Lecture Notes on Algebraic K-Theory John Rognes April 29th 2010 Contents 1 Introduction 1 1.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Monoid structures . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 Group completion . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.8 Loop space completion . . . . . . . . . . . . . . . . . . . . . . . . 16 1.9 Grothendieck–Riemann–Roch . . . . . . . . . . . . . . . . . . . . 17 1.10 Vector fields on spheres . . . . . . . . . . . . . . . . . . . . . . . 18 1.11 Wall’s finiteness obstruction . . . . . . . . . . . . . . . . . . . . . 19 1.12 Homology of linear groups . . . . . . . . . . . . . . . . . . . . . . 22 1.13 Homology of symmetric groups . . . . . . . . . . . . . . . . . . . 24 1.14 Ideal class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.15 Automorphisms of manifolds . . . . . . . . . . . . . . . . . . . . 27 2 Categories and functors 29 2.1 Sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4 Isomorphisms and groupoids. . . . . . . . . . . . . . . . . . . . . 39 2.5 Ubiquity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.6 Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.7 Representations of groups and rings . . . . . . . . . . . . . . . . 47 2.8 Few objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.9 Few morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3 Transformations and equivalences 56 3.1 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Natural isomorphisms and equivalences . . . . . . . . . . . . . . 60 3.3 Tannaka–Krein duality . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4 Adjoint pairs of functors . . . . . . . . . . . . . . . . . . . . . . . 69 3.5 Decategorification . . . . . . . . . . . . . . . . . . . . . . . . . . 77 i CONTENTS ii 4 Universal properties 81 4.1 Initial and terminal objects . . . . . . . . . . . . . . . . . . . . . 81 4.2 Categories under and over . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Colimits and limits . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.4 Cofibered and fibered categories. . . . . . . . . . . . . . . . . . . 96 5 Homotopy theory 99 5.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2 CW complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3 Compactly generated spaces . . . . . . . . . . . . . . . . . . . . . 113 5.4 Cofibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5 The gluing lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.6 Homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.7 Weak homotopy equivalences . . . . . . . . . . . . . . . . . . . . 126 5.8 Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6 Simplicial methods 130 6.1 Combinatorial complexes . . . . . . . . . . . . . . . . . . . . . . 130 6.2 Simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.3 The role of non-degenerate simplices . . . . . . . . . . . . . . . . 147 6.4 The role of degenerate simplices. . . . . . . . . . . . . . . . . . . 156 6.5 Bisimplicial sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.6 The realization lemma . . . . . . . . . . . . . . . . . . . . . . . . 165 6.7 Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.8 Realization of fibrations . . . . . . . . . . . . . . . . . . . . . . . 168 7 Homotopy theory of categories 171 7.1 Nerves and classifying spaces . . . . . . . . . . . . . . . . . . . . 171 7.2 The bar construction . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.3 Quillen’s theorem A . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.4 Theorem A* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.5 Quillen’s theorem B . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.6 The simplex category. . . . . . . . . . . . . . . . . . . . . . . . . 185 7.7 ∞-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 8 Waldhausen K-theory 191 8.1 Categories with cofibrations . . . . . . . . . . . . . . . . . . . . . 192 8.2 Categories of weak equivalences . . . . . . . . . . . . . . . . . . . 202 8.3 The S -construction . . . . . . . . . . . . . . . . . . . . . . . . . 206 • 8.4 Algebraic K-groups. . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.5 The additivity theorem. . . . . . . . . . . . . . . . . . . . . . . . 218 8.6 Delooping K-theory . . . . . . . . . . . . . . . . . . . . . . . . . 224 8.7 The iterated S -construction . . . . . . . . . . . . . . . . . . . . 227 • 8.8 The spectrum level rank filtration. . . . . . . . . . . . . . . . . . 229 8.9 Algebraic K-theory of finite sets . . . . . . . . . . . . . . . . . . 236 9 Abelian and exact categories 240 9.1 Additive categories . . . . . . . . . . . . . . . . . . . . . . . . . . 240 9.2 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . 240 9.3 Exact categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 CONTENTS iii 10 Quillen K-theory 244 10.1 The Q-construction . . . . . . . . . . . . . . . . . . . . . . . . . . 244 10.2 The cofinality theorem . . . . . . . . . . . . . . . . . . . . . . . . 244 10.3 The resolution theorem . . . . . . . . . . . . . . . . . . . . . . . 244 10.4 The devissage theorem . . . . . . . . . . . . . . . . . . . . . . . . 244 10.5 The localization sequence . . . . . . . . . . . . . . . . . . . . . . 244 Foreword These are notes intended for the author’s algebraic K-theory lectures at the University of Oslo in the spring term of 2010. The main references for the course will be: • Daniel Quillen’s seminal paper “Higher algebraic K-theory. I” [55], sec- tions 1 though 5 or 6, including his theorems A and B concerning the homotopy theory of categories, the definition of the algebraic K-theory of an exact category using the Q-construction, the additivity, resolution, devissage and localization theorems, and probably the fundamental theo- rem; • Friedhelm Waldhausen’s foundational paper [68] “Algebraic K-theory of spaces”, sections 1.1 through 1.6 and 1.9, including the definition of the algebraic K-theory of a category with cofibrations and weak equivalences using the S -construction, the additivity, generic fibration and approxi- • mation theorems, and the relation with the Q-construction; • Saunders MacLane’s textbook “Categories for the Working Mathemati- cian” [40] on category theory, where parts of chapters I though IV, VII, VIII, XI and XII are relevant; • Allen Hatcher’s textbook “Algebraic Topology” [26] on homotopy theory, drawingonpartsofsections4.1onhigherhomotopygroups,4.Konquasi- fibrations and the appendix; • Theauthor’sPhDthesis“AspectrumlevelrankfiltrationinalgebraicK- theory” [57], for the iterated S -construction and a proof of the Barratt– • Priddy–Quillen theorem. Backgroundmaterialandotherconnectivetissuewillbeprovidedinthesenotes. As the list above shows, the selection of material may be a bit subjective. Comments and corrections are welcome—please write to [email protected] . iv Chapter 1 Introduction What is algebraic K-theory? Here is a preliminary discussion, intended to lead the way into the subject and to motivate some of the constructions involved. Such a preamble may be useful,sincemodernalgebraicK-theoryreliesonquitealargebodyoftechnical foundations, and it is easily possible to get sidetracked by developing one or more of these foundations to their fullest, such as the model category theory of simplicial sets, before reaching the natural questions to be studied by algebraic K-theory. Theremaynotevenbeacommonagreementaboutwhatthesenaturalques- tionsare. AlgebraicK-theoryisinsomesenseameetinggroundforseveralother mathematical subjects, including number theory, geometric topology, algebraic geometry, algebraic topology and operator algebras, relating to constructions like the ideal class group, Whitehead torsion, coherent sheaves, vector bundles and index theory. It is quite possible to give a course that outlines all of these neighboring subjects. However, the aim for this course will instead be to focus on algebraic K-theory itself, rather than on these applications of algebraic K-theory. In particular, we will focus directly on “higher algebraic K-theory”, the definition of which requires more categorical and homotopy theoretic subtlety than the simpleralgebraicgroupcompletionprocessthatismostimmediatelyneededfor some of the applications. After giving a first overview of the subject matter, we will therefore spend some time on necessary background, starting with category theory and contin- uing with homotopy theory. The aim is to spend the minimal amount of time on this that is needed for an honest treatment, but not less. Then we turn to the construction and fundamental theorems of higher algebraic K-theory. Here we will reverse the historical order, at least as it is visible in the published record, by first working with Waldhausen’s simplicial construction of algebraic K-theory, called the S -construction, and only later will we specialize this to • Quillen’s purely categorical construction, known as the Q-construction. Thespecializationmayturnouttoonlybeanapparentrestriction, asongo- ingworkbyClarkBarwickandtheauthorextendstheQ-constructiontoaccept ∞-categories as input, but this is work in progress. 1 CHAPTER 1. INTRODUCTION 2 1.1 Representations Manymathematicalobjectscometolifethroughtheirrepresentationsbyactions on other, simpler, mathematical objects. Historically this was very much so for groups, which were at first realized as permutation groups, with each group element acting by an invertible substitution on some fixed set. We now say that the group acts on the given set, and this gives a discrete representation of the group. Similarly, one may consider the action of a group through linear isomorphismsonavectorspace,andthisleadstothemoststandardmeaningof a representation. Concentrating on the additive structure of the vector space, we may also consider actions of rings on abelian groups, which leads to the additive representations of a ring through its module actions. [[Retractive spaces over X.]] Example 1.1.1. In more detail, given a group G with neutral element e we may consider the class of left G-sets, which are sets X together with a function G×X →X taking (g,x) to g·x = gx, such that (gh)·x = g·(h·x) and e·x = x for all g,h∈G and x∈X. These are discrete representations of groups. Example 1.1.2. Similarly, given a ring R with unit element 1 we may con- sider the class of left R-modules, which are abelian groups M together with a homomorphism R⊗M →M taking r⊗m to r·m = rm, such that (rs)·m = r·(s·m) and 1·m = m for all r,s∈R and m∈M. These are additive representations of rings. Example 1.1.3. Given a group G and a field k, we can form the group ring k[G], and a left k[G]-module M is then the same as a k-linear representation of G, since the scalar action by k ⊆ k[G] on M makes M a k-vector space. Most of the time G and k will come with topologies, and it will then be natural to focus on topological modules with continuous actions. Example 1.1.4. [[Retractive spaces over X.]] 1.2 Classification Abasicproblemistoorganize,orclassify,thepossiblerepresentationsofagiven mathematical object. This way, if such a representation appears “in nature”, perhaps arising from a separate mathematical problem or construction, then we may wish to understand how this representation fits into the classification scheme for these mathematical objects. Inthiscontext,weareusuallywillingtoviewcertainpairsofrepresentations as being equivalent for all practical purposes. For example, two G-sets X and Y,withactionfunctionsG×X →X andG×Y →Y,aresaidtobeisomorphic if there is an invertible function f: X →Y such that g·f(x)=f(g·x) in Y for all g ∈G and x∈X. Functions respecting the given G-actions in this way are said to be G-equivariant. This way any statement about the elements of X and its G-action can be translated into a logically equivalent statement about the CHAPTER 1. INTRODUCTION 3 elementsofY anditsG-action, byeverywherereplacingeachelementx∈X by the corresponding element f(x)∈Y, and likewise replacing the G-action on X by the G-action on Y. Since f is assumed to be invertible, we can equally well go the other way, replacing elements y ∈ Y by their images f−1(y) ∈ X under the inverse function f−1: Y →X. We are therefore usually really asking for a classification of all the possible mathematical objects of a given kind, up to isomorphism. That is, we are askingforanunderstandingofthecollectionofisomorphismclassesofthegiven mathematical object. Example 1.2.1. IfG={e}isthetrivial group, thenaG-setisthesamething as a set, and the classification of G-sets up to isomorphism is the same as the classification of sets up to one-to-one correspondence of their elements, i.e., up to bijection. More-or-less by definition, this classification problem is solved by the theory of cardinalities. As a key special case, if we are only interested in finite sets, then two finite sets X and Y can be put in bijective correspondence if and only if they have the same number of elements, i.e., if #X =#Y, where #X ∈N ={0,1,2,...}denotesthenon-negativeintegerobtainedbycounting 0 the elements of X. In this case the counting process establishes a one-to-one correspondencebetweentheelementsofX andtheelementsofthestandardset n={1,2,...,n} with n elements, so the classification of finite sets up to bijection is given by this identification between the collection of isomorphism classes and the set of non-negative integers. Example 1.2.2. Returning to the case of a general group G, to each G-set X and each element x∈X, we can associate a subset Gx={g·x∈X |g ∈G} of X, called the G-orbit of x∈X, and a subgroup G ={g ∈G|g·x=x} x of G, called the stabilizer subgroup of x∈X. There is a natural isomorphism f : G/G −∼=→Gx x x from the set of left cosets gG of G in G to the G-orbit of x, taking the coset x x gG to the element g ·x ∈ X. Here G/G is a left G-set, with the G-action x x G × G/G → G/G that takes (g,hG ) to ghG , and the isomorphism f x x x x x respects the G-actions, as required for an isomorphism of G-sets. Given two elements x,y ∈ X, the G-orbits Gx and Gy are either equal or disjoint, and in general the G-set X can be canonically decomposed as the disjoint union of its G-orbits. In the special case when there is only one G- orbit, so that Gx = X for some x ∈ X, we say that the G-action is transitive. To classify G-sets we first classify the transitive G-sets, and then apply this classification one orbit at a time, for general G-sets. IfX isatransitiveG-set,choosinganelementx∈X wegetanisomorphism f : G/G →Gx=X,asabove. HencewewouldliketosaythatX corresponds x x CHAPTER 1. INTRODUCTION 4 to the subgroup G ⊆ G. However, the stabilizer subgroup G will in general x x depend on the choice of element x. If y ∈ X is another element, then there is also an isomorphism f : G/G → Gy = X, so we should also say that X y y corresponds to the subgroup G . What is the relation between G and G ? y x y Well, since the G-action is transitive, we know that y ∈Gx, so there must exist an element h∈G with h·x=y. Then G =hG h−1 y x sinceg·y =y isequivalenttogh·x=h·x,hencealsoequivalenttoh−1gh∈G x or g ∈ hG h−1. Hence it is the conjugacy class (G ) of G as a subgroup of x x x G that is a well-defined invariant of the transitive G-set X. Checking a few details, the conclusion is that the classification of transitive G-sets is given by this identification with the set of conjugacy classes of subgroups of G. The inverseidentificationtakestheconjugacyclass(H)ofasubgroupH ⊆Gtothe isomorphism class of the transitive left G-set X =G/H. Forexample,ifG=C iscyclicofprimeorderp,thepossiblesubgroupsare p H ={e} and H =G, and the transitive G-sets are G/{e}∼=G and G/G∼=∗ (a one-point set). Exercise 1.2.3. Let G be a finite group. Let ConjSub(G) be the set of con- jugacy classes of subgroups of G. Show that the isomorphism classes of finite G-sets X are in one-to-one correspondence with the functions ν: ConjSub(G)→N . 0 The correspondence takes such a function ν to the isomorphism class of the G-set ν(H) X(ν)= G/H. (aH) a What about the case when G is not finite? [[Classify k-linear G-representations, at least in the semi-simple case when G is finite and #G is invertible in k. Maybe focus on k =R and C.]] Example 1.2.4. For a general ring R, the classification of all R-modules up to R-linear isomorphism is a rather complicated matter. For later purposes we are at least interested in the finitely generated free R-modules M, with are isomorphic to the finite direct sums Rn =R⊕···⊕R withncopiesofRontherighthandside,andthefinitelygeneratedprojective R- modulesP,whichariseasdirectsummandsoffinitelygeneratedfreeR-modules, so that there is a sum decomposition P ⊕Q∼=Rn of R-modules. Note that the composite R-linear homomorphism Rn ∼=P ⊕Q−p→r P −i→n P ⊕Q∼=Rn CHAPTER 1. INTRODUCTION 5 is represented by an n×n matrix B, which is idempotent in the sense that B2 =B. Projective modules are therefore related to idempotent matrices. We are also interested in finitely generated R-modules M, for which there exists a surjective R-linear homomorphism f: Rn →M This notion is most interesting for Noetherian rings R, since for such R the kernel ker(f) will also be a finitely generated R-module. [[Reference to coherence for non-Noetherian R.]] When R = k is a field, an R-module is the same as a k-vector space, and the notions of finitely generated, finitely generated free and finitely generated projective all agree with the condition of being finite dimensional. In this case the classification of finite dimensional vector spaces is given by the dimension function,establishingaone-to-onecorrespondencebetweenisomorphismclasses of finite dimensional vector spaces and non-negative integers. When R is a PID (principal ideal domain), the classification of finitely gen- erated R-modules is well known. In this case, a finitely generated R-module is projectiveifandonlyifitisfree, sofinitelygeneratedprojectiveR-modulesare classified by their rank, again a non-negative integer. When R is a Dedekind domain, e.g. the ring of integers in a number field, the classification of finitely generated projective R-modules is due [[Check]] to Ernst Steinitz, see John Milnor’s “Introduction to algebraic K-theory” [48, §1]. Every nonzero projective module P of rank n is isomorphic to a direct sum Rn−1⊕I, where I is a non-zero ideal in R, and I ∼= ΛnP is determined up to isomorphism by P. Example 1.2.5. [[For retractive spaces over X, classification up to homotopy equivalence may be more realistic than classification up to topological isomor- phism, or homeomorphism.]] 1.3 Symmetries The classification question, as posed above, only asks about the existence of isomorphisms f: X → Y between two mathematical objects X and Y. Taken in isolation, this may be the question one is principally interested in, but as we shall see, when trying to relate several such classification questions to one another, it turns out also to be useful to ask about the degree of uniqueness of such isomorphisms. After all, if X is somehow built out of X and X along 1 2 a common part X , and similarly Y is built out of Y and Y along a common 0 1 2 part Y , then we might hope that having an isomorphism f : X →Y and an 0 1 1 1 isomorphism f : X → Y will suffice to construct an isomorphism f: X → Y 2 2 2 that extends f and f . In many cases, however, this will require that both f 1 2 1 and f restrict to isomorphisms from X to Y , along the common parts, and 2 0 0 furthermore, that these restrictions agree, i.e., that f |X =f |X : X −∼=→Y . 1 0 2 0 0 0 This means that we do not just need to know that X and Y are isomorphic, 0 0 by some unknown isomorphism, but we also need to be able to compare the different possible isomorphisms connecting these two objects.