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 Set................ 11 2.2 Classifying Spaces of a Small Category..................... 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 Universal Property 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 cover ∼ Swan's Theorem, the Q-construction, the isomorphism 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) ring with identity 1 = 1R; M be a finitely generated R-module. We say M is projective if it is a direct summand of a free module. We denote P (R) the category of projective R-modules with module homomorphisms, then the Grothendieck group 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 2 P (R) whenever P , Q 2 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 abelian group 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 ⊕ " = V ⊕ " for some n n n, where " 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: fV ! Xg ! Γ(V ) := fcontinuous sections of V g : If V is a vector bundle, then there exists W such that V ⊕ W = "n, so Γ(V ) ⊕ Γ(W ) = Γ("n) = 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 integer 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 : "n ! "n 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)=fs 2 Γ j s(X) = 0g 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 Grothendieck group 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 exact sequence 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 commutative ring R with identity, we define ab K1(R) := GL (1;R) ab = colim GL (n; R) ; n K2(R) := group of non-trivial relations among elementary matrices. λ Recall an elementary matrix eij 2 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 Dedekind domain 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 simplicial set X• provides a combinatorial description of a topological space. 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 2 X1, its \endpoints" are the \vertices" d0(e); d1(e) 2 X0. si's correspond to repeating a vertex in a simplex. For instance, if v is a vertex, then (v; v) is a degenerate 1-simplex. More formally, if σ 2 Xn is a simplex with an ordered set of vertices σ = [v0; v1; ··· ; vn], then di(σ) = [v0; v1; ··· ; v^i; ··· ; vn] 2 Xn−1; and si(σ) = [v0; ··· ; vi; vi; ··· ; vn] 2 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) := ff : ∆ ! X j f is continuousg ; ( n ) n n+1 X where ∆ := (x0; ··· ; xn) 2 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] := f0 < 1 < ··· < ng, 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 = f0; 1g ; I1 = f[0; 1]; [0; 0]; [1; 1]g ; I2 = f[0; 0; 1]; [0; 1; 1]g ; 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 jX•j := Xn × ∆ =∼ : n≥0 The relation \∼" identifies faces and collapses degeneracies: n (τ; δi(v)) ∼ (di(τ); v) for τ 2 Xn+1 and v 2 ∆ ; n (τ; σi(v)) ∼ (si(τ); v) for τ 2 Xn−1 and v 2 ∆ : 12 CHAPTER 2. HIGHER K-THEORY Theorem 2.1.2 (Milnor). (i)[Mil57, Theorem 1] jX•j is a CW-complex with one n-cell for each non-degenerate simplex τ 2 Xn. (ii)[Mil57, Theorem 2] jX• × Y•j and jX•j × jY•j are weakly equivalent, and are home- omorphic when jX•j or jY•j 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 jX• × Y•j = jX•j × jY•j. Note that (σ; τ) 2 Xn × Yn is degenerate only when σ and τ are both degenerate.