Algebraic K-theory

Daniel Ramras

Typed by Virgil Chan

29 October 2004 – 11 November 2004 2 Contents

Preface 5

1 Projective Modules7 1.1 Equivalent Definitions...... 7 1.2 Topological K-theory...... 7 1.3 K0 for Schemes...... 8

2 Higher K-theory9 2.1 Simplicial Sets...... 9 2.1.1 Canonical Example–Singular Sets...... 10 2.1.2 Formal Definition...... 11 2.1.3 Geometric Realisation of a Simplicial ...... 11 2.2 Classifying Spaces of a Small ...... 12 2.3 Definition of Higher K-theory...... 14 2.3.1 The Q-construction...... 14 2.3.2 Exact Categories and the Q-construction...... 16 2.3.3 of the Q-construction...... 18  2.3.4 An Explicit Map α : K0(R) → π1 Q P ...... 23  2.3.5 Another Approach to π1 Q M ...... 26

3 Bloch’s Formula for Chow Groups 31 3.1 Product...... 35

4 Theorem A and D´evissage 39

3 4 CONTENTS Preface

These are notes from a series of three lectures I gave at Stanford in Fall 2004. They  ∼ Swan’s Theorem, the Q-construction, the π1 Q P (R) = K0(R), Bloch’s Formula, and D´evissage. Thanks to Virgil Chan for turning my handwritten notes into a readable document.

5 6 CONTENTS Chapter 1

Projective Modules

1.1 Equivalent Definitions

Let R be a (commutative) with identity 1 = 1R; M be a finitely generated R-. We say M is projective if it is a direct summand of a . We denote P (R) the category of projective R-modules with module homomorphisms, then the Grothendieck K0(R) is defined as follows: Let F be the free abelain group with a generator hP i for each isomorphism class of projective modules over R, and let R be the subgroup of F generated by hP ⊕ Qi − hP i − hQi . (Note: P ⊕ Q ∈ P (R) whenever P , Q ∈ P (R)). We define

K0(R) := F/R and write [P ] for hP i R. In other words, we formally add inverse to the monoid M of isomorphism classes. K0(R) is universal in the sense that any monoid map M → A with A an factors through K0(R) uniquely. Remark [P ] = [Q] if and only if P is stably isomorphic to Q, that is, there exists n such n ∼ n that P ⊕ R = Q ⊕ R . So K0(R) is the ”group of modules up to stable isomorphism”. ∼ Example 1.1.1. If R is a field, then K0(R) = Z, generated by [R].

1.2 Topological K-theory

Let X be a compact Hausdorff space. We form a group K0(X) just as before, replacing projective modules by vector bundles over X. (We can use complex, real, or quaternion, but 0 0 n ∼ 0 n let us use C.) Again, in K (X) we have [V ] = [V ] if and only if V ⊕ ε = V ⊕ ε for some n n n, where ε is the n-dimensional trivial bundle X × C → X.

7 8 CHAPTER 1. PROJECTIVE MODULES

0 ∼ Theorem 1.2.1 ([Swa62]). K (X) = K0(C(X)), where C(X) is the ring of continuous functions X → C.

Sketch of Proof. We need to show that vector bundles correspond to projective C(X)-modules:

{V → X} ←→ Γ(V ) := {continuous sections of V } . If V is a , then there exists W such that V ⊕ W = εn, so Γ(V ) ⊕ Γ(W ) = Γ(εn) = C(X)n. i.e., Γ is projective. If M is a projective, finitely generated C(X)-module, then M ⊕ P ∼= C(X)n for some C(X)-module P and some n. So there exists f : C(X)n → C(X)n with f 2 = f. (f(C(X)) = M.) Now f corresponds to a map g : εn → εn with g2 = g, and we can check that im (g) is a vector bundle with sections Γ(im (g)) = M. (To get g, note Γ(ξ)/{s ∈ Γ | s(X) = 0} is fibre over X. So f induces map on each fibre, which is a bundle map.)

1.3 K0 for Schemes

Let X be a N¨otherianscheme. We define K0(X) to be the of the category of coherent sheaves on X (sheaves which are locally finitely generated modules). This is the group with generators [F] for each isomorphism classes of coherent sheaves on X, and relations

[F] = [F 0] + [F 00] for each short

0 → F 0 → F → F 00 → 0. Note we are not assume such sequences split in the category of coherent sheaves, by in K-theory, [F] = [F 0] + [F 00] = [F 0 ⊕ F 00] for each short exact sequence

0 → F 0 → F → F 00 → 0. This is the case originally studied by Grothendieck in his proof of the Riemann-Roch Theo- rem. Chapter 2

Higher K-theory

For a R with identity, we define

ab K1(R) := GL (∞,R)  ab = colim GL (n, R) , n

K2(R) := group of non-trivial relations among elementary matrices.

λ Recall an elementary matrix eij ∈ GL (n, Λ) has 1’s on the main diagonal, λ in the (i, j)- entry, and 0’s else where. The groups K0, K1, and K2 are related by exact sequences. For example, if Λ is a with fraction field F , there is a localisation sequence:

M M K2(F ) K1(Λ/p) K1(Λ) K1(F ) K0(Λ/p) K0(Λ) K0(F ) 0 p p

Quillen extended this to the left by defining higher K-groups (even to odd other K2-terms one needs a completely new way of thinking about K0, K1, K2).

2.1 Simplicial Sets

A X• provides a combinatorial description of a . It is a sequence of sets X0,X1,X2, ···, and structure maps

di : Xn → Xn−1,

si : Xn → Xn+1 for each i with 0 ≤ i ≤ n, satisfying the following relations

9 10 CHAPTER 2. HIGHER K-THEORY

didj = dj−1di if i < j, disj = sj−1di if i < j, djsj = dj+1sj = id, (2.1.1) disj = sjdi−1 if i > j + 1, sisj = sj+1si if i ≤ j.

We think of Xi as the collection of i-simplices of X•, with face di and degeneracy si operators. For example, given an “edge” e ∈ X1, its “endpoints” are the “vertices” d0(e), d1(e) ∈ X0. si’s correspond to repeating a in a . For instance, if v is a vertex, then (v, v) is a degenerate 1-simplex. More formally, if σ ∈ Xn is a simplex with an ordered set of vertices σ = [v0, v1, ··· , vn], then

di(σ) = [v0, v1, ··· , vˆi, ··· , vn] ∈ Xn−1, and

si(σ) = [v0, ··· , vi, vi, ··· , vn] ∈ Xn+1. From this readers can verify Equation (2.1.1) easily.

2.1.1 Canonical Example–Singular Sets Let X be any topological space, we define the singular set of X to be the simplicial set

n Sn(X) := {f : ∆ → X | f is continuous} , ( n )

n n+1 X where ∆ := (x0, ··· , xn) ∈ R 0 ≤ xi ≤ 1, xi = 1 is the standard n-simplex. To i=0 define face and degeneracy operators, we first define

n−1 n δi : ∆ → ∆ (inclusion as i-th face) , n+1 n σi : ∆ → ∆ (collapse onto i-th face) by

δi(x0, ··· , xn−1) = (x0, ··· , xi−1, 0, ··· , xn−1)

σi(x0, ··· , xn+1) = (x0, ··· , xi + xi+1, ··· , xn+1). Then define

di(f) := f ◦ δi,

si(f) := f ◦ σi. 2.1. SIMPLICIAL SETS 11

2.1.2 Formal Definition

op op A simplicial set is a functor X• : ∆ → Sets, where ∆ is the category of finite totally ordered sets and order-preserving morphisms. So if [n] := {0 < 1 < ··· < n}, then Xn := X•([n]) is the set of n-simplices, and the face and degeneracy faces are induced by the maps

δi :[n] → [n + 1] σi :[n] → [n − 1]  j if j < i,  j if j ≤ i, δ (j) = (skip i) σ (j) = (repeat i) i j + 1 if j ≥ i i j − 1 if j > i

Example 2.1.1 (The Simplicial Interval). The simplicial interval I• := ∆[1] is the simplicial set

I0 = {0, 1} ,

I1 = {[0, 1], [0, 0], [1, 1]} ,

I2 = {[0, 0, 1], [0, 1, 1]} ,

with

d0([0, 1]) = 1,

d1([0, 1]) = 0

and et cetera. For a formal definition of I•, see Example 2.2.2.

2.1.3 Geometric Realisation of a Simplicial Set

Given a simpicial set X•, we want to create a topological space ! G n |X•| := Xn × ∆ /∼ . n≥0 The relation “∼” identifies faces and collapses degeneracies:

n (τ, δi(v)) ∼ (di(τ), v) for τ ∈ Xn+1 and v ∈ ∆ , n (τ, σi(v)) ∼ (si(τ), v) for τ ∈ Xn−1 and v ∈ ∆ . 12 CHAPTER 2. HIGHER K-THEORY

Theorem 2.1.2 (Milnor).

(i)[Mil57, Theorem 1] |X•| is a CW-complex with one n-cell for each non-degenerate simplex τ ∈ Xn.

(ii)[Mil57, Theorem 2] |X• × Y•| and |X•| × |Y•| are weakly equivalent, and are home- omorphic when |X•| or |Y•| is locally finite.

Here, X• × Y• has n-simplices Xn × Yn and si, di operate coordinate-wise.

Remark 2.1.3.1. Earlier definitions of simplicial sets ignored degeneracies and hence we did not get |X• × Y•| = |X•| × |Y•|. Note that (σ, τ) ∈ Xn × Yn is degenerate only when σ and τ are both degenerate.

Remark 2.1.3.2. A map f : X• → Y• (i.e., fn : Xn → Yn commutes with di, si) defines a continuous map |f| : |X•| → |Y•| by f(xn, t) = (f(xn), t).

2.2 Classifying Spaces of a Small Category

Let C be a small category (i.e., Obj (C), MorC (X,Y ) are all sets). We think of C as a concrete combinatorial object, not as an abstraction of some entire field of math. For us, categories will serve as a mean of describing complicated spaces algebraically/combinatorially.

Definition 2.2.1 (Nerve of a Category). Let C be a small category. We construct a simplicial set BC : ∆op → Sets, called the nerve of C, by

(BC)n = BC([n]) := {A0 → · · · → An} .

For Φ ∈ Mor∆op ([n], [m]), its image (BC)(Φ) is defined as

 ϕ0 ϕn−1   ϕΦ(0) ϕΦ(n−1)  A0 −→· · · −−−→ An 7→ AΦ(0) −−−→· · · −−−−→ AΦ(n)

Face operators are given by composing morphisms, and degeneracy operators insert identity morphisms.

For instance,

 ϕ0 ϕ1 ϕ2   ϕ0 ϕ2◦ϕ1  d2 A0 −→ A1 −→ A2 −→ A3 = A0 −→ A1 −−−→ A3 ,

 ϕ0 ϕ1   ϕ0 id ϕ1  s1 A0 −→ A1 −→ A2 = A0 −→ A1 −→ A1 −→ A2 2.2. CLASSIFYING SPACES OF A SMALL CATEGORY 13

Example 2.2.2 (Formal Definition of Simplicial Interval). Let C be the category with two objects 0, 1, and

MorC (0, 1) := {0 → 1, id0, id1} . Then |BC| = ∆[1]. What is BC then? We have

BC0 = {0, 1} ,

BC1 = {0 → 1, degenerate simplices} ,

BCi = {degenerate simplices} for i > 1.  G  So |BC| = {0, 1} × ∆0 {0 → 1} × ∆1 /∼ is the following

0 1 • •

which is just the standard 1-simplex.

Proposition 2.2.3. A functor F : C → D induces a continuous map |F | : |BC| → |BD|. A natural transformation η : F → G of functors F,G : C → D induces a from |F | to |G|. Notice we assume that all functors here are covariant.

Proof. A functor F : C → D induces a simplicial map BC → BD, i.e., maps BCi onto BDi for each i which commute with all face and degeneracy maps. These maps are obvious:

F (A0 → · · · → An) = F (A0) → · · · → F (An). Now a natural transformation η : F → G is really a collection of commuting diagrams:

F (ϕ) F (A) F (B)

ηA ηB

G(A) G(B), G(ϕ)

ϕ one for each map A −→ B in C. Another description of η is that by considering the category I × C (I is the category in Example 2.2.2), we define

η˜ : I × C → D by 14 CHAPTER 2. HIGHER K-THEORY

η˜(0,A) := F (A) η˜(1,A) := G(A), ϕ η˜(id0,A −→ B) := F (ϕ), ϕ η˜(id1,A −→ B) := G(ϕ), ϕ η˜(0 → 1,A −→ B) := G(ϕ) ◦ ηA =η ˜B ◦ F (ϕ)

So we get a functorη ˜ : I × C → D, hence a map

|I × C| D

|I| × |C|

by Milnor’s Theorem 2.1.2, and I = [0, 1], so this is exactly a homotopy from |F | to |G|.

2.3 Definition of Higher K-theory

In order to define the higher K-groups of a ring, we will work with the category P (R) of

projective R-modules. Unfortunately, one cannot just consider the space BP (R) . Instead, we will need to modify the morphisms to construct a category Q P (R). Then the homotopy groups

 Ki(R) := πi+1Q P (R) . will be the higher K-groups. (I dropped |B(−)| on the left.) In particular, I will prove that

 π1Q P (R) = K0(R) = Grothendieck group.

2.3.1 The Q-construction Let P be one of the categories above (or any “”). Define Q P  to be the category of subquotients of P , that is,

Obj Q P  := Obj P  ,   j i , 0  M N 0  Mor (M,M ) := M N ∈ Obj P . Q(P )   2.3. DEFINITION OF HIGHER K-THEORY 15

So a map M → M 0 is simply a way of identifying M as a quotient (with projective kernel) of a subobject (with projective quotient) of M 0. The isomorphisms we consider are those inducing idM , idM 0 :

j i M N M 0

∼

= = =

M N M 0. j0 i0

j j0 0 i 0 0 0 i 00 The composition of M N M following M N M is given by the morphism

0 i ◦ p1 i ◦ p2 0 00 M N ×M 0 N M (2.3.1) via the diagram

M 00

i0

p2 0 0 N ×M 0 N N

0 p1 j

M N M 0. i j

One needs to check that the composition is well-defined up to isomorphism, and is associative. id id M M M Note that the identity map idM is given by , because the compo- sition

M 0

N ∼= M ×M N N

M M M

0 is isomorphic to M N M . 16 CHAPTER 2. HIGHER K-THEORY

 Definition 2.3.1 (Higher K-groups). Ki(R) := πi+1 BQ P .

I will prove that

 π1Q P (R) = K0(R) = Grothendieck group.

2.3.2 Exact Categories and the Q-construction

Definition 2.3.2. An exact category is a category C which is additive and is embedded as a full, additive of an A, closed under extension.

“Closed under extension” means if M 0, M 00 ∈ C, and

0 → M 0 → M → M 00 → 0 is exact in A, then M is in C as well. For example, projective modules are closed under extension. Free sheaves or coherent sheaves are also closed under extension.

Definition 2.3.3 (Q (C)). Q (C) is the category with objects Obj (C), and with mor- phisms from M → M 0 in Q (C) given by equivalence classes of diagrams

q i 0 M P M , where q, i are admissible (i.e., appear in exact sequence in C).    0  q i q i0 M P M 0  M P 0 M 0  Say   ∼   if there is a diagram

q i M P M 0

∼

= = =

M P 0 M 0. q0 i0

Composition of morphisms is given in the same way as in (2.3.1):

 0    q i0 q i  M 0 P 0 M 00  M P M 0   ◦   2.3. DEFINITION OF HIGHER K-THEORY 17

 00  = M P ×M 0 P M

via the diagram

M 00

i0

0 ei 0 P ×M 0 P P

0 qe0 q

M P M 0. q i

We need to check the following

0 (1) P ×M 0 P ∈ C.   Note that qe0 is surjective, because q0 is surjective, and ker qe0 ∼= ker (q0). So we have a short exact sequence

0 0 0 → ker (q ) → P ×M 0 P → P → 0, 0 and hence P ×M 0 P ∈ C.

(2) qe0, ei are admissible. Just note as above that we have short exact sequences

0 0 0 → ker (q ) → P ×M 0 P → P → 0, 0 0 0 0 → P ×M 0 P → P → M /P → 0.

q0 To see the second one, the map P 0 −→ M 0 → M 0/P has kernel

{p0 ∈ P 0 | q0(p0) ∈ i(P )} , 0 0 0 0 0 which is exactly P ×M 0 P via the map p 7→ (q (p ), p ).

(3) Composition of admissibles is admissible. Consider the following short exact sequences

q 0 → M 0 → M −→ M 00 → 0, 18 CHAPTER 2. HIGHER K-THEORY

q0 0 → N 0 → M 00 −→ N 00 → 0.

We want q0q to be admissible, i.e., we want ker (q0q) ∈ C. We have the short exact sequence

q∗ 0 → ker (q) → ker (q0q) −→ ker (q0) → 0. Similarly, if we have

i j M M 0 M 00,

then we get

ji 0 M M 00 M 00/M 0,

and M 00/M ∈ C. This is because we have

0 M 0/M M 00/M M 00/M 0 0,

and M 0/M, M 00/M 0 ∈ C by assumption.

(4) composition is well-defined on equivalence classes, is associative, and has identity (given id id by M M M ). This is easy.

2.3.3 Universal Property of the Q-construction

0 i 0 Given M M in C, let i! ∈ MorQC (M ,M) be the map

id i M 0 M 0 M.

q 00 ! 00 Given M M in C, let q ∈ MorQC (M ,M) be the map

id M 00 M M.

q i M M 0 M 00 ! Now, note that any morphism factors as i! ◦ q , because 2.3. DEFINITION OF HIGHER K-THEORY 19

M 00

i id M M 0

id

M M 0 M 0. id

Proposition 2.3.4. The operations

i 7→ i!, q 7→ q!

satisfy

(1)

(ji)! = j!i!, (pq)! = q!p!.

(2) If the diagram

i N M 0

j j0

M N 0 i0

0 0 0 ! 0 ! 0 is a bi-Cartesian (i.e., N = M ×N 0 M , and N = M ⊕N M ), then i!j = (j ) i!.

Note that if the square is Cartesian, then it is automatically co-Cartesian also.

Proof. (1) Firstly, we have 20 CHAPTER 2. HIGHER K-THEORY

M 00

j i M M 0

id id

M M M 0. id i

This proves (ji)! = j!i!. Similarly, we have

M 00

id id M 00 M 00

q q

M 0 M M. p id

This proves (pq)! = q!p!.

(2) From the diagram

M 0

i id N N

id id

M N N, j id

we get

  j i ! M N M 0 i! ◦ j =   . 2.3. DEFINITION OF HIGHER K-THEORY 21

On the other hand, the diagram

M 0

id i N M 0

j j0

M M N 0, id i0

gives

  j i 0 ! 0 M N M 0 (j ) ◦ i! =   .

! 0 ! 0 This proves i! ◦ j = (j ) ◦ i!.

Now we can state the universal property.

Theorem 2.3.5 (Universal Property of the Q-construction). Say we are given, q for each M ∈ C, an object F (M) ∈ D; and for each M i M 0 (resp. M M 0) in C, a 0 0 morphism F1(i): F (M) → F (M ) (resp. F2(q): F (M ) → F (M)). Assume the operations

i 7→ F1(i),

q 7→ F2(q)

satisfy the properties in Proposition 2.3.4. Then there exists a unique functor

QC → D M 7→ F (M),  q  0 i 00  !  M M M q 0 i! 00   7→ F (M) −→ F (M ) −→ F (M ) .

Proof. Long, but easy to check. 22 CHAPTER 2. HIGHER K-THEORY

Corollary 2.3.6. If f : C → D is an of exact categories (i.e., f is additive, and take exact sequences to exact sequences), then f induces a functor QC → QD via

f(i!) = f(i)!, f(j!) = f(j)!. Proof. We claim that the assignment

Obj (C) → Obj (QD) M 7→ f(M),   i 0 M M 7→ f(i)!,

 j  M M 0 7→ f(j)! satisfy the properties in Proposition 2.3.4. Property (1): 0 0 ii 7→ f(ii )! 0 = (f(i)f(i ))! 0 = f(i)!f(i )!, and

jj0 7→ f(jj0)! = (f(j)f(j0))! = f(j0)!f(j)!. Property (2): Suppose we have a bi-Cartesian square in C.

i N M 0

j j0

M N 0. i0

! 0 ! 0 Then we have i!j = (j ) i!. Because f is an exact functor, we have

! 0 ! 0 f(i)if(j) = f(j ) f(i )!. This is Property (2) in QD. 2.3. DEFINITION OF HIGHER K-THEORY 23  2.3.4 An Explicit Map α : K0(R) → π1 Q P  For each generator [M] ∈ K0(R), we have a canonical loop in Q P : there are two canonical maps 0 → M in Q P :

  iM 0 0 M   = (iM )! ,

  jM 0 M M !   = (jM ) .

So these give two oriented edges from 0 to M

 ! −1 ! in Q P . The loop (jM ) ◦ iM will be the image of M under α. To see that α is a homomorphism, one needs to check that

α(M ⊕ N) ' α(M) + α(N), then α defines a map on the generated by isomorphism classes, which factors through K0(R). Recall

  id i M 0 M 0 M i! =   ,

 q id  ! M 00 M M q =   .

A straight verification shows that

i! ◦ (iM 0 )! = (iM )!, ! ! ! q ◦ jM 00 = jM . So we have two 2-simplices as shown (in red): 24 CHAPTER 2. HIGHER K-THEORY

Next, consider the map u : 0 → M given by

0 i 0 M M . We claim that

! u = i! ◦ jM ! = q ◦ (iM 00 )!.

! First we compute i! ◦ jM 0

M

M 0 M 0

0 M 0 M 0

! and get u. Next, we compute q ◦ (iM 00 )!

M

0 ×M 00 M M

0 0 M 00, and see

! 0 0 ×M 00 M M q ◦ (iM 00 )! = . 2.3. DEFINITION OF HIGHER K-THEORY 25

To show the desired equality, one needs the diagram

00 0 0 ×M M M

∼ = = =

0 0 M M. (2.3.2) But we have the following commutative diagram

M 0 M

M 00, 0 (2.3.3) which is an isomorphism by exactness of

q 0 → M 0 −→i M −→ M 00 → 0. The commutativity of diagram (2.3.2), (2.3.3) are equivalent. Consequently, we have the desired equality. We then get the following space

which is a sphere minus three disks:

The arrowed loop is α(M 0) ◦ α(M 00), and is homotopic to 26 CHAPTER 2. HIGHER K-THEORY

which is α(M) as desired. Note that α is surjective. Any morphism f : M → N in QP has the form

q i M P N , ! and f = i! ◦ q .  2.3.5 Another Approach to π1 Q M  ∼   Let M be an exact category. We want to show that π1 BQ M = K0 M , where K0 M is the group with generators the isomorphism classes [M] in M, and a relation

[M] = [M 0] + [M 00] whenever there is an exact sequence

0 → M 0 → M → M 00 → 0 in M.    How do we show that π1 BQM = K0 M ? We will study the category Cov BQM of covering spaces of BQM.

Claim 1: Let F be the category of functors F : QM → Sets such that F sends all morphisms to isomorphisms, Then Cov BQM ∼= F. ∼ Claim 2: F = K0(M) − Sets   Assuming these claims, we want to see that we get π1 BQM = K0 M .    Well, π1 BQM = AutCov(BQM) BQM^ . where BQM^ is the universal cover; and    K0 M = AutK0(M)−Sets K0 M with left action . (Generally, automorphisms of G acting on itself by left multiplication, so G acting on itself by right multiplication.)  So we just need to see that BQM^ corresponds to K0 M with left action under the equivalence of categories. Let U = BQM^ . Then we know that U admits maps to every other elements of Cov BQM by the definintion of universal cover. But U is not inital. 2.3. DEFINITION OF HIGHER K-THEORY 27

  So if U corresponds to a K0 M -set X, then X too must admit maps to all other K0 M sets. This means that

   X = K0 M t K0 M t · · · t K0 M ,  because X → K0 M shows that all stabilisers are trivial in X.   But K0 M t K0 M is a coproduct in this category, while U is not. Now we want to understand the claims.

Understanding Claim 1

Given a covering p : Xe → BC, we have, for each object c in C the fibre p−1(c) ⊆ Xe. Let F (c) := p−1(c), this is a functor because a morphism c → c0 gives a path in BC from c to c0, and the Homotopy Lifting Path Property says we get (canonically) an isomorphism

F (c) = p−1(c) → p−1(c0) = F (c0). So we get a map Cov BQM → F. Note that a map

Xe Ye

q p BC

of coverings induces a map on each fibre, and clearly it gives a natural transformation of morphism inverting functors. Now, how do we reconstruct the covering Xe from F . We will build a category out of F , whose is Xe.

Definition 2.3.7 (The Category F/C). F/C is the category with objects (c, x), where c ∈ C, x ∈ F (C), with a morphism (c, x) → (c0, x0) given by a map f : c → c0 such that F (f): F (c) → F (c0) takes x to x0.

Then assuming B(F/C) is a covering space (this result is somewhat delicate), with natural map

p : B(F/C) → BC as covering map (this natural map is induced by

F (C) → C, (c, x) 7→ c.

Note that natural transformation η : F → G gives functor 28 CHAPTER 2. HIGHER K-THEORY

η F/C e G/C

C, hence induce covering transformations.) we can immediately see that the functor associated to B(F/C) is just F : The pre-image of a vertex c ∈ BC will just be all vertices (c, x) ∈ F/C, which is naturally F (c), and a map f : c → c0 will to morphisms (c, x) → (c0,F (f)(x)) in F/C, i.e., this path c → c0 will lift to the path (c, x) → (c0,F (f)(x)) yielding exactly the map F (f) from F (c) to F (c0). So we have

β Cov BQM −→Fα −→ Cov BQM with α ◦ β ' id. (So α is full, and it is also clearly faithful.) We want to explain, still, why B(F/C) is the same as Xe when F comes from Xe. But covering spaces are determined by the actions of the by dek transforma- tions, and (by simplicial approximation) this is determined by the information we have on the 1-skeleta.

Understanding Claim 2 ∼ We want to show that F = K0(M) − Sets. F models the fibres of covering spaces (over sections). We want to think of reducing BQM by modding out the maximal tree consisting of all arrows

! iM : 0 → M. ! Then all the (isomorphic) fibres are stratified and iM corresponds to the identity map. Algebraically, we consider the full subcategory F 0 ⊆ F consisting of functors F such that ! F (M) = F (0), and F (iM ) = idF (0) for all M ∈ M. 0 0 0 ! −1 ! (Any F ∈ F is isomorphic to F where F (M) = F (0), and F (u) = F (iM ) F (u)F (iM ):

F (M 0) ! F (iM 0 ) F (u)

F (0) F (M); ! F (iM )

Clearly F 0 ∈ F 0, so F 0 ⊆ F is an equivalence of categories.) 0 ∼ We need to show F = K0(M) − Sets. 0  Given F ∈ F , F (0) has a natural K0 M action given by 2.3. DEFINITION OF HIGHER K-THEORY 29

! [M] 7→ F (jM ). ! F (jM ) is by definition an isomorphism from F (0) to F (M) = F (0). To see that this map is well-defined, we proceed as in the “hands-on” construction: Given

j 0 → M 0 −→i M −→ M 00 → 0, we need

! ! ! F (jM ) = F (jM )F (jM 00 ). ! ! ! ! ! ! We saw before that i! ◦ jM 0 = j ◦ iM 00 : 0 → M, so F (i!) ◦ F (jM 0 ) = F (j ) ◦ F (iM 00 ). Actually,

i ◦ iM 0 = iM

⇒i! ◦ iM! = iM!

⇒F (i!) = id,

! ! so F (jM ) = F (j ). Now from the diagram

M

id id M M

0 M 00 M 00 id

! ! ! we get jM ◦ jM 00 = jM . Thus

! ! ! ! F (jM 0 ) ◦ F (jM 00 ) = F (j ) ◦ F (jM 00 ) ! ! = F (jM ◦ jM 00 ) ! = F (jM ).

0 This yields F → K0(M) − Sets. (We need to check that natural transformation induce equivalent maps.)  0 In the reverse direction, given a K0 M -set S, we define FS ∈ F by

FS(M) = S, 30 CHAPTER 2. HIGHER K-THEORY

! FS(i ) = idS, ! FS(j ) = action by [ker (j).

To see that this is a functor QM → F 0, we just need to use the universal property, i.e., check that

1) (action by [ker (j)]) ◦ (action by [ker (j0)]) = (action by [ker (j0j)]). This is just because

0 → ker (j) → ker (j0j) → ker (j0) → 0.

2) Given a commutative square

0 0 i M ×N M M

j0 j

M 0 N, i

we want

i0 ◦ (action by [ker (j0)]) = (action by [ker (j)]) ◦ i0. This follows because ker (j0) ∼= ker (j).

Again, equivalent maps induce natural transformations. These are inverse equivalence:

S 7→ FS 7→ FS(0) = S,

F 7→ F (0) 7→ FF (0). Chapter 3

Bloch’s Formula for Chow Groups

We want to interpret Chow groups in terms of algebraic K-theory. Given a scheme X, we ˜ can forma pre- Kp(X) via

U 7→ Kp (Γ(U, OX ); this is a presheaf because if U  V is the inclusion map, then we have restriction Γ(V, OX ) → Γ(U, OX ), which is a ring homomorphism, and hence induces a map

Kp(Γ(V, OX )) → Kp(Γ(U, OX )) as desired. ˜ Let Kp(X) be the sheafification of Kp(X).

Theorem 3.0.1 (Quillen). If X is a regular scheme of finite typ over a field, with pure dimension n, then

∼ p An−p(X) = H (X; Kp(X)).

∼ 1 Consider the case p = 1. Then we wwant to show that An−1(X) = H (X; K1(X)). We have:

Example 3.0.2. ∼ (1) An−1(X) = Pic (X). ∼ ∗ ∗ ∗ (2) K1(X) = OX (where OX (U) = Γ(U, OX ) is the collection of invertible sections). 1 ∗ ∼ (3) H (X; OX ) = Pic (X). 2 For (1), since X is regular (local rings (R, m) satisfies dimR/m m/m = dim (R)), it is also locally factorial (regular local rings are UFD’s). So

∼ An−1(X) = Pic (X) (EGA) .

31 32 CHAPTER 3. BLOCH’S FORMULA FOR CHOW GROUPS

Generally, there is a map Div(X) → Zn−1X given by

X D 7→ ordV D[V ] (order of divisor D at V ) V

(ordV D = ordV (fα), where fα is defining equation of D near V ) which descents to Pic (X) → An−1(X). To prove (2), we note that the determinant gives a map GL (Γ (X,U)) → Γ(X,U), and hence a map

ab ∗ K1(Γ (X,U)) := GL (Γ (X,U)) → Γ(X,U) , since Γ (X,U)∗ is Abelian. This is clearly functorial, hence induces a map

∗ ∗ f : K1(X) → Γ(X,U) . We need to show that f is an isomorphism. It suffices to prove this at the level of stalks. Since K-theory commutes with limits (clear for K1), we have

Kp(X)x = colim K1 (Γ (U, OX )) x∈U   = K1 colim Γ(U, OX ) x∈U

= K1(OX,x) ∗ = OX,x, ∼ ∗ where the last step follows from the general fact (below) that K1(R) = R when R is local. So the stalks are isomorphic, and in fact the map on stalks is just the determinant map

∗ K1(OX,x) → OX,x, which (below) is an isomorphism.

Lemma 3.0.3. If R is a , then the determinant map

∗ det : K1(R) → R is an isomorphism.

Proof. Clearly det is surjetive, with kernel SL(R)/E(R), where

SL(R) := colim SL (n, R) , n→∞

E(R) = subgroup generated by elementary matrices 33

= [GL(R), GL(R)]. (Whitehead Theorem)

We need to show that SL(R) is generated by elementary matrices. Let A ∈ SL(R) be given. Choose a representative (called A) in SL (n, R).

(i) In each row/column of A, soe entry lies in R − m, where m is the maximal (hence −1 −1 this element is a unit). To see this, write out A A or AA . Let A = (aij), and −1 −1 A = (˜aij). Then, since A A = I, we have

n X a˜jiaij = 1 6∈ m. i=1 −1 Hence, aij 6∈ m for some i.Likewise, since AA = I, we conclude aji ∈ m for some i. (ii) Start with the first column. We use the invertible entry to eliminate all other entries in the first column, then move the invertible entry to the diagonal. This can be done with elementary row operations. This last goes like

 0   v  7→ 1 v1 v1  v  7→ 1 . 0

So we have

 0  a00 a01 ··· a0n  0  e e ··· e A =   , 1 2 m1  . 0   . A  0

0 with ei ∈ E(R) for i = 1, ··· , m1. Now the top row of A has a unit, so we can write

 0  a00 a01 a02 ··· a0n  0 a0 0 ··· 0   11  e ··· e Ae ··· e =  0 a21  , 1 m1 m1+1 m2    . . 00   . . A  0 an1 0 and now a11 is a unit, so we use it to eliminate the ai1 for i 6= 1. Preceeding like this, we get

0 0 ei ··· ekAek+1 ··· e` = diag (a00, ··· , ann) 0 with aii 6∈ m. 34 CHAPTER 3. BLOCH’S FORMULA FOR CHOW GROUPS

To finish the proof, we multiply by matrices of the form diag 1, ··· , 1, a, a−1, 1, ··· , 1. Really, the two a’s are symmetrically placed some distance apart. We can always assume our matrices have even size, since we start in SL(R). If A ∈ SL(R) is a unit, then A = diag (a, 1, ··· , 1). In particular, a = 1. This suffices since

 B 0   IB   I 0   IB   0 −I  = 0 B−1 0 I −B−1 I 0 I I 0  1 −1   1 0   1 −1  = 0 1 1 1 0 1

Any upper/lower triangular matrices with 1’s one the diagonal is in E(R) by row- reduction.

(3) is [Har97, Exercise 4.5 on page 224].

0 Example 3.0.4. Consider the case p = 0. We want to show that An(X) = H (X, K0(X)). What is K0(X)? I claim it is the constant sheaf Z. We need to define a map of presheaves

˜ constant sheaf → K0(X), i.e., we need maps Z → K0(Γ (U, OX ) for each U. These maps always exist.

Lemma 3.0.5. Let R be a commutative ring. Then there exists a mp Z → K0(R) which sends 1 to [R].

∼ Proof. The map 1 7→ 1 induces a map Z → R, hence a map K0(Z) → K0(R). But K0(Z) = Z, by the Structure Theorem for Modules over PID. So we have

Z K0(Z) K0(R)

1 [Z] [R] as desired.

So this gives a map of presheaves, which of course induces a map of sheaves. To check that it is an isomorphism, we just need to examine the stalks, i.e., we need to show that for the local rings OX,x, K0(OX,x) = h[OX,x]i.

∼ Lemma 3.0.6. If R is a local ring, then K0(R) = Z, generated by [R]. 3.1. PRODUCT 35

Proof. We need to show that every finitely generated projective R-module P is actually free. Since P is projective, there exists an R-module Q such that P ⊕ Q ∼= Rk for some k. Now, let m ⊆ R be the . Then

(P ⊕ Q)/m(P ⊕ Q) ∼= (P/mP ) ⊕ (Q/mQ) ∼= (R/mR)k = (R/m)k .

So (P/mP ) ⊕ (Q/mQ) is a over R/m, and we can lift a of it up to P ⊕ Q yielding elements pi ∈ P , qj ∈ Q, for 1 ≤ i ≤ n, and 1 ≤ j ≤ m. ∼ k We claim {p1, ··· , pn, q1, ··· , qm} is a basis for P ⊕ Q = R . k Writing the pi, qi in therms of the standard basis for R , this becomes a question of invert- ibility of a matrix A over R. But we know A is invertible, modulo m, so det (A) 6≡ 0 (mod m). i.e., det (A) 6∈ m, i.e., det (A) is invertible.

So {p1, ··· , pn, q1, ··· , qm} is a basis, and hence P = span ({p1, ··· , pn}), Q = span ({q1, ··· , qn}) are free.

0 So K0(X) is the constant sheaf Z, which means that the global sections H (X; K0(X)) are just a copy of Z for each connected component of X. This is clearly the same as An(X).

M p M Theorem 3.0.7 (Grayson). If f : H (X; Kp(X)) → An−p(X) is the (additive) p p isomorphism given by the Bloch’s Formula, then

f(α · β) = (−1)deg(α) deg(β)f(α) · f(β), where

deg(α) = p = codim (f(α))

p for α ∈ H (X; Kp(X)).

3.1 Product

Where do products in K-theory come from? Given a map X ∧ X → X, we get a map p q πp(X) ⊗ πq(X) → πp+q(X), because two spheroids α : S → X, β : S → X will give the spheroid

p+q p q α∧β S = S ∧ S −−→ X ∧ X → X. 36 CHAPTER 3. BLOCH’S FORMULA FOR CHOW GROUPS

What one needs, essentially, is a bi-exact functor f : M ×M → M. i.e., f(A, −) and f(−,A) should be exact (another technical condition is also required). This will induce a “product”

Ω QM ∧ Ω QM → Ω QM . (Not so simple...) For example, tensor product of modules gives a bi-exact functor

⊗ : P(R) × P(R) → P(R) for the category P(R) of finitely generated projective R-module. The functor A⊗− : P(R) → M P(R) is exact, because projectives are flat (because Ai is flat if and only if Ai is flat for i every i). Actually, if we have a short exact sequence

0 → M 0 → M → M 00 → 0, with M 00 projective, then this sequence actually splits (this is one definition of projectivity, essentially), so the sequence

0 → M 0 ⊗ A → M ⊗ A → M 00 ⊗ A → 0 also splits, hence is exact. A⊗− is always right exact, so only injectivity of M 0 ⊗A → M ⊗A is in question. But the splitting s : M → M 0 satisfies si = id, so

(s ⊗ id) ◦ (i ⊗ id) = id ⊗ id, meaning i ⊗ id is injective. So A ⊗ − : P(R) → Mod(R) for any A. M The groups Ap(X) form a graded ring Ap(X) under the intersection product of p algebraic cycles (as described in Ravi’s class). We want to understand how to get this product from the K-theory side. M If R is a ring, we actually have a product on Kp(R). This is essentially induced by p tensor product on the category P(R). So we have maps

K0(R) ⊗ Kq(R) → Kp+q(R), meaning that for a scheme X, we get

Kp(X) ⊗ Kq(X) → Kp+q(X) (everything is natural, so the local products extend to maps of sheaves). This now yields a map

p+q p+q H (Kp(X) ⊗ Kq(X)) → H (Kp+q(X)) = An−(p+q)(X), so to get the desired product

p q p+q An−p(X) ⊗ An−q(X) = H (Kp(X)) ⊗ H (Kq(X)) → H (X)) = An−(p+q)(X), 3.1. PRODUCT 37 we just need to describe a map

Hp(F) ⊗ Hq(G) → Hp+q(F ⊗ G) for every sheaves F and G on a scheme X. Do this explicitly via Cechˇ . 38 CHAPTER 3. BLOCH’S FORMULA FOR CHOW GROUPS Chapter 4

Theorem A and D´evissage

One main tool for proving facts about QC is Quillen’s Theorem A. Let f : C → D be a functor between small categories. Define the over-categories f/M, M ∈ D as follows:

(1) Obj(f/M) are pairs (C, u) with u : f(C) → M.

(2) A morphism (C, u) → (C0, u0) is a map α : C → C0 in C such that the diagram

f(α) f(C) f(C0)

0 u u M

commutes.

Theorem 4.0.1 ([Qui73, Theorem A on page 93], Quillen’s Theorem A). If B (f/M) is contractible for each M ∈ D, then f induces a homotopy equivalence Bf : BC → BD.

Theorem 4.0.2 ([Qui73, Theorem 4 on page 112], D´evissage). Let B ⊆ A be Abelain categories with B closed under taking subobjects, quotients, and finite products in A. Assume that every M ∈ A has a filtration

M0 ⊂ M1 ⊂ · · · ⊂ Mn = M,

where n depends on M, with Mi/Mi−1 ∈ B. Then the inclusion functor induces a

39 40 CHAPTER 4. THEOREM A AND DEVISSAGE´

homotopy equivalence QB → QA.

Proof. By Theorem A, we just need to show that if f : QB → QA inclusion, then all the over-categories f/M, for which M ∈ A, are contractible. f/M is the category with objects (N, u), with N ∈ B, u : N → M a map in QA. N M 0 M Associate u = to the ”layer” of M,(M0,M1), where

0 M1 := im (M  M) ⊆ M, M0 := ker (M1  N) ⊆ M1 ⊆ M. ∼ Of course, M1/M0 = N ∈ B. A morphism in f/M is a commutative diagram

α N N 0

0 u u M,

which is equivalent to the following diagram

B ∈ N P N 0

0 M1 M1

M

in QB, for which

ker (M1  N) = M0, and

0 0 0 ker (M1  N ) = M0 = ker (M1  P ) . α tells us that

0 0 M0 ⊆ M0 ⊆ M1 ⊆ M1 ⊂ M. 41

So f/M is the poset of layers (M0,M1) ⊆ M, with M1/M0 ∈ B, and

0 0 0 0 (M0,M1) ≤ (M0,M1) ⇔ M0 ⊆ M0 ⊆ M1 ⊆ M1. Call this J(M). We then have filtration

M ⊆ Mn−1 ⊆ · · · ⊆ M0 with Mi/Mi−1 ∈ B. We just need to show that

∼ ∼ ∼ J(M) = J(Mn−1) = ··· = J(M0) = J(0) ' trivial category. Let i : J(M 0) → J(M) be the canonical inclusion, where M 0 ≤ M; and define

r : J(M) → J(M 0) 0 0 (M0,M1) 7→ (M0 ∩ M ,M1 ∩ M ),

s : J(M) → J(M) 0 (M0,M1) 7→ (M0 ∩ M ,M1).

Note that

0 0 0 (M1 ∩ M ) /(M0 ∩ M ) ⊆ M1/(M0 ∩ M ) 0 ⊆ (M1/M0) × (M/M ),

0 and M1/M0, M/M are in B as their product is too.

(1) r is order-preserving because

0 0 0 0 (M0,M1) ≤ (M0,M1) ⇒ M0 ⊆ M0 ⊆ M1 ⊆ M1 0 0 0 0 0 0 ⇒ M0 ∩ M ⊆ M0 ∩ M ⊆ M1 ∩ M ⊆ M1 ∩ M 0 0 0 0 0 0 ⇒ (M0 ∩ M1,M1 ∩ M ) ≤ (M0 ∩ M ,M ∩ M )

(2) s is order-preserving because

0 0 0 0 (M0,M1) ≤ (M0,M1) ⇒ M0 ⊆ M0 ⊆ M1 ⊆ M1 0 0 0 0 ⇒ M0 ∩ M ⊆ M0 ∩ M ⊆ M1 ⊆ M1 42 CHAPTER 4. THEOREM A AND DEVISSAGE´

0 0 0 More importantly, why are the quotients (M1 ∩ M )/(M0 ∩ M ), M1/(M0 ∩ M ) in B? We jbiw ri = id. We want ir ' id. But

ir ' s ' id because

0 0 0 (M0 ∩ M ,M1 ∩ M ) ≤ (M0M , M∞)

≥ (M0,M1) Index

BC, nerve of a category, 12 q!, 18 F/C, 27 admissible morphisms, 16 I•, simplicial interval, 11 0 K (X),7 D´evissage, 39 K0 for scheme,8 K0(R),7 exact category, 16 K (R), 16 i geometric realisation, 11 Q (C), 16 Sn(X), singular set of X, 10 localisation sequence,9 |X•|, geometric realisation of the simplicial set X•, 11 Milnor’s Theorem, 12 ∆n, standard n-simplex, 10 nerve of a category, 12 Γ(V ), continuous sections of V ,8 P(R), category of finitely generated ,7 projective R-module, 36 Cov BQM, category of covering spaces Quillen’s Theorem A, 39 of BQM, 26 simplicial set, 11 Kp(X), 31 ˜ singular set, 10 Kp(X), 31 Swan’s Theorem,8 f/M, over-category, 39

i!, 18 universal property of Q-construction, 21

43 44 INDEX Bibliography

[Har97] Robin Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics 52. Springer, 1997. isbn: 0387902449. [Mil57] . “The Geometric Realization of a Semi-Simplicial Complex”. In: 65.2 (1957), pages 357–362. issn: 0003486X. url: http: //www.jstor.org/stable/1969967. [Qui73] Daniel Quillen. “Higher algebraic K-theory: I”. In: Higher K-Theories: Proceedings of the Conference held at the Seattle Research Center of the Battelle Memorial Institute, from August 28 to September 8, 1972. Edited by Hyman Bass. Berlin, Heidelberg: Springer Berlin Heidelberg, 1973, pages 85–147. isbn: 978-3-540-37767- 2. doi: 10.1007/BFb0067053. url: http://dx.doi.org/10.1007/BFb0067053. [Swa62] Richard G. Swan. “Vector Bundles and Projective Modules”. In: Transactions of the American Mathematical Society 105.2 (1962), pages 264–277. issn: 00029947. url: http://www.jstor.org/stable/1993627.

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