K THEORY TALK

INNA ZAKHAREVICH AND INNA ZAKHAREVICH

1. Basic Definition

1.1. Introduction and Motivation. Note: in this talk we will ignore the existence of an explicit definition for K_{2}. Everything that we say will still morally be true.

Our initial problem in algebraic K-theory is classifying finitely generated projective modules over a ring R. We have the following two definitions:

K_{0}(R) = F{f.g.projectivemodulesoverR}/([M ⊕ M′] − [M] − [M′]).
K_{1}(R) = GL(R)^{ab}.

Note that with these definitions K_{0}(R) keeps track only of the different isomorphism classes of modules, with its group structure classifying how modules are put together. Note, also, that K_{0} keeps track only of “stable equivalence”: if M ⊕ N ≅ M′ ⊕ N then [M] = [M′]. On the other hand, K_{1} keeps track of the different automorphism information for projective modules. This is of a similar flavor to how the 0-th homotopy group keeps track of the component information of a topological space, while the 1-st one keeps track of the different connectivities. In addition, for any Dedekind domain R with field of fractions F we have the following exact sequence:

[figure: ⨁_{m} K_{1}(R/m); K_{1}R; K_{1}F; ⨁ K_{0}(R/m); K_{0}R; K_{0}F]

which is reminiscent of the long exact sequence in homotopy. With this kind of structure already present in the system it may seem reasonable (to a topologically inclined mind) to try to define K-groups as homotopy groups of some kind of space.

1.2. Group Completion. Let C be the category of finitely generated projective R-modules. Note that this category is a symmetric monoidal category under the operation ⊕, with 0 as its unit. Consider the definition of K_{0}. We can write

K_{0}(R) = F{isoclassesofC}/[M ⊕ M′] − [M] − [M′]
= Gr(isoclassesofC, ⊕)
= Gr(π_{0}(B(iso C)), ⊕).

Thus to get K_{0} we took a group completion of the monoid π_{0}(B(iso C)). But B(iso C) has the homotopy type of a topological monoid and thus its higher homotopy groups should tell us more about its structure. However, we are not just interested in the structure of this monoid; we would like to be able to get K-groups out of it directly. So we are looking for a topological construction that would group complete on the 0-th level and then do as little to change the rest of the homotopy type of the space above that. In fact, such a process is also called group completion:

Definition 1.1. Let X be a homotopy commutative topological monoid. CHECK THIS. Its group completion is a homotopy commutative topological monoid Y together with a map φ : X → Y such that

(1) φ_{∗} : π_{0}(X) → π_{0}(Y) is a group completion of π_{0}(X), and

(2) the induced map φ_{∗} : [π_{0}(X)^{−1}]H_{∗}(X) → H_{∗}(Y) is an isomorphism.

Note that from this definition group completions, if they exist, are unique up to weak equivalence.

One candidate for such a completion would be the space ΩBB(iso C). We know that for this π_{0} is a group (since it is π_{1}(BBiso C)), but because of the path-loop fibration we know that applying classifying spaces just shifts the higher homotopy groups up one (and looping shifts them down). In fact, we have the following theorem:

Theorem 1.2 (Group Completion Theorem). Let M be a topological monoid. If π_{0}(M) is in the center of H_{∗}(M) then H_{∗}(M)[π_{0}(M)^{−1}] ≅ H_{∗}(ΩBM). In particular, ΩBM is the group completion of M.

We know that for a symmetric monoidal groupoid such as B(iso C) the conditions of this theorem will work, so we have a candidate for the kind of space we want. We know from the definition that K_{0}(R) = π_{0}(ΩBB(iso C)) so one possible definition of the K-groups of a ring would be

K_{i}(R) = π_{i}(ΩBB(iso {f.g.proj.modulesoverR)).

1.3. Q Construction. On the other hand, the nice topological nature of the above construction is actually somewhat damaging, since such topological constructions are very difficult to compute. Thus it would be nice if we could make a construction which would be simpler to compute with.

If we consider the original category of finitely generated projective modules over R we can encode the monoidal structure of the category in a different way. If M = M′ ⊕ M′′ we have an exact sequence

[figure: 0; M′; ⊂; M; M′′; 0]

Since in C all exact sequences split this category encodes the same information on the monoidal structure. However, it also has the advantage that these do not forget automorphisms; given any automorphism of M there will be a new short exact sequence consisting of M, M′, M′′. We will construct a category whose classifying space will have the desired homotopy type. In order to do this we want to encode stable equivalence in the structure of the category itself. Thus we will construct a category where there is a morphism M → N whenever M is a subquotient of N.

Definition 1.3. An admissible monomorphism is one that can appear as the first map in a short exact sequence. An admissible epimorphism is one that can appear as the second map in a short exact sequence.

We define the objects of QC to just be the objects of C. We define the morphisms A → B to be equivalence classes of diagrams [figure: A; C; ⊂; B]. Two such diagrams will be equivalent whenever there is an isomorphism between their centers that makes the obvious diagrams commute. Morphisms compose by pulling back along the obvious diagrams. Note that there is a morphism M → N iff N = M ⊕X. In addition, notice that we have a notion of “admissible monomorphism” and “admissible epimorphism” inside of this category: they are simply the morphisms that can be written to have the epimorphism (resp. monomorphism) the identity. Any morphism can be factored as an epimorphism followed by a monomorphism.

For any module M we have two different morphisms 0 → M:

[figure: 0; M; ⊂; id; M; 0; 0; ⊂; M]

Thus any module M we have a homotopy class of loops [τ_{M}] defined by going to M by the monomorphism and then back along the epimorphism. We want to show that this defines an isomorphism K_{0}(R) → π_{1}(BQC). To show that it is a homomorphism we need to show that for any exact sequence M′ → M → M′′ we have [τ_{M}] = [τ_{M′}][τ_{M′′}]. Now notice that the exact sequences M′ → M′ ⊕M′′ → M′′ and M′′ → M′ ⊕M′′ → M′ show that the two classes on the right commute, so that the map is well-defined. In QC we have maps M′ → M and M′′ → M defined by composing, respectively, with the monomorphism and the epimorphism. But these maps give the maps 0 → M by composing with the monomorphism 0 → M′ and the epimorphism 0 → M′′, respectively. It turns out (through simple definition chasing) that the other two compositions (epi 0 → M′ and mono 0 → M′′) are equal. So in BC the image of this diagram is a sphere with three disks removed (sharing a point) exactly making the desired statement work. Similar computations will show that this homomorphism is in fact an isomorphism. Thus we see that K_{0}(R) = π_{1}(BQC). Therefore we may be motivated to define

K_{i}(R) = π_{i+1}(BQ{f.g.proj.modulesoverR).

1.4. Generalizing the Definitions. As often happens we may want to be able to generalize the above definitions to contexts other than projective modules over a ring. For the first definition, note that we did not use any facts other than that we had a symmetric monoidal groupoid to work with (with a set of isomorphism classes). In our case we had f.g. projective R-modules, but the definition would work just as well with any other symmetric monoidal groupoid.

In our second definition we needed a notion of exact sequence. We could say that we need to be working in an abelian category, but in fact we don’t need the existence of kernels and cokernels in general; what we needed instead was closure under extensions. So we could work in an exact category, by which we mean a subcategory of an abelian category closed under extensions. (There is actually an intrinsic definition of an exact category, but it is long and unnecessary for the major examples.) Given such a category we also immediately get a symmetric monoidal groupoid, given by the maximal subgroupoid of the exact category (with the operation being ⊕).

Now the obvious question arises: did our two definitions agree? It would be very sad if the two different approaches gave different answers. In fact, they don’t, as we will now move on to proving.

A note about notation: in the following discussion, C will denote an exact category and S = iso C will be its maximal subgroupoid. Our goal will be to show that

π_{i}(ΩBB(iso C)) ≅ π_{i+1}(BQC).

2. Equivalence of Definitions

2.1. The Group Completion Theorem. In order to prove that these definitions are equivalent we will first construct a categorical approximation to group completion. Then (since group completions uniquely determine homotopy type) we will be able to use theorems about the structure of categories to construct weak equivalences between their classifying spaces. This will be an advantage because constructing functors between categories (and proving things about them) is easier than constructing maps of classifying spaces.

Definition 2.1. Given a symmetric monoidal groupoid S and a category D with a functor ⊕ : S × D → D such that

s_{1} ⊕ (s_{2} ⊕ d) ≅ (s_{1} ⊕ s_{2}) ⊕ d

we say that S acts on D.

For example, any symmetric monoidal groupoid acts on itself by (s, t) ↦ s⊕t. In addition, for any additive category A, iso A acts on A in the same way.

Definition 2.2. If S acts on D we define S^{−1}D to be the category with objects S × D, and morphisms (s_{1}, d_{1}) → (s_{2}, d_{2}) will be equivalence classes of triples (t, f, g) with

t ∈ S f : t ⊕ s_{1} → s_{2} g : t ⊕ d_{1} → d_{2}.

We say that two such triples (t, f, g) and (t′, f′, g′) are equivalent if there is an isomorphism t → t′ which makes the relevant diagrams commute.

The idea of this construction is to imitate the construction of a field of frations on a category. We want the topological realization of this to be “group completing”, so since the field of fractions is the “group completion” of a ring this idea makes sense. In addition, notice that we have a functor D → S^{−1}D given by d ↦ (0, d). We have the following theorem:

Theorem 2.3 (Quillen). If S is a symmetric monoidal groupoid such that Aut(d) → Aut(s⊕ d) is injective for all s ∈ S and d ∈ D, then BDrightarrowBS^{−1}D induces an isomorphism

[π_{0}(BS)^{−1}]H_{∗}(BD) ≅ H_{∗}(BS^{−1}D).

In particular, this along with the discussion above of the map BS → BS^{−1}S means that this map is a group completion.

Putting these together, we see that for all i,

π_{i}(BS^{−1}S) ≅ π_{i}(ΩBBS).

2.2. S^{−1}S and the Q Construction. So now we have a nice categorical construction of group completion. Now if we could show that when S = iso C, π_{i}(BS^{−1}S) ≅ π_{i+1}(BQC) for our exact category C we would be done. We will do this in two steps. First, we will construct a contractible space BS^{−1}EC which will come equipped with maps

BS^{−1}S ⟶ BS^{−1}EC ⟶ BQC.

Then we will use Quillen’s theorem B to prove that this sequence of maps leads to a long exact sequence in homotopy, from which we will get the desired result.

Definition 2.4. We define the category EC to consist of the exact sequences in C. In particular, the objects in EC are exact sequences [figure: A; ⊂; B; C] in C; morphisms between exact sequences will be equivalence classes of diagrams

[figure: A; ⊂; B; C]

[figure: A′; ∪; ⊂; B; ‖; C′′]

[figure: A′; ‖; ⊂; B′; ∩; C′; ∩]

(where the morphism is from the top row to the bottom row). Two such diagrams are equal if there is a natural transformation between them that is the identity everywhere other than at C′′.

Now S = iso C acts on EC by the map

(s, [figure: A; ⊂; B; C]) ⟼ [figure: s ⊕ A; ⊂; s ⊕ B; C]

so we can form the category S^{−1}EC.

Proposition 2.5. BS^{−1}EC is contractible.

Proof. Note that by the group completion theorem we have that the homology of BS^{−1}EC is 0 iff the homology of BEC is 0. So it suffices to show this for BEC. In order to show this we will use a theorem that is important in its own right, so we take some time out of this proof to discuss it.

Quillen’s Theorem A. Let F : C → D be a functor. If for every Y ∈ D (Y ↓ F) is contractible then BF is a homotopy equivalence. Dually, if every (F ↓ Y) is contractible then BF is a homotopy equivalence.

In our current problem consider the functor m : EC → iQC, defined by

(A → B → C) ⟼ B,

where iQC is the subcategory of QC consisting only of those maps where the admissible epimorphism is the identity. Now for any B ∈ iQC note that (m ↓ B) is contractible, since for any object

(B′, f : m(A′ → B′ → C′) → B) ∈ (m ↓ B)

we have a unique square

[figure: A; ⊂; α; B′; β; C′]

[figure: 0; !_{∪}; ⊂; !; B′; ‖; = !; B′]

[figure: 0; ‖; ⊂; B; f; ∩; = = = = =; B; f; ∩]

so 0^{⊂} B = = = = = B is terminal in (m ↓ B). Thus each (m ↓ B) is contractible, so BEC ≃ BiQC. On the other hand, iQC has initial object 0, so it is also contractible. So EC is contractible, as desired. □

Notice, however, that we have a functor S → EC given by

A ⟼ ([figure: A; = = = = =; A; 0])

and a functor T : EC → QC given by

([figure: A; ⊂; B; C]) ⟼ C.

Note that, since the action of S on EC does not touch the last component, T actually induces a functor T̂ : S^{−1}EC → QC.

Theorem 2.6. Suppose that C is split exact and S = iso C. Then the sequence BS^{−1}S → BS^{−1}EC → BQC produces a long exact sequence in homotopy.

Once we have this we will be done, since a long exact sequence in homotopy where the middle term is contractible produces exactly the kind of isomorphism we are interested in. Note that the long exact sequence only works above level 0, but this does not matter since both constructions were explicitly made so that the 0-levels would agree.

Proof of theorem. The proof of this theorem uses another of Quillen’s theorems that is important enough to state separately.

Quillen’s Theorem B. Let F : C → D be a functor such that for every morphism X → Y ∈ D, the induced functor (Y ↓ F) → (X ↓ F) is a homotopy equivalence. Then for any Y ∈ C the pullback square

[figure: (Y ↓ F); C; (Y ↓ D); D; F]

is a homotopy pullback Thus for any X ∈ F^{−1}(Y) we have a long exact sequence

[figure: ⋯; π_{i+1}(D, Y); π_{i}((Y ↓ F), X); π_{i}(C, X); π_{i}(D, Y); ⋯]

We will be applying this theorem to the functor T̂, so that C = S^{−1}EC and D = QC. First, assume that the theorem applies. Setting Y = 0 and X = 0 = = = = = 0 = = = = = 0 we get the long exact sequence

[figure: ⋯; π_{i+1}(BS^{−1}EC); π_{i+1}(BQC); π_{i}((0 ↓ T̂)); π_{i}(BS^{−1}EC); ⋯]

Now we know that π_{k}(BS^{−1}EC) = 0, so this gives us isomorphisms

π_{i+1}(BQC) ≅ π_{i}(B(0 ↓ T̂)) i > 0.

Even though these don’t work at level 0 this is OK since the two constructions were designed to agree at level 0. If we could prove that B(0 ↓ T̂) ≃ BS^{−1}S we would be done. For this, two facts (both of which are proved using simple explicit computation) are needed:

• For all C ∈ QC, B(C ↓ T̂) ≃ BT^{−1}(C), induced by the inclusion T̂^{−1}(C) → (C ↓ T̂)

(s, A → B → C) ⟼ (A → B → C, id).
• T̂^{−1}(0) = S^{−1}S.

Now all that remains to show is that the theorem actually applies. To show this, we need to show that any f : X → Y ∈ QC acts by homotopy equivalences on the under categories. But since each such f is a composition of an admissible monomorphism and an admissible epimorphism, it suffices to show it just for those. Since C is split exact, we can just show it for 0 → C and C → 0 for all C ∈ C. This is easy to do with one additional fact:

• If C is split exact then for all C ∈ C we have a homotopy equivalence BS^{−1}S ≃ BT̂^{−1}(C) induced by the functor S → T^{−1}(C)

A ⟼ (A → A ⊕ C → C).

□