪ Y from Q^{+}. Then the canonical square for the morpism 0 → Y corresponding to the subquotient Y/Y, identifies it with 0 ↪ Y/Z. This loop equals π_{0}ν(Y/Z) ∈ K_{0}(M), so K_{0}(M) is generated by those elements. The relations come from composition of morphisms in Q. The corresponding canonical squares compose in an evident way, and this yields the usual relations coming from exact sequences. □

C. Comparison. Suppose that M is the category of finitely-generated projective A-modules. Then ν fits into the setting of the Lemma from Section A, so we get

ν_{i} : K_{i}(A) → K_{i}(M).

Theorem. These are isomorphisms.

Idea of the proof. (a) The first step is to describe ν in a slightly different way.

Let us decompose the embedding Q^{+} ↪ Q into a composition Q^{+} →^{i} Q̃^{+} →^{p} Q where i is fully faithful, Q̃^{+} is contractible, and p is surjective on morphisms.

Objects of Q̃^{+} are injections X ↪ Y in M. A morphism (X ↪ Y) → (X′ ↪ Y′) in Q̃^{+} is a pair of injections X′ ↪ X, Y ↪ Y′ such that the composition X′ ↪ X ↪ Y ↪ Y′ is equal to our morphism X′ ↪ Y′ (so we get a commutative “trapezoid”). The functors are i(Y) := (0 ↪ Y) and p(X ↪ Y) := Y/X. The contractibility of Q̃^{+} comes from canonical morphisms (X ↪ Y) → (0 ↪ Y) ← (0 ↪ 0) in Q̃^{+}.

Notice that the fiber p^{−1}(0) ⊂ Q̃^{+} is formed by X →^{∼} Y, i.e., it identifies canonically with M^{iso} = P. Contracting this fiber in Q̃^{+} into a point (say, using the above contraction), we embed a cone over BP into BQ̃^{+}. Projecting it down by p (so that the bottom of our cone contracts to the point), we get a map SBP → BQ. The map BP → ΩBQ defined by adjunction is homotopic to ν.

(b) If p : BQ̃^{+} → BQ were a fibration then, by contractibility of Q̃^{+}, our ν would have been a homotopy equivalence. This is not the case, but one can modify the picture to make it become true.

Notice that direct sum makes Q̃^{+} a symmetric monoidal category, and p is a symmetric monoidal functor. The monoidal category p^{−1}(0), which is our old friend P from Section A, acts then along the fibers of p. Quillen explains how one can modify canonically an action of any monoidal category on another category so that the translation functors become homotopy equivalences. Applying this to our situation (the monoidal category P acting on Q̃^{+}) we get a new category over Q. It is still contractible, and the projection to Q is a fibration. The morphism of H-spaces BP → T where T is the new fiber over 0 has the property that the corresponding maps ν_{i} from the Lemma from Section A are isomorphisms. We are done. □