An Introduction to K-theory

An Introduction to K-theory
Eric M. Friedlander∗
Department of Mathematics, Northwestern University, Evanston, USA
Lectures given at the School on Algebraic K-theory and its Applications Trieste, 14 - 25 May 2007
LNS0823001
∗[email protected]

Contents
0 Introduction 5 1 K0(−), K1(−), and K2(−) 7 1.1 Algebraic K0 of rings . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Topological K0 . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Quasi-projective Varieties . . . . . . . . . . . . . . . . . . . . 10 1.4 Algebraic vector bundles . . . . . . . . . . . . . . . . . . . . . 12 1.5 Examples of Algebraic Vector Bundles . . . . . . . . . . . . . 13 1.6 Picard Group Pic(X) . . . . . . . . . . . . . . . . . . . . . . 14 1.7 K0 of Quasi-projective Varieties . . . . . . . . . . . . . . . . . 15 1.8 K1 of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.9 K2 of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Classifying spaces and higher K-theory 19 2.1 Recollections of homotopy theory . . . . . . . . . . . . . . . . 19 2.2 BG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Quillen’s plus construction . . . . . . . . . . . . . . . . . . . . 22 2.4 Abelian and exact categories . . . . . . . . . . . . . . . . . . 23 2.5 The S−1S construction . . . . . . . . . . . . . . . . . . . . . 24 2.6 Simplicial sets and the Nerve of a Category . . . . . . . . . . 26 2.7 Quillen’s Q-construction . . . . . . . . . . . . . . . . . . . . . 28
3 Topological K-theory 29 3.1 The Classifying space BU ×Z . . . . . . . . . . . . . . . . . . 29 3.2 Bott periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Spectra and Generalized Cohomology Theories . . . . . . . . 33 3.4 Skeleta and Postnikov towers . . . . . . . . . . . . . . . . . . 36 3.5 The Atiyah-Hirzebruch Spectral sequence . . . . . . . . . . . 37 3.6 K-theory Operations . . . . . . . . . . . . . . . . . . . . . . . 39 3.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Algebraic K-theory and Algebraic Geometry 42 4.1 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Algebraic cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3 Chow Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 Smooth Varieties . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Chern classes and Chern character . . . . . . . . . . . . . . . 51
4.6 Riemann-Roch . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Some Diﬃcult Problems 55 5.1 K∗(Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Bass Finiteness Conjecture . . . . . . . . . . . . . . . . . . . 57 5.3 Milnor K-theory . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.4 Negative K-groups . . . . . . . . . . . . . . . . . . . . . . . . 59 5.5 Algebraic versus topological vector bundles . . . . . . . . . . 60 5.6 K-theory with ﬁnite coeﬃcients . . . . . . . . . . . . . . . . . 60 5.7 Etale K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.8 Integral conjectures . . . . . . . . . . . . . . . . . . . . . . . . 63 5.9 K-theory and Quadratic Forms . . . . . . . . . . . . . . . . . 65
6 Beilinson’s vision partially fulﬁlled 65 6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2 Statement of conjectures . . . . . . . . . . . . . . . . . . . . . 66 6.3 Status of Conjectures . . . . . . . . . . . . . . . . . . . . . . . 67 6.4 The Meaning of the Conjectures . . . . . . . . . . . . . . . . 69 6.5 Etale cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.6 Voevodsky’s sites . . . . . . . . . . . . . . . . . . . . . . . . . 74
References 75
An Introduction to K-theory 5
0 Introduction
These notes are a reasonably faithful transcription of lectures which I gave in Trieste in May 2007. My objective was to provide participants of the Algebraic K-theory Summer School an overview of various aspects of algebraic K-theory, with the intention of making these lectures accessible to participants with little or no prior knowledge of the subject. Thus, these lectures were intended to be the most elementary as well as the most general of the six lecture series of our summer school. At the end of each lecture, various references are given. For example, at the end of Lecture 1 the reader will ﬁnd references to several very good expositions of aspects of algebraic K-theory which present their subject in much more detail than I have given in these lecture notes. One can view these present notes as a “primer” or a “course outline” which oﬀer a guide to formulations, results, and conjectures of algebraic K-theory found in the literature. The primary topic of each of my six lectures is reﬂected in the title of each lecture: 1. K0(−), K1(−), and K2(−) 2. Classifying spaces and higher K-theory 3. Topological K-theory 4. Algebraic K-theory and Algebraic Geometry 5. Some Diﬃcult Problems 6. Beilinson’s vision partially fulﬁlled Taken together, these lectures emphasize the connections between algebraic K-theory and algebraic geometry, saying little about connections with number theory and nothing about connections with non-commutative geometry. Such omissions, and many others, can be explained by the twin factors of the ignorance of the lecturer and the constraints imposed by the brevity of these lectures. Perhaps what is somewhat novel, especially in such brief format, is the emphasis on the algebraic K-theory of not necessarily aﬃne schemes. Another attribute of these lectures is the continual reference to topological K-theory and algebraic topology as a source of inspiration and intuition.
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We very brieﬂy summarize the content of each of these six lectures. Lecture 1 introduces low dimensional K-theory, with emphasis on K0(X), the Grothendieck group of ﬁnitely generated projective R-modules for a (commutative) ring R if SpecR = X, of topological vector vector bundles over X if X is a ﬁnite dimensional C.W. complex, and of coherent, locally free OX-modules if X is a scheme. Without a doubt, a primary goal (if not the primary goal) of K-theory is the understanding of K0. The key concept discussed in Lecture 2 is that of “homotopy theoretic group completion”, an enriched extension of the process introduced by Alexander Grothendieck of taking the group associated to a monoid. We brieﬂy consider three versions of such a group completion, all due to Daniel Quillen: the plus-construction, the S−1S-construction, and the Qconstruction. In this lecture, we remind the reader of simplicial sets, abelian categories, and the nerve of a category. The early development of topological K-theory by Michael Atiyah and Fritz Hirzebruch has been a guide for many algebraic K-theorists during the past 45 years. Lecture 3 presents some of machinery of topological K-theory (spectra in the sense of algebraic topology, the Atiyah-Hirzebruch spectral sequence, and operations in K-theory) which reappear in more recent developments of algebraic K-theory. In Lecture 4 we discuss the relationship of algebraic K-theory to the study of algebraic cycles on (smooth) quasi-projective varieties. In particular, we remind the reader of the deﬁnition of Chow groupsof algebraic cycles modulo rational equivalence. The relationship between algebraic K-theory and algebraic cycles was realized by Alexander Grothendieck when he ﬁrst introduced algebraic K-theory; indeed, algebraic K0 ﬁgures in the formulation of Grothendieck’s Riemann-Roch theorem. As we recall, one beautiful consequence of this theorem is that the Chern character from K0(X) to CH∗(X) of a smooth, quasi-projective variety X is a rational equivalence. In order to convince the intrigued reader that there remain many fundamentalquestionswhichawait solutions, wediscussinLecture5afewdiﬃcult open problems. For example, despite very dramatic progress in recent years, we still do not have a complete computation of the algebraic K-theory of the integers Z. This lecture concludes somewhat idiosyncratically with a discussion of integral analogues of famous questions formulated in terms of the “semi-topological K-theory” constructed by Mark Walker and the author. The ﬁnal lecture could serve as an introduction to Professor Weibel’s lectures on the proof of the Bloch-Kato Conjecture. The thread which orga
An Introduction to K-theory 7
nizestheeﬀortofmanymathematiciansisalistof7conjecturesbyAlexander Beilinson which proposes to explain to what extent algebraic K-theory possesses properties analogous to those enjoyed by topological K-theory. We brieﬂy discuss the status of these conjectures (all but the Beilinson-Soul´ e vanishing conjecture appear to be veriﬁed) and discuss brieﬂy the organizational features of the motivic spectral sequence. We conclude this Lecture 6, and thus our series of lectures, with a very cursory discussion of etale cohomology and Grothendieck sites introduced by Vladimir Voevodsky in his dazzling proof of the Milnor Conjecture.
1 K0(−), K1(−), and K2(−) Perhaps the ﬁrst new concept that arises in the study of K-theory, and one which recurs frequently, is that of the group completion of an abelian monoid. The basic example to keep in mind is that the abelian group of integers Z is the group completion of the monoid N of natural numbers. Recall that an abelian monoid M is a set together with a binary, associative, commutative operation + : M×M → M and a distinguished element 0∈M which serves as an identify (i.e., 0+ m = m for all m ∈ M). Then we deﬁne the group completion γ : M → M+ by setting M+ equal to the quotient of the free abelian group with generators [m],m ∈ M modulo the subgroup generated by elements of the form [m] + [n]−[m + n] and deﬁne γ : M → M+ by sending m ∈ M to [m]. We frequently refer to M+ as the Grothendieck group of M. The group completion map γ : M →M+ satisﬁes the following universalp roperty. For any homomorphism φ : M → A from M to a group A, theree xistsauniquehomomorphismφ+ : M+ →A such that φ+◦γ = φ : M → A. 1.1 Algebraic K0 of rings Thisleadsalmostimmediatelyto K-theory. Let R bearing(always assumed associative with unit, but not necessarily commutative). Recall that an (always assumed left) R-module P is said to be projective if there exists another R-module Q such that P ⊕Q is a free R-module. Deﬁnition 1.1. Let P(R) denote the abelian monoid (with respect to ⊕) of isomorphism classes of ﬁnitely generated projective R-modules. Then we deﬁne K0(R) to be P(R)+.
8 E.M. Friedlander
Warning: The group completion map γ :P(R)→ K0(R) is frequently not injective. Exercise 1.2. Verify that if j : R →S is a ring homomorphism and if P is a ﬁnitely generated projective R-module, then S⊗R P is a ﬁnitely generated projective S-module. Using the universal property of the Grothendieck group, you should also check that this construction determines j∗ : K0(R)→K0(S). Indeed, we see that K0(−) is a (covariant) functor from rings to abeliang roups. Example 1.3. If R = F is a ﬁeld, then a ﬁnitely generated F-module is just aﬁnitedimensional F-vector space. Twosuchvector spacesareisomorphicif and only if they have the same dimension. Thus,P(F)(N and K0(F) = Z. Example 1.4. Let K/Q be a ﬁnite ﬁeld extension of the rational numbers (K is said to be a number ﬁeld) and let OK ⊂ K be the ring of algebraic integers in K. Thus,O is the subring of those elements α∈K which satisfy a monic polynomial pα(x) ∈ Z[x]. Recall that OK is a Dedekind domain. The theory of Dedekind domains permits us to conclude that K0(OK) = Z⊕Cl(K) where Cl(K) is the ideal class group of K. A well-known theorem of Minkowski asserts that Cl(K) is ﬁnite for any number ﬁeld K (cf. [5]). Computing class groups is devilishly diﬃcult. We do know that there only ﬁnitely many cyclotomic ﬁelds (i.e., of the form Q(ζn) obtained by adjoining a primitive n-th root of unity to Q) with class group {1}. The smallest n with non-trivial class group is n = 23 for which Cl(Q(ζ23)) = Z/3. A check of tables shows, for example, that Cl(Q(ζ100)) = Z/65. The reader is referred to the book [4] for an accessible introduction to this important topic. The K-theory of integral group rings has several important applications in topology. For a group π, the integral group ring Z[π] is deﬁned to be the ring whose underlying abelian group is the free group on the set [g],g ∈ π and whose ring structure is deﬁned by setting [g]•[h] = [g•h]. Thus, if π is not abelian, then Z[π] is not a commutative ring. Application 1.5. Let X be a path-connected space with the homotopy type of a C.W. complex and with fundamental group π. Suppose that X is a
An Introduction to K-theory 9
retract of a ﬁnite C.W. complex. Then the Wall ﬁniteness obstruction is an element of K0(Z[π]) which vanishes if and only if X is homotopy equivalent to a ﬁnite C.W. complex.
1.2 Topological K0 We now consider topological K-theoryfor a topological space X. Thisisalso constructed as a Grothendieck group and is typically easier to compute than algebraic K-theory of a ring R. Moreover, results ﬁrst proved for topological K-theory have both motivated and helped to prove important theorems in algebraic K-theory. A good introduction to topological K-theory can be found in [1]. Deﬁnition 1.6. Let F denote either the real numbers R or the complex numbers C. An F-vector bundle on a topological space X is a continuous open surjective map p : E →X satisfying (a) For all x∈X, p−1(x) is a ﬁnite dimensional F-vector space. (b) There are continuous maps E×E → E,F×E →E which provide the vector space structure on p−1(x), all x∈X. (c) For all x∈X, there exists an open neighborhood Ux ⊂X, an F-vector space V, and a homeomorphism ψx : V ×Ux →p−1(Ux) over Ux (i.e., pr2 = p◦ψx : V ×Ux →Ux) compatible with the structure in (b). Example 1.7. Let X = S1, the circle. The projection of the M¨ obius band M to its equator p : M →S1 is a rank 1, real vector bundle over S1. Let X = S2, the 2-sphere. Then the projection p : TS2 → S2 of thet angent bundle is a non-trivial vector bundle. Let X = S2, but now view X as the complex projective line, so that points of X can be viewed as complex lines through the origin in C2 (i.e., complex subspaces of C2 of dimension 1). Then there is a natural rank 1, complex line bundle E → X whose ﬁbre above x ∈ X is the complex line parametrized by x; if E − o(X) → X denotes the result of removing the origin of each ﬁbre, then we can identify E−o(X) with C2 −{0}. Deﬁnition 1.8. Let V ectF(X) denote the abelian monoid (with respect to ⊕) of isomorphism classes of F-vector bundles of X. We deﬁne K0 top(X) = V ectC(X)+, KO0 top(X) = V ectR(X)+.
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(This deﬁnition will agree with our more sophisticated deﬁnition of topological K-theory given in a later lecture provided that the X has the homotopy type of a ﬁnite dimensional C.W. complex.) Thereasonweuseasuperscript0ratherthanasubscript0fortopological K-theoryisthatitdeterminesacontravariant functor. Namely, iff : X →Y is a continuous map of topological spaces and if p : E → Y is an F-vector bundle on Y, then pr2 : E ×Y X → X is an F-vector bundle on X. This determines f∗ : K0 top(Y)→K0 top(X). Example 1.9. Let nS2 denote the “trivial” rank n, real vector bundle over S2 (i.e., pr2 : Rn ×S2 → S2) and let TS2 denote the tangent bundle of S2. Then TS2 ⊕1S2 ( 3S2. We conclude that V ectR(S2) → KO0 top(S2) is not injective in this case. Here is one of the early theorems of K-theory, a theorem proved by Richard Swan. You can ﬁnd a full proof, for example, in [5]. Theorem 1.10. (Swan) Let F = R (respectively, = C), let X be a compact Hausdorﬀ space, and let C(X,F) denote the ring of continuous functions X →F. For any E ∈V ectF(X), deﬁne the F-vector space of global sections Γ(X,E) to be Γ(X,E) ={s : X →E continuous;p◦s = idX}. Then sending E to Γ(X,E) determines isomorphisms
KO0 top(X)→K0(C(X,R)), K0 top(X)→K0(C(X,C)).
1.3 Quasi-projective Varieties We brieﬂy recall a few basic notions of classical algebraic geometry; a good basic reference is the ﬁrst chapter of [3]. Let us assume our ground ﬁeld k is algebraically closed, so that we need only consider k-rational points. For more general ﬁelds k, we could have to consider “points with values in some ﬁnite ﬁeld extension L/k.” Recollection 1.11. Recall projective space PN, whose k-rational points are equivalence classes of N +1-tuple, *a0,...,aN+, some entry of which is non-zero. Two N +1-tuples (a0,...,aN),(b0,...,bN) are equivalent if there exists some 0,= c∈k such that (a0,...,aN) = (cb0,...,cbN).
An Introduction to K-theory 11
If F(X0,...,XN) is a homogeneous polynomial, then the zero locus Z(F)⊂PN is well deﬁned. Recall that PN is covered by standard aﬃne opens Ui = PN\Z(Xi). Recall the Zariski topology on PN, a base of open sets for which are the subsets of the form UG = PN\Z(G). Recollection 1.12. Recall the notion of a presheaf on a topological space T: a contravariant functor from the category whose objects are open subsets of T and whose morphisms are inclusions. Recall that a sheaf is a presheaf satisfying the sheaf axiom: for T compact, this axiom can be simply expressed as requiring for each pair of open subsets U,V that F(U ∪V) = F(U)×F(U∩V ) F(V). Recall the structure sheaf of “regular functions” OPN on PN, sections of OPN(U) on any open U are given by quotients P(X0,...,XN) Q(X0,...,XN) of homogeneous polynomials of the same degree satisfying the condition that Q has no zeros in U. In particular, OPN(UG) ={F(X)/Gj,j ≥0;F homgeneous of deg = j•deg(G)}. Deﬁnition 1.13. A projective variety X is a space with a sheaf of commutative ringsOX which admits a closed embedding into some PN, i : X ⊂PN, sothatOX isthequotientofthesheafOPN bytheidealsheafofthoseregular functions which vanish on X. A quasi-projective variety U is once again a space with a sheaf of commutative rings OU which admits locally a closed embedding into some PN, j : U ⊂ PN, so that the closure U ⊂ PN of U admits the structure of a projective variety and so that OU equals the restriction of OU to U ⊂U. A quasi-projective variety U is said to be aﬃne if U admits a closed embedding into some AN = PN\Z(X0) so that OU is the quotient of OAN by the sheaf of ideals which vanish on U. Remark 1.14. Any quasi-projective variety U has a base of (Zariski) open subsets which are aﬃne. Most quasi-projective varieties are neither projective nor aﬃne. There is a bijective correspondence between aﬃne varieties and ﬁnitely generated commutative k-algebras. If U is an aﬃne variety, then Γ(OU) is the corresponding ﬁnitely generated k-algebra. Conversely, if A is written
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as a quotient k[x1,...,xN] → A, then SpecA → Spec(k[x1,...,xN]) = AN is the corresponding closed embedding of the aﬃne variety SpecA. Example 1.15. Let F be a polynomial in variables X0,...,XN homogeneous of degree d (i.e., F(ca0,...,caN) = cdF(a0,...,aN). Then the zero locus Z(F)⊂PN is called a hypersurface of degree d. For example if N = 2, then Z(F) is 1-dimensional (i.e., a curve). If k = C and if the Jacobian of F does not vanish anywhere on C = Z(F) (i.e., if C is smooth), then C is a projective, smooth, algebraic curve of genus (d−1)(d−2) 2 .
1.4 Algebraic vector bundles Deﬁnition 1.16. Let X be a quasi-projective variety. A quasi-coherent sheaf F on X is a sheaf ofOX-modules (i.e., an abelian sheaf equipped with a pairingOX⊗F →F of sheaves satisfying the condition that for each open U ⊂X this pairing gives F(U) the structure of anOX(U)-module) with the property that there exists an open covering {Ui ⊂ X;i ∈ I} by aﬃne open subsets so thatF|Ui is the sheaf associated to an OX(Ui)-module Mi for each i. If each of the Mi can be chosen to be ﬁnitely generated as an OX(Ui)module, then such a quasi-coherent sheaf is called coherent. Deﬁnition 1.17. Let X be a quasi-projective variety. A coherent sheaf E on X is said to be an algebraic vector bundle if E is locally free. In other words, if there exists a (Zariski) open covering {Ui;i ∈ I} of X such that E|Ui (Oei X|Ui for each i. Remark 1.18. Ifaquasi-projective varietyisaﬃne,thenanalgebraicvector bundle on X is equivalent to a projective Γ(OX)-module. Construction 1. If M is a free A-module of rank r, then the symmetric algebra Sym• A(M) is a polynomial algebra of r generators over A and the structure map π : SpecSym• A(M) → SpecA is just the projection Ar ×S pecA → SpecA. This construction readily globalizes: if E is an algebraicv ector bundle over X, then πE : V(E)≡SpecSym• OX (E)∗ → X is locally in the Zariski topology a product projection: if {Ui ⊂ X;i ∈} is an open covering restricted to which E is trivial, then the restriction of πE above each Ui is isomorphic to the product projection Ar ×Ui → Ui. In
An Introduction to K-theory 13
the above deﬁnition of πE we consider the symmetric algebra on the dual E∗ = HomOX(E,OX), so that the association E 0→ V(E∗) is covariantly functorial. Thus, we may alternatively think of an algebraic vector bundle on X as a map of varieties πE : V(E∗) → X satisfying properties which are the algebraic analogues of the properties of the structure map of a topological vector bundle over a topological space. Remark 1.19. We should be looking at the maximal ideal spectrum of a variety over a ﬁeld k, rather than simply the k rational points, whenever k is not algebraically closed. We suppress this point, for we will soon switch to prime ideal spectra (i.e., work with schemes of ﬁnite type over k). However, we do point out that the reason it suﬃces to consider the maximal ideal spectrumrather the spectrumof all prime ideals is the validity of the Hilbert Nullstellensatz. One form of this important theorem is that the subset of maximal ideals constitute a dense subset of the space of prime ideals (with the Zariski topology) of a ﬁnitely generated commutative k-algebra.
1.5 Examples of Algebraic Vector Bundles Example 1.20. Rank 1 vector bundles OPN(k),k ∈Z on PN. The sections of OPN(j) on the basic open subset UG = PN\Z(G) are given by the formula OPN(k)(UG) = k[X0,...,XN,1/G](j) (i.e., ratios of homogeneous polynomials of total degree j). In terms of the trivialization on the open covering Ui,0 ≤ i ≤ N, thep atching functions are given by Xj i /Xj i". Γ(OPN(j)) has dimension!N+j j "if j > 0, dimension 1 if j = 0, and 0o therwise. Thus, using the fact that OPN(j)⊗OX OPN(j$) = OPN(j + j$),w e conclude that Γ(OPN(j)) is not isomorphic to Γ(OPN(j$)) provided that j$ ,= j. Proposition 1.21. (Grothendieck) Each vector bundle on P1 has a unique decomposition as a ﬁnite direct sum of copies of OP1(k),k ∈Z. Example 1.22. Serre’s conjecture (proved by Quillen and Suslin) asserts that every algebraic vector bundle on AN (or any aﬃne open subset of AN) is trivial. In more algebraic terms, every ﬁnitely generated projective k[x1,...,xn]-module is free.
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Example 1.23. Let X = Grassn,N, the Grassmann variety of n−1-planes in PN (i.e., n-dimensional subspaces of kN+1). We can embed Grassn,N as a Zariski closed subset of PM−1, where M =!N+1 n ", by sending the subspaceV ⊂ kN+1 to its n-th exterior power ΛnV ⊂ Λn(kN+1). There is a naturalr ank n algebraic vector bundle E on X provided with an embedding in thet rivial rank N + 1 dimensional vector bundle ON+1 X (in the special case n = 1, this is OPN(−1) ⊂ ON+1 PN ) whose ﬁbre above a point in X is the corresponding subspace. Of equal importance is the natural rank N −ndimensional quotient bundle Q=ON+1 PN /E.T his example readily generalizes to ﬂag varieties. Example 1.24. Let A be a commutative k-algebra and recall the module ΩA/k of Ka¨ hler diﬀerentials. These globalize to a quasi-coherent sheaf ΩX on a quasi-projective variety X over k. If X is smooth of dimension r, then ΩX is an algebraic vector bundle over X of rank r.
1.6 Picard Group Pic(X) Deﬁnition 1.25. Let X be a quasi-projective variety. We deﬁne Pic(X) to be the abelian group whose elements are isomorphism classes of rank 1 algebraic vector bundles on X (also called “invertible sheaves”). The group structure on Pic(X) is given by tensor product. So deﬁned, Pic(X) is a generalization of the construction of the Class Group (of fractional ideals modulo principal ideal) for X = SpecA with A a Dedekind domain. Example 1.26. By examining patching data, we readily verify that H1(X,O∗ X) = Pic(X) whereO∗ X isthesheaf ofabelian groupson X with sections Γ(U,O∗ X)deﬁned to be the invertible elements of Γ(U,OX) (with group structure given by multiplication). If k = C, then we have a short exact sequence of analytic sheaves of abelian sheaves on the analytic space X(C)an, 0→Z→OX exp → O∗ X →0. We useidentiﬁcation dueto Serreofanalytic and algebraic vector bundleson a projective variety. If X = C is a smooth curve, this identiﬁcation enables
An Introduction to K-theory 15
us to conclude the short exact sequence 0→Cg/Z2g →Pic(C)→H2(C,Z) since H1(C,OC) ( H0(C,ΩC) = Cg (where g is the genus of C). In particular, we conclude that for a curve of positive genus, Pic(C) is very large, having a “continuous part” (which is an abelian variety). Example 1.27. A K3 surface S over the complex numbers is characterized by the conditions that H0(S,Λ2(ΩS)) = 0 = H1 sing(S,Q). Even though the homotopy type of a smooth K3 surface does not depend upon the choice of such a surface S, the rank of Pic(S) can vary from 1 to 20. [The dimension of H2 sing(S,Q) is 22.]
1.7 K0 of Quasi-projective Varieties Deﬁnition 1.28. Let X be a quasi-projective variety. We deﬁne K0(X) to be the quotient of the free abelian group generated by isomorphism classes [E] of (algebraic) vector bundles E on X modulo the equivalence relation generated pairs ([E],[E1]+[E2]) for each short exact sequence 0→E1 →E → E2 →0 of vector bundles. Remark 1.29. Let A be a ﬁnitely generated k-algebra. Observe that every short exact sequence of projective A-modules splits. Thus, the equivalence relation deﬁning K0(A) is generated by pairs ([E1 ⊕E2],[E1]+ [E2]). Every elementofK0(A)canbewrittenas[P]−[m]forsomenon-negativeinteger m; moreover, projective modules P,Q determine the same class in K0(A) if and only ifthere exists some non-negative integer m such that P⊕Am (Q⊕Am. Proposition 1.30. K0(PN) is a free abelian group of rank N + 1. Moreover, for any k ∈Z, the invertible sheaves OPN(k),...,OPN(k+N) generate K0(PN). Proof. Oneobtainsarelationamong N+2invertiblesheavesfromtheKoszul complex on N +1 dimensional vector space V: 0→ΛN+1V ⊗S∗−N−1(V)→•••→V ⊗S∗−1(V)→ S∗(V)→k →0. One shows that the invertible sheaves OPN(j),j ∈ Z generate K0(PN)u singSerre’stheoremthatforanycoherentsheafF onPN thereexistintegers m,n > 0 and a surjective map of OPN-modules OPN(m)n →F. One way to show that the rank of K0(PN) equals N +1 is to use Chern classes.
16 E.M. Friedlander
1.8 K1 of rings So far, we have only considered degree 0 algebraic and topological K-theory. Before we consider Kn(R),n ∈ N,Kn top(X),n ∈ Z, we look explicitly at K1(R). This was ﬁrst investigated in depth in the classic book by Bass [2].
Deﬁnition 1.31. Let R be a ring (assumed associative, as always and with unit). We deﬁne K1(R) by the formula K1(R) ≡ GL(R)/[GL(R),GL(R)], where GL(R) = lim − →n GL(n,R)andwhere[GL(r),GL(R)] isthecommutators ubgroup of the group GL(R). Thus, K1(R) is the maximal abelian quotient of GL(R), K1(R) = H1(GL(R),Z). The commutator subgroup [GL(R),GL(R)] equals the subgroup E(R)⊂ GL(R)deﬁnedasthesubgroupgeneratedbyelementarymatricesEi,j,(r),r ∈ R,i ,= j (i.e., matrices which diﬀer by the identity matrix by having r in the (i,j) position). This group is readily seen to be perfect (i.e., E(R) = [E(R),E(R)]); indeed, it is an elementary exercise to verify that E(n,R) = E(R)∩GL(n,R) is perfect for n≥3. Proposition 1.32. If R is a commutative ring, then the determinant map det : K1(R)→R× from K1(R) to the multiplicative group of units of R provides a natural splitting of R× = GL(1,R)→GL(R)→ K1(R). Thus, we can write K1(R) = R××SL(R) where SL(R) = ker{det}. If R is a ﬁeld or more generally a local ring, then SK1(R) = 0.
The following theorem is not at all easy, but it does tell us that nothing surprising happens for rings of integers in number ﬁelds. Theorem 1.33. (Bass-Milnor-Serre) IfOK is the ring of integers in a number ﬁeld K, then SK1(OK) = 0.
An Introduction to K-theory 17
Application 1.34. The work of Bass-Milnor-Serre was dedicated to solving the following question: is every subgroup H ⊂ SL(OK) of ﬁnite index a “congruent subgroup” (i.e., of the form ker{SL(OK) → SL(OK/pn)} for some prime ideal p ⊂OK. The answer is yes if the number ﬁeld F admits a real embedding, and no otherwise. The Bass-Milnor-Serre theorem is complemented by the following classical result due to Dirichlet (cf. [5]). Theorem 1.35. (Dirichlet’s Theorem) Let OK be the ring of integers in a number ﬁeld K. Then O∗ K = µ(K)⊕Zr1+r2−1 where µ(K) ⊂ K denotes the ﬁnite subgroup of roots of unity and where r1 (respectively, r2) denotes the number of embeddings of K into R (resp., number of conjugate pairs of embeddings of K into C). We conclude this brief commentary on K1 with the following early application to topology. Application 1.36. Let π be a ﬁnitely generated group and consider the Whitehead group Wh(π) = K1(R)/{±g;g ∈π}. A homotopy equivalence of ﬁnite complexes with fundamental group π has an invariant (its “Whitehead torsion”) in Wh(π) which determines whether or not this is a simple homotopy equivalence (given by a chain of “elementary expansions” and “elementary collapses”). The interested reader can ﬁnd a wealth of information about K0 and K1 in the books [2] and [6].
1.9 K2 of rings One can think of K0(R) as the “stable group” of projective modules “modulo trivial projective modules” and of K1(R) of the stabilized group of automorphisms of the trivial projective module modulo “trivial automorphisms” (i.e., the elementary matrices up to isomorphism. This philosophy can be extended to the deﬁnition of K2, but has not been extended to Ki,i > 2. Namely, K2(R) can be viewed as the relations among the trivial automorphisms (i.e., elementary matrices) modulo those relations which hold universally.
18 E.M. Friedlander
Deﬁnition 1.37. Let St(R), the Steinberg group of R, denote the group generated by elements Xi,j(r),i ,= j,r ∈ R subject to the following commutator relations: [Xi,j(r),Xk,"(s)] =     1 if j ,= k,i,= ' Xi,"(rs) if j = k,i,= ' Xk,j(−rs) if j ,= k,i = ' We deﬁne K2(R) to be the kernel of the map St(R) → E(R), given by sending Xi,j(r) to the elementary matrix Ei,j(r), so that we have a short exact sequence 1→K2(R)→St(R)→E(R)→1. Proposition 1.38. The short exact sequence 1→K2(R)→St(R)→ E(R)→1 is the universal central extension of the perfect group E(R). Thus, K2(R) = H2(E(R),Z), the Schur multiplier of E(R). Proof. One can show that a universal central extension of a group E exists if andonly E isperfect. Inthiscase, agroup S mappingonto E istheuniversal central extension if and only if S is also perfect and H2(S,Z) = 0. Example 1.39. If R is a ﬁeld, then K1(F) = F×, the non-zero elements of the ﬁeld viewed as an abelian group under multiplication. By a theorem of Matsumoto, K2(F) is characterized as the target of the “universal Steinberg symbol”. Namely, K2(F) is isomorphic to the free abelian group with generators “Steinberg symbols” {a,b}, a,b∈F× and relations i. {ac,b} = {a,b}{c,b}, ii. {a,bd} = {a,b}{a,d}, iii. {a,1−a}= 1, a,= 1,= 1−a. (Steinberg relation) Observe that for a∈F×, −a = 1−a 1−a−1 , so that {a,−a}={a,1−a}{a,1−a−1}−1 ={a,1−a−1}−1 ={a−1,1−a−1}= 1. Then we conclude the skew symmetry of these symbols: {a,b}{b,a}={a,−a}{a,b}{b,a}{b,−b} ={a,−ab}{b,−ab}={ab,−ab}= 1. Milnor used this presentation of K2(F) as the starting point of his deﬁnition of the Milnor K-theory KMilnor ∗ of a ﬁeld F, discussed brieﬂy in Lecture 5.
An Introduction to K-theory 19
2 Classifying spaces and higher K-theory
2.1 Recollections of homotopy theory Much of our discussions will require some basics of homotopy theory. Two standard references are [8] and [14]. Deﬁnition 2.1. Two continuous maps f,g : X → Y between topological spaces are said to be homotopic if there exists some continuous map F : X × I → Y with F|X×{0} = f,F|X×{1} = g (where I denotes the unit interval [0,1]). If x ∈ X,y ∈ Y are chosen (“base points”), then two (“pointed”) maps f,g : (X,{x}) → (Y,{y}) are said to be homotopic if there exists some continuous map F : X ×I → Y such that F|X×{0} = f,F|X×{1} = g, and F|{x}×I = {y} (i.e., F must project {x}×I to {y}. We use the notation [(X,x),(Y,y)] to denote the pointed homotopy classes of maps from (X,x) (previously denoted (X,{x})) to (Y,{y}). We shall employ the usual notation, [X,Y] to denote homotopy classes of continuous maps from X to Y. Another basic deﬁnition is that of the homotopy groups of a topological space. Deﬁnition 2.2. For any n≥0 and any pointed space (X,x), πn(X,x)≡[(Sn,∞),(X,x)]. For n = 0, πn(X,x) is a pointed set; for n≥1, a group; for n≥2, an abelian group. If (X,x) is “nice”, then πn(X,x) ( [Sn,X]; moreover, if X is path connected, then the isomorphism class of πn(X,x) is independent of x∈X. A relative C.W. complex is a topological pair (X,A) (i.e., A is a subspace of X) such that there exists a sequence of subspaces A = X−1 ⊂ X0 ⊂•••⊂ Xn ⊂••• of X with union equal to X such that Xn is obtained from Xn−1 by “attaching” n-cells (i.e., possibly inﬁnitely many copies of the closed unit disk in Rn, where “attachment” means that the boundary of the disk is identiﬁed with its image under a continuous map Sn−1 →Xn−1 ) and such that a subset F ⊂X is closed if and only if X∩Xn ⊂Xn is closed for all n. A space X is a C.W. complex if (X,∅) is a relative C.W. complex. A pointed C.W. complex (X,x) is a relative C.W. complex for (X,{x}). C.W.complexeshavemanygoodproperties,oneofwhichisthefollowing.
20 E.M. Friedlander
Theorem 2.3. (Whitehead theorem) If f : X → Y is a continuous map of connected C.W. complexes such that f∗ : πn(X,x) → πn(Y,f(x)) is an isomorphism for all n≥1, then f is a homotopy equivalence. Moreover, C.W. complexes are quite general: If (T,t) is a pointed topological space, then there exists a pointed C.W. complex (X,x) and a continuous map g : (X,x) → (T,t) such that g∗ : π∗(X,x) → π∗(T,t) is an isomorphism. Recall that a continuous map f : X →Y is said to be a ﬁbration if it hast he homotopy lifting property: given any commutative square of continuous maps A×{0}
!!
"" X
!!
A×I "" Y then there exits a map A×I →X whose restriction to A×{0} is the upper horizontal map and whose composition with the right vertical map equals the lower horizontal map. A very important property of ﬁbrations is that if f : X → Y is a ﬁbration, then there is a long exact sequence of homotopy groups for any xo ∈X,y ∈Y: •••→ πn(f−1(y),x0)→πn(X,x0)→πn(Y,y0)→ πn−1(f−1(y),x0)→••• If f : (X,x) → (Y,y) is any pointed map of spaces, we can naturallyc onstruct a ﬁbration ˜ f : ˜ X → Y together with a homotopy equivalence X → ˜ X over Y. We denote by htyfib(f) the ﬁbre ˜ f−1(y) of ˜ f.
2.2 BG Deﬁnition 2.4. Let G be a topological group and X a topological space. Then a G-torsor over X (or principal G-bundle) is a continuous map p : E →X together with a continuous action of G on E over X such that there exists an open covering {Ui} of X homeomorphisms G×Ui → E|Ui for each i respecting G-actions (where G acts on G×Ui by left multiplication on G). Example 2.5. Assume that G is a discrete group. Then a G-torsor p : E → X is a normal covering space with covering group G. Theorem 2.6. (Milnor) Let G be a topological group with the homotopy type of a C.W. complex. There there exists a connected C.W. complex BG and a
An Introduction to K-theory 21
G-torsor π : EG → BG such that sending a continuous function X → BG to the G-torsor X ×BG EG→X over X determines a 1-1 correspondence [X,BG] & → {isom classes of G-torsors over X} Moreover, the homotopy type of BG is thereby determined; furthermore, EG is contractible.
The topology on G when considering the classifying space BG is crucial. One interesting construction one can consider is the map on classifying spaces induced by the continuous, bijective function Gδ → G where G is a topological group and Gδ is the same group but provided with the discrete topology. Corollary 2.7. If G is discrete, then π1(BG,∗) = G and πn(BG,∗) = 0 for all n > 0 (where ∗ is some choice of base point). Moreover, these properties characterize the C.W. complex BG up to homotopy type.
Proof. Sketch of proof. If n > 0, then the facts that π1(Sn) = 0 and EG is contractible imply that [Sn,BG] = {0}. The fact that π1(BG,∗) = G is classical covering space theory.
The proof of the following proposition is fairly elementary, using a standard projection resolution of Z as a Z[π]-module. Proposition 2.8. Let π be a discrete group and let A be a Z[π]-module. Then H∗(Bπ,A) = Ext∗ Z[π](Z,A)≡H∗(π,A) H∗(Bπ,A) = TorZ[π] ∗ (Z,A)≡H∗(π,A). Now, vector bundles are not G-torsors but rather ﬁbre bundles for the topological groups O(n) (respectively, U(n)) in the case of a real (resp., complex) vector bundle of rank n. Nevertheless, because O(n) (resp., U(n)) acts faithfully and transitively on Rn (resp., Cn), we can readily conclude using Theorem 2.6 [X,BO(n)] ={isom classes of real rank n vector bundles over X} [X,BU(n)] ={isom classes of complex rank n vector bundles over X}.
22 E.M. Friedlander
2.3 Quillen’s plus construction DanielQuillen’soriginaldeﬁnitionofKi(R),i > 0, wasintermsofthefollowing “Quillen plus construction”. A detailed exposition of this construction can be found in [7]. Theorem 2.9. (Plus construction) Let G be a discrete group and H ⊂ G be a perfect normal subgroup. Then there exists a C.W. complex BG+ and a continuous map γ : BG→BG+ such that ker{π1(BG)→π1(BG+)}= H and such that ˜ H∗(htyfib(γ),Z) = 0. Moreover, γ is unique up to homotopy. The classical “Whitehead Lemma” implies that the commutator subgroup [GL(R),GL(R)] of GL(R) is perfect. (One veriﬁes that an n × n elementary matrix is itself a commutator of elementary matrices provided that n≥4.) Deﬁnition 2.10. For any ring R, let γ : BGL(R)→BGL(R)+ denote the Quillen plus construction with respect to [GL(R),GL(R)] ⊂ GL(R). We deﬁne Ki(R)≡πi(BGL(R)+), i > 0. This construction is closely connected to the group completions of our ﬁrst lecture. In some sense,'n BGL(n,R) is “up to homotopy, a commutative topological monoid” and BGL(R)+×Z is a group completion in an appropriate sense. There are several technologies which have been introduced in part to justify this informal description (e.g., the “S−1S construction” discussed below). Remark 2.11. Essentiallybydeﬁnition, K1(R)asdeﬁnedintheﬁrstlecture agrees with that of Deﬁnition 2.10. Moreover, for any K1(R)-module A, H∗(BGL(R)+,A) = H∗(BGL(R),A). Moreover, one can verify that K2(R) as introduced in the ﬁrst lecture agrees with that of Deﬁnition 2.10 for any ring R by identifying this second homotopy group with the second homology group of the perfect group [GL(R),GL(R)].
An Introduction to K-theory 23
When Quillen formulated his deﬁnition of K∗(R), he also made the following fundamental computation. Indeed, this computation was a motivating factor for Quillen’s deﬁnition (cf. [10]). Theorem 2.12. (Quillen’s computation for ﬁnite ﬁelds) Let Fq be a ﬁnite ﬁeld. Then the space BGL(Fq)+ can be described as the homotopy ﬁbre of a computable map. This leads to the following computation for i > 0: Ki(Fq) = Z/qj −1 if i = 2j−1 Ki(Fq) = 0 if i = 2j. As you probably know, homotopy groups are notoriously hard to compute. So Quillen has played a nasty trick on us, giving us very interesting invariants with which we struggle to make the most basic calculations. For example, a fundamental problem which is still not fully solved is to compute Ki(Z). Early computations of higher K-groups of a ring R often proceeded by ﬁrst computing the group homology groups of GL(n,R) for n large, then relating these homology groups to the homotopy groups of BGL(R)+.
2.4 Abelian and exact categories Much of our discussion in these lectures will require the language and concepts of category theory. Indeed, working with categories will give us a method to consider various kinds of K-theories simultaneously. I shall assume that you are familiar with the notion of an abelian category. RecallthatinanabeliancategoryA,thesetofmorphismsHomA(B,C) for any A,B ∈ Obj A has the natural structure of an abelian group; moreover, for each A,B ∈ Obj A, there is an object B ⊕ C which is both a product and a coproduct; moreover, any f : A → B in HomA(A,B) has both a kernel and a cokernel. In an abelian category, we can work with exact sequences just as we do in the category of abelian groups. Example 2.13. Here are a few “standard” examples of abelian categories. • the category Mod(R) of (left) R-modules. • the category mod(R) of ﬁnitely generated R-modules (in which case we must take R to be Noetherian). • the category QCoh(X) of quasi-coherent sheaves on a variety X.
24 E.M. Friedlander
• the category Coh(X) of coherent sheaves on a Notherian variety X. Warning. ThefullsubcategoryP(R) ⊂ mod(R)isnotanabeliancategory. For example, if R = Z, then n : Z → Z is a homomorphism of projective R-modules whose cokernel is not projective and thus is not in P(Z). Deﬁnition 2.14. An exact categoryP isa fulladditive subcategory ofsome abelian category A such that (a)ThereexistssomesetS ⊂ObjAsuchthateveryA∈ObjAisisomorphic to some element of S. (b) If 0 → A1 → A2 → A3 → 0 is an exact sequence in A with both A1,A3 ∈Obj P, then A2 ∈Obj P. Anadmissiblemonomorphism(respectively, epimorphism)inPisamono-m orphism A1 → A2 (resp., A2 → A3) in P which ﬁts in an exact sequenceo f the form of (b). Deﬁnition 2.15. IfP is an exact category, we deﬁne K0(P) to be the group completion of the abelian monoid deﬁned as the quotient of the monoid of isomorphism classes of objects of P (with respect to ⊕) modulo the equivalence relation [A2]−[A1]−[A3] for every exact sequence of the form (I.5.b). Exercise 2.16. Show that K0(R) equals K0(P(R)), where P(R) is the exact category of ﬁnitely generated projective R-modules. More generally, show that K0(X) equals K0(Vect(X)), where Vect(X) ist he exact category of algebraic vector bundles on the quasi-projective variety X. Deﬁnition 2.17. Let P be an exact category in which all exact sequences split. Consider pairs (A,α) where A∈Obj P and α is an automorphism of A. Direct sums and exact sequences of such pairs are deﬁned in the obvious way. Then K1(P) is deﬁned to be the group completion of the abelian monoid deﬁned as the quotient of the monoid of isomorphism classes of such pairs modulo the relations given by short exact sequences.
2.5 The S−1S construction Recall that a symmetric monoidal category S is a (small) category with a unit object e∈S and a functor !: S×S →S which is associative and commutative up to coherent natural isomorphisms. For example, if we consider
An Introduction to K-theory 25
the category P of ﬁnitely generated projective R-modules, then the direct sum ⊕:P×P →P is associative but only commutative up to natural isomorphism. The symmetric monoidal category relevant for the K-theory of a ring R is the category Iso(P) whose objects are ﬁnitely generated projective R-modules and whose morphisms are isomorphisms between projective R-modules. Quillen’s construction of S−1S for a symmetric monoidal category S is appealing, modelling one way we would introduce inverses to form the group completion of an abelian monoid. A good reference for this is [13]. Deﬁnition 2.18. Let S be a symmetric monoidal category. The category S−1S is the category whose objects are pairs {a,b} of objects of S and whose maps from {a,b} to {c,d} are equivalence classes of compositions of the following form: {a,b} s!− → {s!a,s!b) (f,g) → {c,d} where s is some object of S, f,g are morphisms in S. Another such composition {a,b} s"!− → {s$!a,s$!b) (f",g") → {c,d} is declared to be the same map in S−1S from {a,b} to {c,d} if and only if there exists some isomorphism θ : s → s$ such that f = f$◦(θ!a),g = g$◦(θ!b). Heuristically, we view {a,b} ∈ S−1S as representing a − b, so that {s!a,s!b} also represents a−b. Moreover, we are forcing morphisms in S to be invertible in S−1S. If we were to apply this construction to the natural numbersNviewed as a discrete category with addition as the operation, then we get N−1N = Z. The following theorem of Quillen shows how the S−1S construction can provide a homotopy-theoretic group completion Theorem 2.19. (Quillen) Let S be a symmetric monoidal category with the property that for all s,t∈S the map s!−: Aut(t)→Aut(s!t) is injective. Then the natural map BS → B(S−1S) of classifying spaces (see the next section) is a homotopy-theoretic group completion. In particular, if S denotes the category whose objects are ﬁnite dimensional projective R-modules and whose maps are isomorphisms (so that BS = '[P] BAut(P)), then K(R) is homotopy equivalent to B(S−1S).
26 E.M. Friedlander
2.6 Simplicial sets and the Nerve of a Category The reader is referred to [9] for a detailed introduction to simplicial sets. Deﬁnition 2.20. The category of standard simplices, ∆, has objects n = *0,1,...,n+ indexed by n∈N and morphisms given by Hom∆(m,n) ={non-decreasing maps *0,1,...,n+→*0,1,...,m+}. The special morphisms ∂i : n-1→n (skip i); σj : n+1→n (repeat j) in∆generate (undercomposition)allthemorphismsof∆andsatisfycertain standard relations which many topologists know by heart. A simplicial set S• is a functor ∆op →(sets). In other words, S• consists of a set Sn for each n≥0 and maps di : Sn → Sn−1,sj : Sn →Sn+1 satisfying the relations given by the relations satisﬁed by ∂i,σj ∈∆. Example 2.21. Let T be a topological space. Then the singular complex Sing•T is a simplicial set. Recall that SingnT is the set of continuous maps ∆n → T, where ∆n ⊂ Rn+1 is the standard n-simplex: the subspace consisting of those points x = (x0,...,xn) with each xi ≥ 0 and(xi = 1. Since any map µ : n → m determines a (linear) map ∆n → ∆m, it also determines µ : SingmT →SingnT, so that we may easily verify that Sing•T : ∆op →(sets) is a well-deﬁned functor. Deﬁnition 2.22. (Milnor’s geometric realization functor)For any simplicial set X•, we deﬁne its geometric realization as the topological space|X.|given as follows: |X•|=) n≥0 Xn ×∆n/∼ where the equivalence relation is given by (x,µ ◦ t) ( (µ ◦ x,t) whenever x ∈ Xm,t ∈∆n,µ : n→m a map of ∆. This quotient is given the quotient topology, where each Xn×∆n is topologized as a disjoint union indexed by x∈Xn of copies of ∆n ⊂Rn+1.
An Introduction to K-theory 27
Now, simplicial sets are a very good combinatorial model for homotopy theory as the next theorem reveals. Theorem 2.23. (Homotopy category) The categories of topological spaces and simplicial sets satisfy the following relationships. • Milnor’s geometric realization functor is left adjoint to the singular functor; in other words, for every simplicial set X• and every topological space T, Hom(s.sets)(X•,Sing•T) = Hom(spaces)(|X•|,T). • For any simplicial set X•, |X•| is a C.W. complex; moreover, for any topological space T, Sing.•(T) is a particularly well behaved type of simplicial set called a Kan complex. • For any topological space T and any point t ∈ T, the adjunction morphism (|Sing•T|,t)→(T,t) induces an isomorphism on homotopy groups. • The adjunction morphisms above induce an equivalence of categories (Kan cxes)/∼hom.equiv ( (C.W. cxes)/ ∼hom.equiv . Now we can deﬁne the classifying space of a (small) category. Deﬁnition 2.24. Let C be a small category. We deﬁne the nerve NC ∈ (s.sets) to be the simplicial set whose set of n-simplices is the set of composable n-tuples of morphisms in C: NCn ={Cn γn →Cn−1 →••• γ1 →C0}. For ∂i : n-1 → n, we deﬁne di : NCn → NCn−1 to send the n-tuple Cn → ••• → C0 to that n − 1-tuple given by composing γi+1 and γi whenever 0 < i < n, by dropping γ1 → C0 if i = 0 and by dropping Cn γn → if i = n. For σj : n → n+1, we deﬁne sj : NCn → NCn+1 by repeating Cj and inserting the identity map. We deﬁne the classifying space BC of the category C to be |NC|, theg eometric realization of the nerve of C.
28 E.M. Friedlander
The reader is encouraged to consult [12] for a discussion and insight into this construction. Example 2.25. Let G be a (discrete) group and let G denote the category with a single object (denoted ∗) and with HomG(∗,∗) = G. Then BG is a model for BG (i.e., BG is a connected C.W. complex with π1(BG,∗) = G and all higher homotopy groups 0). Example 2.26. Let X be a polyhedron and let S(X) denote the category whose objects are simplices of X and maps are the inclusions of simplices. Then BS(X) can be identiﬁed with the ﬁrst barycentric subdivision of X.
2.7 Quillen’s Q-construction What are the higher K-groups of an exact category? In particular, what are the higher K-groups of a quasi-projective variety X (i.e., of the exact category Vect(X)) or more generally of a scheme? Quillen deﬁnes these in terms of another construction, the “Quillen Qconstruction.” This construction as well as many fundamental applications can be found in Quillen’s remarkable paper [11]. Deﬁnition 2.27. LetP beanexactcategory andlet QP bethecategory obtained from P by applying the Quillen Q-construction (as discussed below). Then Ki(P) = πi+1(BQP), i≥0, the homotopy groups of the geometric realization of the nerve of QP. Theorem 2.28. Let X be a scheme and let Vect(X) denote the exact category of ﬁnitely presented, locally free OX-modules. Then Ki(X) ≡ πi(Vect(X)) ≡ πi+1(BQVect(X)) agrees for i = 0 with the Grothendieck group ofVect(X) and for X = SpecA an aﬃne scheme agrees with Ki(A) = πi(BLG(A)+) provided that i > 0.
Quillen proves this theorem using the S−1S construction as an intermediary. Here is the formulation of Quillen’s Q-construction.
An Introduction to K-theory 29
Deﬁnition 2.29. Let P be an exact category. We deﬁne the category QP as follows. We set Obj QP equal to Obj P. For any A,B ∈ Obj QP, we deﬁne HomQP(A,B) ={A p " X i # B;p (resp. i) admissible epi (resp. mono)/∼} where the equivalence relation is generated by pairs A " X # B,A " X$ # B which ﬁt in a commutative diagram
A
=
!!
Xp##
!!
i ""
!!
B
=
!!
A X p$ ## i$ "" B Waldhausen in [15] gives a somewhat more elaborate construction of Quillen’s Q construction which produces “n-fold deloopings” of BQP for every n ≥0: pointed spaces Tn with the property that Ωn(Tn) is homotopy equivalent to BQP.
3 Topological K-theory
Inthislecture, wewilldiscusssomeofthemachinerywhichmakestopological K-theory both useful and computable. Not only does topological K-theory play a very important role in topology, but also it has played the most important guiding role in the development of algebraic K-theory. As general references, the books [17], [18] and [14] are recommended.
3.1 The Classifying space BU ×Z The following statements about topological vector bundles are not valid (in general) for algebraic vector bundles. These properties suggest that topological K-theory is better behaved than algebraic K-theory.
Proposition 3.1. (cf. [1]) Let T be a compact Hausdorﬀ space. If p : E → T is a topological vector bundle on T, then for some N > 0 there is a surjective map of bundles on T, (CN+1 ×T)→E.
30 E.M. Friedlander Any surjective map E → F of topological vector bundles on T admits as plitting over T. The set of homotopy classes of maps [T,BU(n)] is in natural 1-1 correspondence with the set of isomorphism classes of rank n topological vector bundles on T. Proof. The ﬁrst statement is proved using a partition of unity argument. The proof of the second statement is by establishing a Hermitian metric on E (sothat E (F⊕F⊥), which isachieved byonce again usingapartition of unity argument. To prove the last statement, one veriﬁes that if T×I →G is a homotopyr elating continuous maps f,g : T →G and if E isa topological vector bundleo n G, then f∗E ( g∗E as topological vector bundles on T. Once again, ap artition of unity argument is the key ingredient in the proof. Proposition 3.2. For any space T, the set of homotopy classes of maps [T,BU ×Z], BU = lim −→ n BUn admits a natural structure of an abelian group induced by block sum of matrices Un ×Um →Un+m. We deﬁne K0 top(T)≡[T,BU ×Z]. For any compact, Hausdorﬀ space T, K0 top(T) is naturally isomorphic to the Grothendieck group of topological vector bundles on T:
K0 top(T)( Z[iso classes of top vector bundles on T] [E] = [E1]+[E2], whenever E (E1 ⊕E2 . Proof. (External) directsumofmatrices gives a monoid structureon7nBUn which determines a (homotopy associative and commutative) H-space structure on BU ×Z which we view as the mapping telescope of the self map 7nBUn →7nBUn, BUi ×{+∈BU1}→BUi+1. The (abelian) group structure on [T,BU ×Z] is then determined. To show that this mapping telescope is actually an H-space, one must verify that it has a 2-sided identity up to pointed homotopy: one must verify that product on the left with + ∈ BU1 gives a self map of BU ×Z which is related to the identity via a base-point preserving homotopy. (Such a
An Introduction to K-theory 31
veriﬁcation is not diﬃcult, but the analogous veriﬁcation fails if we replace the topological groups Un by discrete groups GLn(A) for some unital ring A.)
Example 3.3. Since the Lie groups Un are connected, the spaces BUn are simply connected and thus
K0 top(S1) = π1(BU ×Z) = 0. It is useful to extend K0 top(−) to a relative theory which applies to pairs( T,A) of spaces (i.e., T is a topological space and A⊂T is a closed subset).I n the special case that A =∅, then T/∅= T+/+, the pointed space obtainedb y taking the disjoint union of T with a point + which we declare to be the basepoint.
Deﬁnition 3.4. If T is a pointed space with basepoint t0, we deﬁne the reduced K-theory of T by ˜ K∗ top(T)≡K∗ top(T,t0). For any pair (T,A), we deﬁne
K0 top(T,A)≡ ˜ K0 top(T/A) thereby extending our earlier deﬁnition of K0 top(T). For any n > 0, we deﬁne
Kn top(T,A)≡ ˜ K0 top(Σn(T/A)). In particular, for any n≥0, we deﬁne K−n top(T)≡K−n top(T,∅)≡ ˜ K0 top(Σn(T+)). Observe that ˜ K0 top(S∧T) = ker{K0 top(S×T)→ K0 top(S)⊕K0 top(T)}, so that (external) tensor product of bundles induces a natural pairing
K−i top(S)⊗K−j top(T)→K−i−j top (S×T).
32 E.M. Friedlander
Just to get the notation somewhat straight, let us take T to be a single point T = {t}. Then T+ = {t,+}, the 2-point space with new point + as base-point. Then Σ2(T+) is the 2-sphere S2, and thus
K−2 top({t}) = ker{K0 top(S2))→K0 top(+)}. We single out a special element, the Bott element β = [OP1(1)]−[OP1]∈K−2 top(pt)), where we have abused notation by identifying (P1)an with S2 and the images of algebraic vector bundles on P1 in K0 top((P1)an) have the same names as in K0(P1).
3.2 Bott periodicity Of fundamental importance in the study of topological K-theory is the following theorem of Raoul Bott. Recall that if (X,x) is pointed space, then the loop space ΩX is the function complex (with the compact-open topology) of continuous maps from (S1,∞) to (X,x). The loop space functor Ω(−) on pointed spaces is adjoint to the suspension functor Σ(−): there is a natural bijection Maps(Σ(X),Y) ( Maps(X,Ω(Y)) ofsets ofcontinuous, pointed (i.e, base pointpreserving)maps. Anextensive discussion of Bott periodicity can be found in [17].
Theorem 3.5. (Bott Periodicity) There are the following homotopy equivalences. • From BO×Z to its 8-fold loop space: BO×Z ∼ Ω8(BO×Z) Moreover, the homotopy groups πi(BO×Z) are given by Z, Z/2, Z/2, 0, Z, 0, 0, 0
depending upon whether i is congruent to 0,1,2,3,4,5,6,7 modulo 8.
An Introduction to K-theory 33
• From BU ×Z to its 2-fold loop space: BU × Z∼ Ω2(BU ×Z) Moreover, πi(BU ×Z) is Z if i is even and equals 0 if i is odd. Atiyah interprets this 2-fold periodicity in terms of K-theory as follows. Theorem 3.6. (Bott Periodicity) For any space T and any i≥0, multiplication by the Bott element induces a natural isomorphism
β : K−i top(T)→K−i−2 top (T). Using the above theorem, we deﬁne Ki top(X) for any topological space X and any integer i as Ki top(X), where i is 0 if i is even and i is -1 if i is odd. In particular, taking T to be a point, we conclude that ˜ K0 top(S2) = Z, generated by the Bott element. Example 3.7. Let S0 denote {∗,+} = ∗+. According to our deﬁnitions, the K-theory Ktop(∗), of a point equals the reduced K-theory of S0. In particular, for n > 0,
K−n top(∗) = ˜ K−n top(S0) = ˜ K0 top(Sn) = πn(BU). Thus, we conclude
Kn top(∗) =*Z if n is even 0 if n is odd We can reformulate this by writing
Ki top(Sn) =*Z if i+n is even 0 if i+n is odd
3.3 Spectra and Generalized Cohomology Theories Thus, both BO×Z and BU ×Z are “inﬁnite loop spaces” naturally determining Ω-spectra in the following sense. Deﬁnition 3.8. A spectrum E is a of pointed spaces {E0,E1,...}, each of which has the homotopy type of a pointed C.W. complex, together with continuous structure maps Σ(Ei)→Ei+1.
34 E.M. Friedlander The spectrum E is said to be an Ω-spectrum if the adjoint Ei →Ω(Ei+1)o feach mapisa homotopy equivalence; in other words, asequence ofpointed homotopy equivalences E0 & →ΩE1 & →Ω2E2 & →••• & →ΩnEn →••• Each spectrum E determines an Ω-spectrum ˜ E deﬁned by setting ˜ En = lim −→ j ΩjΣj−n(En). The importance of Ω-spectra is clear from the following theorem which asserts that an Ω-spectrum determines a “generalized cohomology theory”. Theorem 3.9. (cf. [14]) Let E be an Ω-spectrum. For any topological space X with closed subspace A⊂X, set hn E(X,A) = [(X,A),En], n≥0 Then (X,a) 0→ h∗ E(X,A) is a generalized cohomology theory; namely, this satisﬁes all of the Eilenberg-Steenrod axioms except that its value at a point (i.e., (∗,∅)) may not be that of ordinary cohomology: (a) h∗ E(−) is a functor from the category of pairs of spaces to graded abeliang roups. (b) for each n ≥ 0 and each pair of spaces (X,A), there is a functorial connecting homomorphism ∂ : hn E(A)→ hn+1 E (X,A). (c) the connecting homomorphisms of (b) determine long exact sequences for every pair (X,A). (d) h∗ E(−) satisﬁes excision: i.e., for every pair (X,A) and every subspace U ⊂A whose closure lies in the interior of A, h∗ E(X,A)(h∗ E(X−U,A−U). Observe that in the above deﬁnition we use the notation h∗ E(X) for h∗ E(X,∅) = h∗ E(X+,∗), where X+ is the disjoint union of X and a point ∗. Deﬁnition 3.10. The(periodic)topological K-theories KO∗ top(−), K∗ top(−)a re the generalized cohomology theories associated to the Ω-spectra given by BO×Z and BU ×Z with their deloopings given by Bott periodicity. In particular, whenever X is a ﬁnite dimensional C.W. complex, K2j top(X) = [X,BU ×Z], K2j−1 top (X) = [X,U], so that we recover our deﬁnition of K0 top(X) (and similarly KO0 top(X)).
An Introduction to K-theory 35
Letusrestrictattention to K∗ top(X)whichsuﬃcestomotivate ourfurther discussion in algebraic K-theory. (K0∗ top(X) motivates Hermetian algebraic K-theory.) There are also other interesting generalized cohomology theories (e.g., cobordism theory represented by the inﬁnite loop space MU) which play a role in algebraic K-theory, and there are also more sophisticated equivariant K-theories, none of which will we discuss in these lectures. Tensor product of vector bundles induces a multiplication K0 top(X)⊗K0 top(X)→K0 top(X) for any ﬁnite dimensional C.W. complex X. This can be generalized by observing that tensor product induces group homomorphisms U(m)×U(n)→ U(n+m) and thereby maps of classifying spaces BU(m)×BU(n)→BU(n+m). Withalittle eﬀort, onecanshowthatthesemultiplication mapsarecompatibleuptohomotopywiththestandardembeddingsU(m)⊂U(m+1),U(n)⊂ U(n+1) and thereby give us a pairing (BU ×Z)×(BU ×Z)→ BU ×Z (factoringthroughthesmashproduct). Inthisway, BU×Zhasthestructure ofan H-spacewhichinducesapairingofspectraandthusamultiplication for thegeneralized cohomology theory K∗ top(−). (Acompletely similarargumenta pplies to KO∗ top(−)). Remark: Each of the topological K-groups, K−i top(X), i ∈ N, is given as K0 top(ΣiX) where ΣiX is the ith suspension of X. On the other hand, algebraic K-groups in non-zero degree are not easily related to the algebraic K0 of some associated ring. As an example of how topological K-theory inspired even the early (very algebraic) eﬀort in algebraic K-theory we mention the following classical theorem of Hyman Bass. The analogous result in topological K-theory for rank e vector bundles over a ﬁnite dimension C.W. complex of dimension d < e can be readily proved using the standard method of “obstruction theory”. Theorem 3.11. (Bass stability theorem) Let A be a commutative, noetherian ring of Krull dimension d. Then for any two projective A-modules P,P$ of rank e > d, if [P] = [P$]∈K0(A) then P must be isomorphic to P$.
36 E.M. Friedlander
3.4 Skeleta and Postnikov towers If X is a C.W. complex then we can deﬁne its p-skeleton skp(X) for each p≥0 as the subspace of X consisting of the union of those cells of dimension ≤p. Thus, the C.W. complex can be written as the union (or colimit) of its skeleta, X = ∪pskp(X). There is a standard way to “chop oﬀ” the bottom homotopy groups of a space (or an Ω-spectrum) using an analogue of the universal covering space of a space (which “chops oﬀ” the fundamental group). Deﬁnition 3.12. Let X be a C.W. complex. For each n ≥ 0, construct a map X → X[n] by attaching cells (proceeding by dimension) to kill all homotopy groups of X above dimension n−1. Deﬁne X(n) to X, htyfib{X → X[n]}. Sodeﬁned, X(n) →X inducesanisomorphismonhomotopygroupsπi, i≥n and πj(X(n)) = 0, j ≤n. The Postinov tower of X is the sequence of spaces X ••• →X(n+1) →X(n) →••• Thus, X can be viewed as the “homotopy inverse limit” of its Postnivkov tower. Algebraic K-theory corresponds most closely the topological K-theory which is obtained by replacing the Ω-spectrum K = BU×Z by kU = bu×Z obtained by taking at stage i the ith connected cover of BU ×Z starting at stage 0. The associated generalized cohomology theory is denoted kU∗(−) and satisﬁes kUi(X) (Ki top(X), i≤0. In studying the mapping complex Mapcont(X,Y) continuous maps from a C.W. complex X to a space Y, one typically ﬁlters this mapping complex using the skeleton ﬁltration of X by its skeleta or the “coskeleton” ﬁltration of Y by its Postnikov tower. We refer to [14] for details of these complementary approaches.
An Introduction to K-theory 37
3.5 The Atiyah-Hirzebruch Spectral sequence The Atiyah-Hirzebruch spectral sequence for topological K-theory has been a strong motivating factor in recent developments in algebraic K-theory. Indeed, perhaps the fundamental criterion for motivic cohomology is that it should satisfy a relationship to algebraic K-theory strictly analogous to the relationship of singular cohomology to topological K-theory. Theorem 3.13. (Atiyah-Hirzebruch spectral sequence [16]) For any generalized cohomology theory h∗ E(−) and any topological space X, there exists a right half-plane spectral sequence of cohomological type Ep,q 2 = Hp(X,hq(∗))⇒hp+q E (X). The ﬁltration on h∗ E(X) is given by FpE∗ ∞ = ker{h∗ E(X)→h∗ E(skp(X)}. In the special case of K∗ top(−), this takes the following form Ep,q 2 = Hp(X,Z(q/2))⇒Kp+q top (X) where Z(q/2) = Z if q is even and 0 otherwise. In the special case of kU∗(−), this takes the following form Ep,q 2 = Hp(X,Z(q/2))⇒kUp+q(X) where Z(q/2) = Z if q is an even non-positive integer and 0 otherwise. Proof. There are two basic approaches to proving this spectral sequence. The ﬁrst is to assume T is a cell complex, then consider T as a ﬁltered space with Tn ⊂T the union of cells of dimension ≤n. The properties of K∗ top(−)s tated in the previous theorem give us an exact couple associated to the long exact sequences •••→⊕Kq top(Sn)(Kq top(Tn/Tn−1)→ Kq top(Tn)→Kq top(Tn−1)→ ⊕Kq+1 top (Sn)→••• where the direct sum is indexed by the n-cells of T. The second approach applies to a general space T and uses the Postnikov tower of BU ×Z. This is a tower of ﬁbrations whose ﬁbers are EilenbergMacLane spaces for the groupswhich occur as the homotopy groupsof BU× Z.
38 E.M. Friedlander
What is a spectral sequence of cohomological type? This is the data of a 2-dimensional array Ep,q r of abelian groups for each r ≥ r0 (typically, r0e quals 0, or 1 or 2; in our case r0 = 2) and homomorphisms
dp,q r : Ep,q r →Ep+r,q−r+1 r such that the next array Ep,q r+1 is given by the cohomology of these homomorphisms: Ep,q r+1 = ker{dp,q r }/im{dp−r,q+r−1 r }. To say that the spectral sequence is “right half plane” is to say Ep,q r = 0 whenever p < 0. We say that the spectral sequence converges to the abutment E∗ ∞ (in our case h∗ E(X)) if at each spot (p,q) there are only ﬁnitely many non-zero homomorphisms going in and going out and if there exists a decreasing ﬁltration {FpEn ∞} on each En ∞ so that En ∞ = + p FpEn ∞, 0 =, p FpEn ∞, FpEn ∞/Fp+1En ∞ = Ep,n−p R , R >> 0.
The Postnikov tower argument together with a knowledge of the kinvariantsofBU×ZshowsthataftertensoringwithQthisAtiyah-Hirzebruch spectral sequence collapses; in other words, that E∗,∗ 2 ⊗Q = E∗,∗ ∞ ⊗Q. Theorem 3.14. ([16]) Let X be a C.W. complex. Then there are isomorphisms kU0(X))⊗Q ( Hev(X,Q), kU−1(X)⊗Q(Hodd(X,Q). These isomorphisms are induced by the Chern character ch =i chi : K0(−) → Hev(−,Q) discussed in Lecture 4. While we are discussing spectral sequences, we should mention the following:
An Introduction to K-theory 39
Theorem 3.15. (Serre spectral sequence; cf. [14]) Let (B,b) be a connected, pointed C.W. complex. For any ﬁbration p : E → B of topological spaces with ﬁbre F = p−1(b) and for any abelian group A, there exists a convergent ﬁrst quadrant spectral sequence of cohomological type Ep,q 2 = Hp(B,Hq(F,A))⇒Hp+q(E,A) provided that π1(B,b) acts trivially on H∗(F,A). The non-existence of an analogue of the Serre spectral sequence in algebraic geometry (for cohomology theories based on algebraic cycles or algebraic K-theory) presents one of the most fundamental challenges to computations of algebraic K-groups.
3.6 K-theory Operations There are several reasons why topological K-theory has sometimes proved to be a more useful computational tool than singular cohomology. • K0 top(−)canbetorsionfree, even though Hev(−,Z)mighthavetorsion.T his is the case, for example, for compact Lie groups. • K∗ top(−) is essentially Z/2-graded rather than graded by the naturaln umbers. • K∗ top(−) has interesting cohomology operations not seen in cohomol-o gy. These operations originate from the observation that the exterior products Λi(P) of a projective module P are likewise projective modules and the exterior products Λi(E) of a vector bundle E are likewise vector bundles. AgoodintroductiontoK-theoryoperationscanbefoundintheappendix of [1]. Deﬁnition 3.16. Let X be a ﬁnite dimensional C.W. complex and E → X be a topological vector bundle of rank r. Deﬁne
λt(E) =
r i=0
[ΛiE]ti ∈ K0 top(X)[t], apolynomialwithconstantterm1andthusaninvertibleelementinK0 top(X)[[t]]. Extend this to a homomorphism λt : K0 top(X)→(1+K0 top(X)[[t]])∗,
40 E.M. Friedlander
(using the fact that λt(E ⊕F) = λt(E)•λt(F)) and deﬁne λi : K0 top(T) → K0 top(T) to be the coeﬃcient of ti of λt. For a general topological space X, deﬁne these λ operations on K0 top(X) forbydeﬁningthemﬁrstontheuniversalvectorbundlesover Grassmannians and using the functoriality of K0 top(−). In particular, J. Frank Adams introduced operations ψk(−) : K0 top(−)→K0 top(−), k > 0 (called Adams operations) which have many applications and which are similarly constructed for algebraic K-theory.
Deﬁnition 3.17. For any topological space T, deﬁne ψt(x) =i≥0 ψi(X)ti ≡rank(x)−t• d dt (logλ−t(x)) for any x∈K0 top(T). The Adams operations ψk satisfy many good properties, some of which we list below. Proposition 3.18. For any topological space T, any x,y ∈ K0 top(T), any k > 0 • ψk(x+y) = ψk(x)+ψk(y). • ψk(xy) = ψk(x)ψk(y). • ψk(ψ"(x) = ψk"(x). • chq(ψk(x)) = kqchq(x)∈H2q(T,Q). • ψp(x) is congruent modulo p to xp if p is a prime number. • ψk(x) = xk whenever x is a line bundle In particular, if E is a sum of line bundles⊕iLi, then ψk(E) =⊕((Li)k),t he k-th power sum. By the splitting principle, this property alone uniquely determines ψk. We introduce further operations, the γ-operations on Ktop 0 (T).
An Introduction to K-theory 41
Deﬁnition 3.19. For any topological space T, deﬁne γt(x) =i≥0 γi(X)ti ≡λt/1−t(x) for any x∈K0 top(T). Basic properties of these γ-operations include the following 1. γt(x+y) = γt(x)γt(y) 2. γ([L]−1) = 1+t([L]−1). 3. λs(x) = γs/1+s(x) Using these γ operations, we deﬁnethe γ ﬁltration on K0 top(T) as follows. Deﬁnition 3.20. For any topological space T, deﬁne Kγ,1 top(T) as the kernel of the rank map Kγ,1 top(T)≡ker{rank : K0 top(T)→K0 top(π0(T))}. For n > 1, deﬁne
K0 top(T)γ,n ⊂Kγ,0 top(T)≡K0 top(T) to be the subgroup generated by monomials γi1(x1)•••γik(xk) with(j ij ≥ n,xi ∈Kγ,1 top(T). 3.7 Applications We can use the Adams operations and the γ-ﬁltration to describe in the following theorem the relationship between K0 top(T), a group which has no natural grading, and the graded group Hev(T,Q). Theorem 3.21. Let T be a ﬁnite cell complex. Then for any k > 0, ψk restricts to a self-map of each Kγ,n top (T) and satisﬁes the property that it induces multiplication by kn on the quotient ψk(x) = kn •x, x∈Kγ,n top(T)/Kγ,n+1 top (T)). Furthermore, the Chern character ch induces an isomorphism chn : Kγ,n top (T)/Kγ,n+1 top (T))⊗Q(H2n(T,Q).
42 E.M. Friedlander
In particular, the preceding theorem gives us a K-theoretic way to deﬁne the grading on K0 top(T)⊗Q induced by the Chern character isomorphism.T he graded piece of (the associated graded of) K0 top(T)⊗Q correspondingt o H2n(T,Q) consists of those classes x for which ψk(x) = knx for some (or all) k > 0. Here is a short list of famous theorems of Adams using topological Ktheory and Adams operations: Application 3.22. Adams used his operations in topological K-theory to solve fundamental problems in algebraic topology. Examples include: • Determination of the number of linearly independent vector ﬁelds on the n-sphere Sn for all n > 1. • Determination of the only dimensions (namely, n = 1,2,4,8) for which Rn admits the structure of a division algebra. (The examples of the real numbers R, the complex numbers C, the quaternions, and the Cayley numbers gives us structures in these dimensions.) • Determination of those (now well understood) elements of the homotopy groups of spheres associated with KO0 top(Sn).
4 Algebraic K-theory and Algebraic Geometry
4.1 Schemes Although our primary interest will be in the K-theory of smooth, quasiprojective algebraicvarieties, forcompletenesswebrieﬂyrecall themoregeneral context of schemes. (A good basic reference is [3].) A quasi-projective variety corresponds to a globalization of a ﬁnitely generated commutative algebra over a ﬁeld; a scheme similarly corresponds to the globalization of a general commutative ring. Recall that if A is a commutative ring we denote by SpecA the set of prime ideals of A. The set X = SpecA is provided with a topology, the Zariski topology deﬁned as follows: a subset Y ⊂ X is closed if and only if there exists some ideal I ⊂ A such that Y = {p ∈ X;I ⊂ p}. We deﬁne the structure sheafOX of commutative rings on X = SpecA by specifying its value on the basic open set Xf ={p ∈ SpecA,f / ∈ p} for some f ∈ A tob e the ring Af obtained from A by adjoining the inverse to f. (Recall that A → Af sends to 0 any element a ∈ A such that fn •a = 0 for some n).
An Introduction to K-theory 43
We now use the sheaf axiom to determine the value of OX on any arbitrary open set U ⊂X, for any such U is a ﬁnite union of basic open subsets. The stalk OX,p of the structure sheaf at a prime ideal p ⊂ A is easily computed to be the local ring Ap ={f / ∈p}−1A.T hus, (X = SpecA,OX) has the structure of a local ringed space: at opological space with a sheaf of commutative rings each of whose stalks is a local ring. A map of local ringed spaces f : (X,OX)→(Y,OY ) is the data of a continuous map f : X → Y of topological spaces and a map of sheaves OY →f∗OX on Y, where f∗OX(V) = OX(f−1(V)) for any open V ⊂Y. If M is an A-module for a commutative ring A, then M deﬁnes a sheaf ˜ M of OX-modules on X = SpecA. Namely, for each basic open subset Xf ⊂ X, we deﬁne ˜ M(Xf) ≡ Af ⊗A M. This is easily seen to determinea sheaf of abelian groups on X with the additional property that for every open U ⊂X, ˜ M(U) is a sheaf of OX(U)-modules with structure compatiblew ith restriction to smaller open subsets U$ ⊂U. Deﬁnition 4.1. Alocalringedspace(X,OX)issaidtobeanaﬃne scheme if it is isomorphic (as local ringed spaces) to (X = SpecA,OX) as deﬁned above. A scheme (X,OX) is a local ringed space for which there exists a ﬁnite open covering {Ui}i∈I of X such that each (Ui,OX|Ui) is an aﬃne scheme. If k is a ﬁeld, a k-variety is a scheme (X,OX) with the property therei s a ﬁnite open covering {Ui}i∈I by aﬃne schemes with the property thate ach (Ui,OX|Ui)((SpecAi,OSpecAi) with Ai a ﬁnitely generated k-algebraw ithout nilpotents. The (SpecAi,OSpecAi) are aﬃne varieties admitting al ocally closed embedding in PN, where N +1 is the cardinality of some set of generators of Ai over k. Example 4.2. TheschemeP1 Z isanon-aﬃnescheme deﬁnedbypatching togethertwocopiesoftheaﬃneschemeSpecZ[t]. SoP1 Z hasacovering{U1,U2}c orresponding to rings A1 = Z[u],A2 = Z[v]. These are “patched together” by identifying the open subschemes Spec(A1)u ⊂ SpecA1, Spec(A2)v ⊂ SpecA2 via the isomorphism of rings (A1)u ((A2)v which sends u to v−1. Note that we have used SpecR to denote the local ringed space (SpecR, OSpecR); we will continue to use this abbreviated notation. Deﬁnition 4.3. Let (X,OX) be a scheme. We denote by Vect(X) the exact category of sheaves F of OX-modules with the property that there exists an open covering {Ui} of X by aﬃne schemes Ui = SpecAi and free,
44 E.M. Friedlander
ﬁnitely generated Ai-modules Mi such that the restriction F|Ui of F to Ui is isomorphic to the sheaf ˜ Mi on SpecAi. In other words,Vect(X) is the exactc ategory of coherent, locally free OX-modules (i.e., of vector bundles over X). We deﬁne the algebraic K-theory of the scheme X by setting K∗(X) = K∗(Vect(X)). 4.2 Algebraic cycles For simplicity, we shall typically restrict our attention to quasi-projective varieties. In some sense, the most intrinsic objects associated to an algebraic variety are the (algebraic) vector bundles E → X and the algebraic cycles Z ⊂X on X. As we shall see, these are closely related. Deﬁnition 4.4. Let X be a scheme. An algebraic r-cycle on X if a formal sum Y nY [Y], Y irreducible of dimension r, nY ∈Z with all but ﬁnitely many nY equal to 0. Equivalently, an algebraic r-cycle is a ﬁnite integer combination of (not necessarily closed) points of X of dimension r. (This is a good deﬁnition even for X a quite general scheme.) If Y ⊂ X is a reduced subscheme each of whose irreducible components Y1,...,Ym is r-dimensional, then the algebraic r-cycle
Z =
m i=1
[Yi]
is called the cycle associated to Y. The group of (algebraic) r-cycles on X will be denoted Zr(X). For example, if X is an integral variety of dimension d (i.e., the ﬁeld of fractions of X has transcendence d over k), then a Weil divisor is an algebraic d−1-cycle. In the following deﬁnition, we extend to r-cycles the equivalence relation we impose on locally principal divisor when we consider thesemoduloprincipaldivisors. Asmotivation, observethat if C isasmooth curve and f ∈frac(C), then f determines a morphism f : C →P1 and (f) = f−1(0)−f−1(∞), where f−1(0),f−1(∞) are the scheme-theoretic ﬁbres of f above 0,∞.
An Introduction to K-theory 45
Deﬁnition 4.5. Two r-cycles Z,Z$ on a quasi-projective variety X are said to be rationally equivalent if there exist algebraic r+1-cycles W0,...,Wn on X ×P1 for some n > 0 with the property that each component of each Wi projectsonto anopensubvariety ofP1 andthat Z = W0[0],Z$ = Wn[∞], and Wi[∞] = Wi+1[0]for0≤i < n. Here, Wi[0](respectively, Wi[∞]denotesthe cycle associated to the scheme theoretic ﬁbre above 0∈P1 (resp., ∞∈P1) of the restriction of the projection X×P1 →P1 to (the components of) Wi. The Chow group CHr(X) is the group of r-cycles modulo rational equivalence.
Observethat intheabove deﬁnitionwe can replace therole of r+1-cycles on X ×P1 and their geometric ﬁbres over 0,∞ by r + 1-cycles on X ×U for any non-empty Zariski open U ⊂ X and geometric ﬁbres over any two k-rational points p,q ∈U. Remark 4.6. Given some r +1 dimensional irreducible subvariety V ⊂ X together with some f ∈ k(V), we may deﬁne (f) =(S ordS(f)[S] where S runsthrough the codimension 1 irreducible subvarieties of V. Here, ordS(−) is the valuation centered on S if V is regular at the codimension 1 point corresponding to S; more generally, ordS(f) is deﬁned to be the length of the OV,S-module OV,S/(f). We readily check that (f) is rationally equivalent to 0: namely, we associate to (V,f) the closure W = Γf ⊂ X ×P1 of the graph of the rational map V $$%P1 determined by f. Then (f) = W[0]−W[∞]. Conversely, givenan r+1-dimensionalirreduciblesubvariety W on X×P1w hich maps onto P1, the composition W ⊂ X ×P1 pr2 → P1 determines f ∈ frac(W) such that (f) = W[0]−W[∞]. Thus, the deﬁnition of rational equivalence on r-cycles of X can be given in terms of the equivalence relation generated by {(f),f ∈frac(W);W irreducible of dimension r +1} In particular, we conclude that the subgroup of principal divisors inside the group of all locally principal divisors consists precisely of those locally principal divisors which are rationally equivalent to 0.
Thereader isreferred to the beginningof [20]for a discussion ofalgebraic cycles and equivalence relations on cycles.
46 E.M. Friedlander
4.3 Chow Groups One should view CH∗(X) as a homology/cohomology theory. These groups are covariantly functorial for proper maps f : X → Y and contravariantly functorial for ﬂat maps W →X, so that they might best be viewed as some sort of Borel-Moore homology theory.
Construction 1. Assume that X is integral and regular in codimension 1. Let L ∈ Pic(X) be a locally free sheaf of rank 1 (i.e., a “line bundle” or “invertible sheaf”) and assume that Γ(L) ,= 0. Then any 0 ,= s ∈ Γ(L) determines a well deﬁned locally principal divisor on X, Z(s)⊂X. Namely, if L|U ( OX|U is trivial when restricted to some open U ⊂ X, then sU ∈ L(U) determines an element of OX(U) well deﬁned up to a unit in OX(U) (i.e., an element of O∗ X(U)) so that the valuation vx(s) is well deﬁned for every x ∈ U(1). We deﬁne Z(s) by the property that Z(s)U = (sU)|U for any open U ⊂ X restricted to which L is trivial, and where (sU) denotes the divisor of an element of OX(U) corresponding to sU under any (OX)|U isomorphism L|U ((OX)|U. Theorem 4.7. (cf. [3]) Assume that X is an integral variety regular in codimension 1. Let D(X) denote the group of locally principal divisors on X modulo principal divisors. Then the above construction determines a well deﬁned isomorphism Pic(X) ( D(X). Moreover, ifOX,x is a unique factorization domain for every x∈X, then D(X) equals the group CH1(X) of codimension 1 cycles modulo rational equivalence. Proof. If s,s$ ∈ Γ(L) are non-zero global sections, then there exists some f ∈ K = frac(OX) such that with respect to any trivialization of L on some open covering {Ui ⊂ X} of X the quotient of the images of s,s$ in OX(Ui) equals f. A line bundleL is trivial if and only if it is isomorphic to OX which is the case if and only if it has a global section s ∈ Γ(X) which never vanishes if and only if (s) = 0. IfL1,L2 are two such line bundles with non-zero global sections s1,s2, then (s1 ⊗s2) = (s1)+(s2). Thus, the map is a well deﬁned homomorphism on the monoid of those line bundles with a non-zero global section. By Serre’s theorem concerning coherent sheaves generated by global sections, for any line bundle L there exists a positive integer n such that L⊗OX OX(n) is generated by global
An Introduction to K-theory 47
sections (and in particular, has non-zero global sections), where we have implicitly chosen a locally closed embedding X ⊂ PM and taken OX(n) to be the pull-back via this embedding of OPM(n). Thus, we can send such an L∈Pic(X) to (s)−(w), where s∈Γ(L⊗OX OX(n)) and w ∈Γ(OX(n)). The fact that Pic(X) →D(X) is an isomorphism is an exercise in un-r avelling the formulation of the deﬁnition of line bundle in terms of local data. Recall that adomain A isa uniquefactorization domain ifand onlyevery prime of height 1 is principal. Whenever OX,x is a unique factorization domain for every x ∈X, every codimension 1 subvariety Y ⊂X is thus locally principal, so that the natural inclusion D(X)⊂CH1(X) is an equality. Remark 4.8. This is a ﬁrst example of relating bundles to cycles, and moreover a ﬁrst example of duality. Namely, Pic(X) is the group of rank 1 vector bundles; the group CH1(X) of is a group of cycles. Moreover, Pic(X) is contravariant with respect X whereas Z1(X) is covariant with respect to equidimensional maps. To relate the two as in the above theorem, some smoothness conditions are required. Example 4.9. Let X = AN. Then any N − 1-cycle (i.e., Weil divisor) Z ∈CHN−1(AN) is principal, so that CHN−1(AN) = 0. More generally, consider the map µ : AN ×A1 → PN ×A1 which sends( x1,...,xn),t to *t•x1,...,t•xn,1+,t. Consider an irreducible subvariety Z ⊂ AN of dimension r > N not containing the origin and Z ⊂ PN be its closure. Let W = µ−1(Z ×A1). Then W[0] = ∅ whereas W[1] = Z. Thus, CHr(AN) = 0 for any r < N. Example 4.10. Arguing in a similar geometric fashion, we see that the inclusionofalinearplanePN−1 ⊂PN inducesanisomorphismCHr(PN−1) = CHr(PN) provided that r < N and thus we conclude by induction that CHr(PN) = Z if r ≤ N. Namely, consider µ : PN ×A1 →PN ×A1 sending *x0,...,xN+,t to *x),...,xN−1,t•xN+,t and set W = µ−1(Z ×A1) for any Z not containing *0,...,0,1+. Then W[0] = prN∗(Z),W[1] = Z. Example 4.11. Let C be a smooth curve. Then Pic(C)(CH0(X). Deﬁnition 4.12. If f : X →Y isa propermap of quasi-projective varieties, then the proper push-forward of cycles determines a well deﬁned homomorphism f∗ : CHr(X) → CHr(Y), r ≥0.
48 E.M. Friedlander
Namely, if Z ⊂X isanirreduciblesubvariety of X ofdimension r, then[Z]is sent to d•[f(Z)]∈CHr(Y) where [k(Z) : k(f(Z))] = d if dim Z = dim f(Z) and is sent to 0 otherwise. If g : W →X is a ﬂat map of quasi-projective varieties of relative dimen-s ion e, then the ﬂat pull-back of cycles induces a well deﬁnedhomomorphism g∗ : CHr(X) → CHr+e(W), r ≥0. Namely, if Z ⊂X is an irreducible subvariety of X of dimension r, then [Z] is sent to the cycle on W associated to Z×X W ⊂W. Proposition 4.13. Let Y be a closed subvariety of X and let U = X\Y. Let i : Y →X,j : U →X be the inclusions. Then the sequence CHr(Y) i∗ →CHr(X) j∗ → CHr(U)→0 is exact for any r ≥0. Proof. If V ⊂ U is an irreducible subvariety of U of dimension r, then the closure of V in X, V ⊂ X, is an irreducible subvariety of X of dimension r with the property that j∗([V]) = [V]. Thus, we have an exact sequence Zr(Y) i∗ →Zr(X) j∗ →Zr(U)→0. If Z =(i ni[Yi] is a cycle on X with j∗(Z) = 0 ∈ CHr(U), then j∗Z = (W,f(f) where each W ⊂U is an irreducible subvarieties of U of dimension r +1 and f ∈ k(W). Thus, Z$ =(i ni[Y i]−(W,f(f) is an r-cycle on Y with the property that i∗(Z$) is rationally equivalent to Z. Exactness of the asserted sequence of Chow groups is now clear. Corollary 4.14. Let H ⊂PN be a hypersurface of degree d. Then CHN−1(PN\H) = Z/dZ. The following “examples” presuppose an understanding of “smoothness” brieﬂy discussed in the next section. Example 4.15. MumfordshowsthatifS isaprojectivesmoothsurfacewith a non-zero global algebraic 2-form (i.e., H0(S,Λ2(ΩS)) ,= 0), then CH0(S) is not ﬁnite dimensional (i.e., must be very large). Bloch’s Conjecture predicts that if S is a projective, smooth surface with geometric genus equal to 0 (i.e., H0(S,Λ2(ΩS)) = 0), then the natural map from CH0(S) to the (ﬁnite dimensional) Albanese variety is injective.
An Introduction to K-theory 49
4.4 Smooth Varieties We restrict our attention to quasi-projective varieties over a ﬁeld k. Deﬁnition 4.16. A quasi-projective variety X is smooth of dimension n at some point x ∈ X if there exists an open neighborhood x ∈ U ⊂ X and k polynomials f1,...,fk in n+k variables (viewed as regular functions on An+k) vanishing at 0 ∈ An+k with Jacobian |∂fi ∂xj|(0) of rank k and an isomorphism of U with Z(f1,...,fk)⊂An+k sending x to 0. In more algebraic terms, a point x∈X is smooth if there exists an openn eighborhood x ∈ U ⊂ X and a map p : U → An sending x to 0 which isﬂ at and unramiﬁed at x. Deﬁnition 4.17. Let X be a quasi-projective variety. Then K$ 0(X) is the Grothendieck group of isomorphism classes of coherent sheaves on X, where the equivalence relation is generated pairs ([E],[E1] + [E2]) for short exact sequences 0→E1 →E →E2 →0 of OX-modules. Example 4.18. Let A = k[x]/x2. Consider the short exact sequence of A-modules 0→k →A→k →0 where k is an A-module via the augmentation map (i.e., x acts as multiplication by 0), where the ﬁrst map sends a ∈ k to ax ∈ A, and the second map sends x to 0. We conclude that the class [A] of the rank 1 free module equals 2[k]. On the other hand, because A is a local ring, K0(A) = Z, generated by the class [A]. Thus, the natural map K0(SpecA) → K$ 0(SpecA) is not surjective. The map is, however, injective, as can be seen by observing that dimk(−) : K$ 0(SpecA)→Z is well deﬁned. Theorem 4.19. If X is smooth, then the natural map K0(X)→ K$ 0(X) is an isomorphism. Proof. Smoothness implies that every coherent sheaf has a ﬁnite resolution by vector bundles, This enables us to deﬁne a map
K$ 0(X) → K0(X) by sending a coherent sheaf F to the alternating sum ΣN i=1(−1)iEi, where0 →EN →•••E0 →F →0 is a resolution of F by vector bundles.
50 E.M. Friedlander
Injectivity follows from the observation that the composition K0(X) → K$ 0(X) → K0(X) istheidentity. SurjectivityfollowsfromtheobservationthatF = ΣN i=1(−1)iEis o that the composition
K$ 0(X) → K0(X) → K$ 0(X)
is also the identity. Perhaps the most important consequence of this is the following observation. Grothendieck explained to us how we can make K$ 0(−) a covariantf unctor with respect to proper maps. (Every morphism between projective varieties is proper.) Consequently, restricted to smooth schemes, K0(−) is notonlyacontravariant functorbutalsoacovariant functorforpropermaps. “Chow’s Moving Lemma” is used to give a ring structure on CH∗(X) on smooth varieties as made explicit in the following theorem. The role of the moving lemma is to verify for an r-cycle Z on X and an s-cycle W on X that Z can be moved within its rational equivalence class to some Z$ such that Z$ meets W “properly”. This means that the intersection of any irreducible component of Z$ with any irreducible component of W is either empty or of codimension d−r−s, where d = dim(X). Theorem 4.20. Let X be a smooth quasi-projective variety of dimension d. Then there exists a pairing CHr(X)⊗CHs(X) • →CHd−r−s(X), d≥r +s, with the property that if Z = [Y],Z$ = [W] are irreducible cycles of dimension r,s respectively and if Y ∩W has dimension ≤d−r−s, then Z•Z$ is a cycle which is a sum with positive coeﬃcients (determined by local data) indexed by the irreducible subvarieties of Y ∩W of dimension d−r−s. Write CHs(X) for CHd−s(X). With this indexing convention, the intersection pairing has the form CHs(X)⊗CHt(X) • → CHs+t(X). Proof. Classically, this was proved by showing the following geometric fact: given a codimension r cycle Z and a codimension s cycle W =(j mjRj with r + s ≤ d, then there is another codimension r cycle Z$ =(i niYi
An Introduction to K-theory 51
rationally equivalent to Z (i.e., determining the same element in CHr(X)) such that Z$ meets W “properly”; in other words, every component Ci,j,k of each Yi ∩Rj has codimension r +s. One then deﬁnes Z$•W =i,j,k nimj •int(Yi ∩Rj,Ci,j,k)Ci,j,k where int(Yi∩Rj,Ci,j,k) is a positive integer determined using local commutative algebra, the intersection multiplicity. Furthermore, one shows that if one chooses a Z$$ rationally equivalent to both Z,Z$ and also intersecting W properly, then Z$•W is rationally equivalent to Z”•W. A completely diﬀerent proof is given by William Fulton and Robert MacPherson (cf. [20]). They use a powerful geometric technique discovered by MacPherson called deformation to the normal cone. For Y ⊂ X closed, the deformation space MY (X) is a variety mapping to P1 deﬁned as the complement in the blow-up of X ×P1 along Y ×∞ of the blow-up of X ×∞ along Y ×∞. One readily veriﬁes that Y ×P1 ⊂M(X,Y ) restricts above∞,= p∈P1 to the given embedding Y ⊂X; and above∞, restricts to the inclusion of Y into the normal cone CY (X) = Spec(⊕n≥0In Y /In1 Y ), where IY ⊂OX is the ideal sheaf deﬁning Y ⊂X. When Y ⊂X is a regular closed embedding, then this normal cone is a bundle, the normal bundle NY (X). This enables a regular closed embedding (e.g., the diagonal δ : X → X×X for X smooth) to be deformed into the embedding of the 0-section of the normal bundle Nδ(X)(X ×X). One deﬁnes the intersection of Z,W as the intersection of δ(X),Z×W and thus one reduces the problem of deﬁning intersection product to the special case of intersection of the 0-section of the normal bundle NX(X×X) with the normal cone N(Z×W)∩δ(X)(Z×W).
4.5 Chern classes and Chern character The following construction of Chern classes is due to Grothendieck (cf. [19]); it applies equally well to topological vector bundles(in which case the Chern classes of a topological vector bundleover a topological space T are elements of the singular cohomology of T). If E is a rank r + 1 vector bundle on a quasi-projective variety X, wed eﬁne P(E) = Proj (SymOXE) → X to be the projective bundle of lines in E. Then P(E) comes equipped with a canonical line bundle OP(E)(1); for X a point, P(E) = Pr and OP(E)(1) =OPr(1).
52 E.M. Friedlander
Construction 2. Let E be a rank r vector bundle on a smooth, quasiprojective variety X of dimension d. Then CH∗(P(E)) is a free module over CH∗(X) with generators 1,ζ,ζ2,...,ζr−1, where ζ ∈ CH1(P(E)) denotes the divisor class associated to OP(E)(1). . We deﬁne the i-th Chern class ci(E)∈CHi(X) of E by the formula CH∗(P(E)) = CH∗(X)[ζ]/ r i=0 (−1)iπ∗(ci(E))•ζr−i. We deﬁne the total Chern class c(E) by the formula c(E) = r i=0 ci(E) and set ct(E) =(r i=0 ci(E)ti. Then the Whitney sum formula asserts that ct(E⊕F) = ct(E)•ct(F). We deﬁne the Chern roots, α1,...,αr of E by the formula ct(E) = r . i=1 (1+αit) where the factorization can be viewed either as purely formal or as occurring in F(E). Observe that ck(E) is the k-th elementary symmetric function of these Chern roots. In other words, the Chern classes of the rank r vector bundle E areg iven by the expression for ζr ∈ CHr(P(E)) in terms of the generators1 ,ζ,...,ζr−1. Thus, the Chern classes depend critically on the identiﬁcation of the ﬁrst Chern class ζ of OP(E)(1) and the multiplicative structure on CH∗(X). The necessary structure for such a deﬁnition of Chern classes is called an oriented multiplicative cohomology theory. The splitting principle guarantees that Chern classes are uniquely determined by the assignment of ﬁrst Chern classes to line bundles. Grothendieck introduced many basic techniques which we now use as a matter of course when working with bundles. The following splitting principle is one such technique, a technique which enable one to frequently reduce constructions for arbitrary vector bundles to those which are a sum of line bundles.
An Introduction to K-theory 53
Proposition 4.21. (Splitting Principle) Let E be a rank r+1 vector bundle on a quasi-projective variety X. Then p∗ 1 : CH∗(X)→CH∗+r(P(E)) is splitinje ctive and p∗ 1(E) =E1 is a direct sum of a rank r bundle and a line bundle.A pplying this construction to E1 on P(E), we obtain p2 : P(E1) → P(E);p roceeding inductively, we obtain p = pr ◦•••◦p1 : F(E) = P(Er−1)→X with the property that p∗ : K0(X)→K0(F(E)) is split injective and p∗(E) is a direct sum of line bundles.
One application of the preceding proposition is the following deﬁnition (due to Grothendieck) of the Chern character. Construction 3. Let X be a smooth, quasi-projective variety, let E be a rank r vector bundle over X, and let π : F(E)→X be the associated bundle of ﬂags of E. Write π∗(E) =L1⊕•••⊕Lr, where each Li is a line bundle on F(E). Then ct(π∗(E)) =/r i=1(1⊕c1(Li))t.W e deﬁne the Chern character of E as ch(E) = r i=1 {1+c1(Li)+ 1 2c1(Li)2 + 1 3!c1(Li)3 +•••}= r i=1 exp(ct(Li)), where this expression is veriﬁed to lie in the image of the injective map CH∗(X)⊗Q → CH∗(F(E))⊗Q. (Namely, one can identify chk(E) as the k-th power sum of the Chern roots, and therefore expressible in terms of the Chern classes using Newton polynomials.) Since π∗ : K0(X) → K0(F(E)), π∗ : CH∗(X) → CH∗(F(E)) are ringh omomorphisms,thesplittingprincipleenablesustoimmediatelyverifythat ch is also a ring homomorphism (i.e., sends the direct sum of bundles to the sum in CH∗(X) of Chern characters, sends the tensor product of bundles to the product in CH∗(X) of Chern characters).
4.6 Riemann-Roch Grothendieck’s formulation of the Riemann-Roch theorem is an assertion of the behaviour of the Chern character ch with respect to push-forward maps induced by a proper smooth map f : X → Y of smooth varieties. It is not the case that ch commutes with the these push-forward maps; one must
54 E.M. Friedlander
modify the push forward map in K-theory by multiplication by the Todd class. This modiﬁcation by multiplication by the Todd class is necessary even when consideration of the push-forward of a divisor. Indeed, the Todd class td : K0(X)→CH∗(X) is characterized by the properties that i. td(L) = c1(L)/(1−exp(−c1(L)) = 1+ 1 2c1(L)+•••; ii. td(E1 ⊕E2) = td(E1)•td(E2); and iii. td◦f∗ = f∗◦td. The reader is recommended to consult [19] for an excellent exposition of Grothendieck’s Riemann-Roch Theorem.
Theorem 4.22. (Grothendieck’s Riemann-Roch Theorem) Let f : X → Y be a projective map of smooth varieties. Then for any x∈K0(X), we have the equality ch(f!(x))•td(TY ) = f∗(ch(x)•td(TX)) where TX,TY are the tangent bundles of X,Y and td(TX),td(TY ) are their Todd classes. Here, f! : K0(X)→K0(Y) is deﬁned by identifying K0(X) with K$ 0(X), K0(Y) with K$ 0(Y), and deﬁning f! : K$ 0(X)→K$ 0(Y) by sending a coherent sheaf F on X to(i(−1)iRif∗(F). The map f∗ : CH∗(X) → CH∗(Y) is proper push-forward of cycles. Just to make this more concrete and more familiar, let us consider a very special case in which X is a projective, smooth curve, Y is a point, and x∈K0(X) is the class of a line bundle L. (Hirzebruch had earlier proved a version of Grothendieck’s theorem in which the target Y was a point.)
Example 4.23. Let C be a projective, smooth curve of genus g and let f : C → SpecC be the projection to a point. Let L be a line bundle on C with ﬁrst Chern class D ∈CH1(C). Then f!([L]) = dimL(C)−dimH1(C,L)∈Z,
An Introduction to K-theory 55
and ch : K0(SpecC) = Z → A∗(SpecC) = Z is an isomorphism. Let K ∈ CH1(C) denote the “canonical divisor”, the ﬁrst Chern class of the line bundle ΩC, the dual of TC. Then td(TC) = −K 1−(1+K + 1 2K2) = 1− 1 2K. Recall that deg(K) = 2g−2. Since ch([L]) = 1+D, we conclude that f∗(ch([L])•td(TC)) = f∗((1+D)•(1− 1 2K)) = deg(D)− 1 2deg(K). Thus, in this case, Riemann-Roch tell us that dimL(C)−dimH1(C,L) = deg(D)+1−g. For our purpose, Riemann-Roch is especially important because of the following consequence. Corollary 4.24. Let X be a smooth quasi-projective variety. Then ch : K0(X)⊗Q→CH∗(X)⊗Q is a ring isomorphism. Proof. The essential ingredient is the Riemann-Roch theorem. Namely, we have a natural map CH∗(X)→ K$ 0(X) sending an irreducible subvariety W to theOX-moduleOW. We show that the composition with the Chern character is an isomorphism by applying Riemann-Roch to the closed immersion W\Wsing →X\Wsing.
5 Some Diﬃcult Problems As we discuss in this lecture, many of the basic problems formulated years ago for algebraic K-theory remain unsolved. This remains a subject in which much exciting work remains to be done.
5.1 K∗(Z) Unfortunately, there are few examples (rings or varieties) for which a complete computation of the K-groups is known. As we have seen earlier, one such complete computation is the K-theory of an arbitrary ﬁnite ﬁeld,
56 E.M. Friedlander
K∗(Fq). Indeed, general theorems of Quillen give us the complete computations K∗(Fq[t]) = K∗(Fq), K∗(Fq([t,t−1]) = K∗(Fq)⊕K∗−1(Fq). Perhaps the ﬁrst natural question which comes to mind is the following: “what is the K-theory of the integers.” In recent years, great advances have been made in computing K∗(OK)o f a ring of integers in a number ﬁeld K (e.g., Z inside Q). • K0(OK)⊗Q is 1 dimensional by the ﬁniteness of the class number of K (Minkowski). • K1(OK)⊗Q has dimension r1 +r2−1, where r1, r2 are the numbers of real and complex embeddings of K. (Dirichlet). • Quillen proved that Ki(OK) is a ﬁnitely generated abelian group for any i. • For i > 1, Borel determined
Ki(OK)⊗Q =
          
0, i≡0 (mod 4) r1 +r2, i≡1 (mod 4) 0, i≡2 (mod 4) r2, i≡3 (mod 4)
(1)
in terms of the numbers r1, r2. • K∗(OK,Z/2) has been computed by Rognes-Weibel as a corollary of Voevodsky’s proof of the Milnor Conjecture. • K∗(Z,Z/p)followsinalldegreesnotdivisibleby4fromtheBloch-Kato Conjecture, now seemingly proved by Rost and Voevodsky.
Here is a table of the values of K∗(Z) whose likely inaccuracy is due to my confusion of indexing of Bernoulli numbers. Many more details can be found in [27].
An Introduction to K-theory 57
Theorem 5.1. The K-theory of Z is given by (according to Weibel’s survey paper):                                K8k = ?0?, 0 < k K8k+1 = Z⊕Z/2, 0 < k K8k+2 = Z/2c2k+1 ⊕Z/2 K8k+3 = Z/2d4k+2, i≡3 K8k+4 = ?0? K8k+5 = Z K8k+6 = Z/c2k+2 K8k+7 = Z/d4k+4 (2) Here, ck/dk is deﬁned to be the reduced expression for Bk/4k, where Bk is the k-th Bernoulli number (deﬁned by
t et −1
= 1+
∞ k=1
Bk (2k)!t2k .
Challenge 5.2. Prove the vanishing of K4i(Z), i > 0.
5.2 Bass Finiteness Conjecture This is one of the most fundamental and oldest conjectures in algebraic Ktheory. Very little progress has been made on this in the past 35 years.
Conjecture 5.3. (Bass ﬁniteness) Let A be a commutative ring which is ﬁnitely generated as an algebra over Z. Is K$ n(A) (i.e., the Quillen K-theory of mod(A)) ﬁnitely generated for all n? In particular, if A is regular as well as commutative and ﬁnitely generated over Z, is each Kn(A) a ﬁnitely generated abelian group? This conjecture seems to be very diﬃcult, even for n = 0. There are similar ﬁniteness conjectures for the K-theory of projective varieties over ﬁnite ﬁelds.
Example 5.4. Here is an example of Bass showing that we must assume A is regular or consider G∗(A). Let A = Z[x,y]/x2. Then the ideal (x) is inﬁnitely additively generated by x,xy,xy2,.... On the other hand, if t∈(x), then 1+t ∈A∗, so that we see that K1(A) is not ﬁnitely generated.
58 E.M. Friedlander
Example 5.5. As pointed out by Bass, it is elementary to show (using general theorems of Quillen and Quillen’s computation of the K-theory of ﬁnite ﬁelds) that if A is ﬁnite, then Gn(A)(Gn(A/radA) is ﬁnite for every n ≥ 0. Subsequently, Kuku proved that Kn(A) is also ﬁnite whenever A is ﬁnite (see [32]). There are many other ﬁniteness conjectures involving smooth schemes of ﬁnite type over a ﬁnite ﬁeld, Z or Q. Even partial solutions to these conjectures would represent great progress.
5.3 Milnor K-theory We recall Milnor K-theory, a major concept in Professor Vishik’s lectures. This theory is motivated by Matsumoto’s presentation of K2(F) (mentioned in Lecture 1), Deﬁnition 5.6. (Milnor) Let F be a ﬁeld with multiplicative group of units F×. The Milnor K-group KMilnor n (F) is deﬁned to be the n-th graded piece of the graded ring deﬁned as the tensor algebra0n≥0(F×)⊗n modulo the ideal generated by elements {a,1−a}∈F∗⊗F∗,a,= 1,= 1−a. In particular, K1(F) = KMilnor 1 (F),K2(F) = KMilnor 2 (F) for any ﬁeld F, and KMilnor n (F) is a quotient of Λn(F×). For F an inﬁnite ﬁeld, Suslin in [24] proved that there are natural maps
KMilnor n (F)→Kn(F)→KMilnor n (F) whose composition is (−1)n−1(n−1)!. This immediately implies, for example, that the higher K-groups of a ﬁeld of high transcendence degree are very large. Remark 5.7. It is diﬃcult to even brieﬂy mention K2 of ﬁelds without also mentioning the deep and import theorem of Mekurjev and Suslin [23]: for any ﬁeld F and positive integer n, K2(F)/nK2(F) ( H2(F,µ⊗2 n ). Inparticular, H2(F,µ⊗2 n )isgeneratedbyproductsofelementsinH1(F,µn) = µn(F). Moreover, if F contains the nth roots of unity, then K2(F)/nK2(F) ( nBr(F),
An Introduction to K-theory 59
where nBr(F) denotes the subgroup of the Brauer group of F consisting of elements which are n-torsion. In particular, nBr(F) is generated by “cyclic central simple algebras”. The most famous success of K-theory in recent years is the following theorem of Voevodsky [26], establishing a result conjectured by Milnor. Theorem 5.8. Let F be a ﬁeld of characteristic ,= 2. Let W(F) denote the Witt ring of F, the quotient of the Grothendieck group of symmetric inner product spaces modulo the ideal generated by the hyperbolic space *1+⊕*−1+ and let I = ker{W(F) → Z/2} be given by sending a symmetric inner product space to its rank (modulo 2). Then the map
KMilnor n (F)/2•KMilnor n (F)→In/In+1, {a1,...,an}0→
n . i=1 (*ai+−1) is an isomorphism for all n ≥ 0. Here, *a+ is the 1 dimensional symmetric inner product space with inner product (−,−)a deﬁned by (c,d)a = acd. Suslinalso proved thefollowing theorem, the ﬁrstconﬁrmation of a series of conjectures which now seem to be on the verge of being settled. Theorem 5.9. Let F be an algebraic closed ﬁeld. If F has characteristic 0 and i > 0, then K2i(F) is a Q vector space and K2i−1(F) is a direct sum of Q/Z and a rational vector space. If F has characteristic p > 0 and i > 0, then K2i(F) is a Q vector space and K2i−1(F) is a direct sum of ⊕",=pQ"/Z" and a rational vector space. Question 5.10. What information is reﬂected in the uncountable vector spaces Kn(C) ⊗ Q? Are there interesting structures to be obtained from these vector spaces?
5.4 Negative K-groups Bassintroducednegativealgebraic K-groups,groupswhichvanishforregular ringsor, more generally, smooth varieties. Thesenegative K-groupsmeasure thefailureof K-theoryin positivedegree toobey“homotopy invariance” and “localization” (i.e., K∗(X) ? = K∗(X ×A1), K∗(X)⊕K∗−1(X) ? = K∗(X ×A1\{0}). Very recently, there has been important progress in computing these negative K-groups by Cortinas, Haesemeyer, Schlicting, and Weibel.
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Question 5.11. Can negative K-groups give useful invariants for the geometric study of singularities?
5.5 Algebraic versus topological vector bundles Let X be a complex projective variety, and let Xan denote the topological space of complex points of X equipped with the analytic topology. Then any algebraic vector bundle E → X naturally determines a topological vector bundle Ean →Xan. This determines a natural map K0(X) → K0 top(Xan). Challenge 5.12. Understand the kernel and image of the above map, especially after tensoring with Q: CH∗(X)⊗Q ( K0(X)⊗Q → K0 top(Xan)⊗ ( Hev(Xan,Q). (3) The kernel of (3) can be identiﬁed with the subspace of CH∗(X)⊗Qc onsisting of rational equivalence classes of algebraic cycles on X which are homologically equivalent to 0. The image of (3) can be identiﬁed with those classes in H∗(Xan,Q) represented by algebraic cycles – the subject of the Hodge Conjecture!
In positive degree, the analogue of our map is uninteresting.
Proposition 5.13. If X is a complex projective variety, then the natural map Ki(X)⊗Q → K−i top(Xan), i > 0 is the 0-map.
5.6 K-theory with ﬁnite coeﬃcients Although the map in positive degrees Ki(X) → K−i top(Xan) is typically of little interest, the situation changes drastically if we consider K-theory mod-n. As an example, we give the following special case of a theorem of Suslin. Recall that (SpecC)an is a point, which we denote by +.
An Introduction to K-theory 61
Theorem 5.14. (cf. [25]) The map Ki(SpecC) → K−i top(+) is the 0-map for i > 0. On the other hand, for any positive integer n and any integer i≥0, the map Ki(SpecC,Z/n) → K−i top(+,Z/n) is an isomorphism. How can the preceding theorem be possibly correct? The point is that K2i−1(SpecC) is a divisible group with torsion subgroup Q/Z. Then, we see that this Q/Z in odd degree integral homotopy determines a Z/n in even degree mod-n homotopy. This is exactly what K−∗ top(+) determines in even mod-n homotopy degree. The K-groups modulo n are deﬁned to be the homotopy groups modulo n of the K-theory space (or spectrum). Deﬁnition 5.15. For positive integers i,n > 1, let M(i,Z/n) denote the C.W. complex obtained by attaching an i-cell Di to Si−1 via the map ∂(Di) = Si−1 →Si−1 given by multiplication by n. For any connected C.W. complex, we deﬁne πi(X,Z/n)≡[M(i,Z/n),X], i,n > 1. If X = Ω2Y, we deﬁne πi(X,Z/n)≡[M(i+2,Z/n),Y ], i≥0,n > 1. Since Si−1 → M(i,Z/n) is the cone on the multiplication by n map Si−1 n →Si−1, we have long exact sequences •••→ πi(X) n →πi(X)→πi(X/Z/n)→πi−1(X)→••• Perhapsthisissuﬃcientto motivate ournextconjecture, which wemight call the Quillen-Lichtenbaum Conjecture for smooth complex algebraic varieties. The special case in which X = SpecC is the theorem of Suslin quoted above. Conjecture 5.16. (Q-L for smooth C varieties) If X is a smooth complex variety of dimension d, then is the natural map Ki(X,Z/n)→Ktop i (Xan,Z/n) an isomorphism provided that i≥d−1≥0?
62 E.M. Friedlander
Remark In “low” degrees, K∗(X,Z/n) should be more interesting and will not be periodic. For example, Ktop ev (X,Z/n) has a contribution from the Brauer group of X whereas K0(X,Z/n) does not.
5.7 Etale K-theory Itisnaturaltotrytoﬁndagood“topological model”forthemod-n algebraic K-theory of varieties over ﬁelds other than the complex numbers. Suslin’s Theorem in its full generality can be formulated as follows Theorem 5.17. If k is an algebraically closed ﬁeld of characteristic p ≥0, then there is a natural isomorphism K∗(k,Z/n) & →Ket ∗ (Speck,Z/n), (n,p) = 1. Moreover, if the characteristic of k is a positive integer p, then Ki(k,Z/p) = 0, for all i > 0.
We have stated the previoustheorem in terms of etale K-theory although this is not the way Suslin formulated his theorem. We did this in order to introduce the etale topology, a Grothendieck topology associated to the etale site. For this site, the distinguished morphismsE are etale morphisms of schemes. A map of schemes f : U → V is said to be etale (or “smooth of relative dimension 0) if there exist aﬃne open coverings {Ui} of U, {Vj} of V such that the restriction to Ui of f lies in some Vj and such that the corresponding map of commutative rings Ai ← Rj is unramiﬁed (i.e., for all homomorphisms from R to a ﬁeld k, A⊗R k ← k is a ﬁnite separable k algebra) and ﬂat. The etale topology was introduced by Grothendieck partly to reinterpret Galois cohomology of ﬁelds and partly to algebraically realize singular cohomology of complex algebraic varieties. The following “comparison theorem” proved by Michael Artin and Alexander Grothendieck is an important property of the etale topology. (See, for example, [21].)
Theorem 5.18. (Artin, Grothendieck) If X is a complex algebraic variety, then H∗ et(X,Z/n)(H∗ sing(Xan,Z/n). Here, H∗ et(X,Z/n) denotes the derived functors of the global section functor applied to the constant sheaf Z/n on the etale site.
An Introduction to K-theory 63
The etale topology not only enables us to deﬁne etale cohomological groups, but also etale homotopy types. Using the etale homotopy type, etale K-theory (deﬁned by Bill Dwyer and myself) can be deﬁned in a manner similar to topological K-theory. For this theory, there is an Atiyah-Hirzebruch spectral sequence
Ep,q 2 = Hp et(X,Kq et(+))⇒Kp+q et (X,Z/n) provided thatOX is a sheaf of Z[1/n]-modules. If we let µn denote the etale sheaf of n-th roots of unity and let µ⊗q/2 n denote µ⊗j n if q = 2j and 0 if j is odd, then this spectral sequence can be rewritten
Ep,q 2 = Hp et(X,µ⊗q/2)⇒Ket q−p
(X,Z/n).
Using etale K-theory, we can reformulate and generalize the QuillenLichtenbaum Conjecture (originally stated for SpecK, where K is a number ﬁeld), putting this conjecture in a quite general context.
Conjecture 5.19. (Quillen-Lichtenbaum) Let X be a smooth scheme of ﬁnite type over a ﬁeld k, and assume that n is a positive integer with 1/n in k or A. Then the natural map Ki(X,Z/n)→Ket i (X,Z/n) is an isomorphism for i−1 greater or equal to the mod-n etale cohomological dimension of X.
Thisconjecture appearsto beproven, or near-proven, thanksto the work of Rost and Voevodsky on the Bloch-Kato Conjecture.
5.8 Integral conjectures There has been much progress in understanding K-theory with ﬁnite coefﬁcients, but much less is known about the result of tensoring the algebraic K-groups with Q. The following theorem of Soul´ e is proved by investigating the group homology of general linear groupsover ﬁelds. Soul´ e proves a vanishingtheorem for more general rings R with a range depending upon the “stable range” of R.
64 E.M. Friedlander
Theorem 5.20. (Soul´ e) For any ﬁeld F, Kn(F)(s) Q = 0, s > n. Here Kn(F)(s) Q is the s-eigenspace with respect to the action of the Adams operations on Kn(F). This motivates the following Beilinson-Soul´ e vanishing conjecture, part of the Beilinson Conjectures discussed in the next lecture. This conjecture is now known if we replace the coeﬃcients Z(n) by their ﬁnite coeﬃcients analogue Z/'(n). Conjecture 5.21. (Beilinson-Soul´ e) For any ﬁeld F, the motivic cohomology groups Hp(SpecF,Z(n)) equal 0 for p < 0. Yet another auxillary K-theory has been developed to investigate Ktheory of complex varieties, especially some aspects involving rational coefﬁcients (cf. [22]). Theorem 5.22. (Friedlander-Walker) Let X be a complex quasi-projective variety. The map from the algebraic K-theory spectrum K(X) to the topological K-theory spectrum Ktop(Xan) factors through the “semi-topological K-theory spectrum Ksst(X). K(X) → Ksst(X) → Ktop(Xan). The ﬁrst map induces an isomorphism inhomotopy groups modulo n, whereas the second map induces an isomorphism for certain special varieties and typically induces an isomorphism after “inverting the Bott element.” This semi-topological K-theory is related to cycles modulo algebraic equivalence is much the same way as usual algebraic K-theory is related to Chow groups (cycles modulo rational equivalence). One important aspect of this semi-topological K-theory is that leads to conjectures which are “integral” whose reduction modulo n give the familiar Quillen-Lichtenbaum Conjecture. We state one precise form of such a conjecture, essentially due to Suslin. Conjecture 5.23. Let X be a smooth, quasi-projective complex variety. Then the natural map Ksst i (X) → K−i top(Xan) is an isomorphism for i ≥ dim(X) − 1 and a monomorphism for i = dim(X)−2.
An Introduction to K-theory 65
Now, we also have a “good semi-topological model” for the K-theory of quasi-projective varieties over R, the real numbers. This is closely related to “Atiyah Real K-theory rather than the topological K-theory we have discussed at several points in these lectures.
Challenge 5.24. Develop a semi-topological K-theory for varieties over an arbitrary ﬁeld.
5.9 K-theory and Quadratic Forms another topic of considerable interest is Hermetian K-theory in which we take into account the presence of quadratic forms. Perhaps this topic is best left to Professor Vishik!
6 Beilinson’s vision partially fulﬁlled
6.1 Motivation In this lecture, we will discuss Alexander Beilinson’s vision of what algebraic K-theoryshouldbeforsmoothvarietiesoveraﬁeldk (cf. [28], [30],and[31]). In particular, we will provide some account of progress towards the solution of these conjectures. Essentially, Beilinson conjectures that algebraic Ktheory can be computed using a spectral sequence of Atiyah-Hirzebruch type using “motivic complexes” Z(n) which satisfy various good properties and whose cohomology plays the role of singular cohomology in the AtiyahHirzebruch spectral sequence for topological K-theory. Although our goal is to describe conjectures which would begin to “explain” algebraic K-theory, let me start by mentioning one (of many) reasons why algebraic K-theory is so interesting to algebraic geometers (and algebraic number theorists). It has been known for some time that there can not bean algebraic theory whosevalues on complex algebraic varieties isintegral (orevenrational)singularhomologyoftheassociatedanalyticspace. Indeed, Jean-Pierre Serre observed that this is not possible even for smooth projective algebraic curves because some such curves have automorphism groups which do not admit a representation which would be implied by functoriality. On the other hand, algebraic K-theory is in some sense integral – we deﬁne it without inverting residue characteristics or considering only mod-n coeﬃcients. Thus, if we can formulate a sensible Atiyah-Hirzebruch type
66 E.M. Friedlander
spectral sequence converging to algebraic K-theory, then the E2-term oﬀers an algebraic formulation of integral cohomology. Before we launch into a discussion ofBeilinson’s Conjectures, let usrecall two results relating algebraic cycles and algebraic K-theory which precede these conjectures. The ﬁrst is the theorem of Grothendieck mentioned earlier relating algebraic K0(X) to the Chow ring of algebraic cycles modulo algebraic equivalence. Theorem 6.1. If X is a smooth variety over a ﬁeld k, then the Chern character determines an isomorphism ch : K0(X)⊗Q ( CH∗(X)⊗Q. The second is Bloch’s formula proved in degree 2 by Bloch and in general by Quillen. Theorem 6.2. Let X be a smooth variety over a ﬁeld and let Ki denote the Zariski sheaf associated to the presheaf U 0→ Ki(U) for an open subset U ⊂X. Then there is a convergent spectral sequence of the form Ep,q 2 = Hp Zar(X,Kq) ⇒Kq−p(X).
6.2 Statement of conjectures We now state Beilinson’s conjectures and use these conjectures as a framework to discuss much interesting mathematics. It is worth emphasizing that one of the most important aspects of Beilinson’s conjectures is its explicit nature: Beilinson conjectures precise values for algebraic K-groups, rather than the conjectures which preceded Beilinson which required the degree to be large or certain torsion to be ignored. Conjecture 6.3. (Beilinson’s Conjectures) For each n ≥ 0 there should be complexes Z(n),n ≥ 0 of sheaves on the Zariski site of smooth quasiprojective varieties over a ﬁeld k, (Sm/k)Zar which satisfy the following: 1. Z(0) = Z, Z(1)(O∗[−1]. 2. Hn(SpecF,Z(n)) = KMilnor n (F) for any ﬁeld F ﬁnitely generated over k. 3. H2n(X,Z(n)) = CHn(X) whenever X is smooth over k.
An Introduction to K-theory 67
4. Vanishing Conjecture: Z(n) is acyclic outside of [0,n]: Hp(X,Z(n)) = 0, p < 0.
5. Motivic spectral sequences for X smooth over k:
Ep,q 2 = Hp−q(X,Z(−q))⇒K−p−q(X),
Ep,q 2 = Hp−q(X,Z/'(−q))⇒K−p−q(X,Z/'), if 1/' ∈k. 6. Beilinson-Lichtenbaum Conjecture: Z(n)⊗L Z/'(τ≤nRπ∗µ⊗n " , if 1/' ∈k where π : etale site → Zariski site is the natural “forgetful continuous map” and τ≤n indicates truncation. 7. Hi(X,Z(n))⊗Q(K2n−i(X)(n) Q . In other words, Beilinson conjectures that there should be a bigraded motivic cohomology groupsHp(X,Z(q))computedastheZariskicohomology of motivic complexes Z(q) of sheaves which satisfy good properties and are related toalgebraic K-theoryassingularcohomology isrelated totopological K-theory.
6.3 Status of Conjectures Bloch’s higher Chow groups CHq(X,n) (cf. [29]) serve as motivic cohomology groups which are known to satisfy most of the conjectures, where the correspondence of indexing is as follows: CHq(X,n) ( H2q−n(X,Z(q)). (1) Furthermore, Suslin and Voevodsky have formulated complexes Z(q), q ≥0 and Voevodsky has proved that the (hyper-)cohomology groups of these complexes satisfy the relationship to Bloch’s higher Chow groups as in (1). Presumably, these constructions will bediscussed in detail in the lectures of Professor Levine. For completeness, I sketch the deﬁnitions. Recall that the standard (algebro-geometric) n-simplex ∆n over a ﬁeld F (which we leave implicit) is given by SpecF[t0,...,tn]/Σiti = 1.
68 E.M. Friedlander
Deﬁnition 6.4. Let X be a quasi-projective variety over a ﬁeld. For any q,n≥0, we deﬁne zq(X,n) to be the free abelian group on the set of cycles W ⊂ X ×∆n of codimension q which meet all faces X ×∆i ⊂ X ×∆n properly. This admits the structure of a simplicial abelian group and thus a chain complex with boundary maps given by restrictions to (codimension 1) faces. The Bloch higher Chow group CHq(X,n) is deﬁned by CHq(X,n) = H2q−n(zq(X,∗)). The values of Bloch’s higher Chow groups are “correct”, but they are not given as (hyper)-cohomology of complexes of sheaves and they are so directly deﬁned that abstract properties for them are diﬃcult to prove. The Suslin-Voevodsky motivic cohomology groups ﬁt in a good formalism as envisioned by Beilinson and agree with Bloch’s higher Chow groups as veriﬁed by Voevodsky. Deﬁnition 6.5. Let X be a quasi-projective variety over a ﬁeld. For any q ≥ 0, we deﬁne the complex of sheaves in the cdh topology (the Zariski topology suﬃces if X is smooth over a ﬁeld of characteristic 0) Z(q) = C∗(cequi(Pn,0)/cequi(Pn−1,0))[−2n] deﬁned as the shift 2n steps to the right of the complex of sheaves whose value on a Zariski open subset U ⊂ X is the complex j 0→ cequi(Pn,0)(∆j)/cequi(Pn−1)(U ×∆j) where cequi(Pn,0)(U ×∆j) is the free abelian group on the cycles on Pn × U ×∆j which are equidimensional of relative dimension 0 over U ×∆j. Conjecture (1) is essentially a normalization, for it speciﬁes what Z(0) andZ(1)mustbe. BlochveriﬁedConjecture2(essentially, aresultofSuslin), Conjecture 3, and Conjecture 7 (the latter with help from Levine) for his higher Chow groups. Bloch and Lichtenbaum produced a motivic spectral sequence for X = Speck; this was generalized to a veriﬁcation of the full Conjecture (5) by Friedlander and Suslin, and later proofs were given by Levine and then Suslin following work of Grayson. The Beilinson-Lichtenbaum conjecture in some sense “identiﬁes” mod-' motivic cohomology in terms of etale cohomology. Suslin and Voevodsky proved that this Conjecture (6) follows from the following:
An Introduction to K-theory 69
Conjecture 6.6. (Bloch-Kato Conjecture) For ﬁelds F ﬁnitely generated over k, KMilnor n ⊗Z/' (Hn et(SpecF,µ⊗n " ). In particular, the Galois cohomology of the ﬁeld F is generated multiplicatively by classes in degree 1. For ' = 2, the Bloch-Kato Conjecture is a form of Milnor’s Conjecture which has been proved by Voevodsky. For ' > 2, a proof of Bloch-Kato Conjecture has apparently been given by Rost and Voevodsky, although not all details have been made available. This conjecture will be the main focus of Professor Weibel’s lectures. This leaves Conjecture (4), one aspect of this is the following Vanishing Conjecture due to Beilinson and Soul´ e. Conjecture 6.7. For ﬁelds F, Kp(F)(q) Q = 0, 2q ≤p,p > 0. Reindexing according to Conjecture (7), this becomes Hi(SpecF,Z(q)) = 0, i≤0,q ,= 0. The status of this Conjecture (4), and in particular the Beilinson-Sou´ e vanishing conjecture, is up in the air. Experts are not at all convinced that this conjecture should be true for a general ﬁeld F. It is known to be true for a number ﬁeld.
6.4 The Meaning of the Conjectures Let us begin by looking a bit more closely at the statement Z(1)(O∗[−1] of Conjecture (1). ConventionIf C∗ isacochain complex(i.e., thediﬀerential increasesdegree by 1, d : Ci → Ci+1), we deﬁne the chain complex C∗[n] for any n ∈ Z as the shift of C∗ “n places to the right”. In other words, (C∗[n])j = C∗−j. In particular, O∗[−1] is the complex (of Zariski sheaves) with only onen on-zero term, the sheafO∗ of units, placed in degree -1 (i.e., shifted 1 placet o the left). In particular, H∗ ZH∗ Zar(X,O∗[−1]) = H∗−1 Zar(X,O∗);

An Introduction to K-theory

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