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2005 | Buch

Handbook of K-Theory

herausgegeben von: Eric M. Friedlander, Daniel R. Grayson

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This handbook offers a compilation of techniques and results in K-theory.

These two volumes consist of chapters, each of which is dedicated to a specific topic and is written by a leading expert. Many chapters present historical background; some present previously unpublished results, whereas some present the first expository account of a topic; many discuss future directions as well as open problems. The overall intent of this handbook is to offer the interested reader an exposition of our current state of knowledge as well as an implicit blueprint for future research. This handbook should be especially useful for students wishing to obtain an overview of K-theory and for mathematicians interested in pursuing challenges in this rapidly expanding field.

Inhaltsverzeichnis

Frontmatter

Foundations and Computations

Frontmatter

I.1. Deloopings in Algebraic K-Theory

Abstract

A crucial observation in Quillen’s definition of higher algebraic K-theory was that the right way to proceed is to define the higher K-groups as the homotopy groups of a space ([23]). Quillen gave two different space level models, one via the plus construction and the other via the Q-construction. The Q-construction version allowed Quillen to prove a number of important formal properties of the K-theory construction, namely localization, devissage, reduction by resolution, and the homotopy property. It was quickly realized that although the theory initially revolved around a functor K from the category of rings (or schemes) to the category Top of topological spaces, K in fact took its values in the category of infinite loop spaces and infinite loop maps ([1]). In fact, K is best thought of as a functor not to topological spaces, but to the category of spectra ([2, 11]). Recall that a spectrum is a family of based topological spaces {X _i}_i≥0, together with bonding maps σ _i : X _i → ΩX _i+1, which can be taken to be homeomorphisms. There is a great deal of value to this refinement of the functor K.

Gunnar Carlsson

I.2. The Motivic Spectral Sequence

Abstract

We give an overview of the search for a motivic spectral sequence: a spectral sequence connecting algebraic K-theory to motivic cohomology that is analogous to the Atiyah–Hirzebruch spectral sequence that connects topological K-theory to singular cohomology.

Daniel R. Grayson

I.3. K-Theory of Truncated Polynomial Algebras

Abstract

In general, if A is a ring and I ⊂ A a two-sided ideal, one defines the K-theory of A relative to I to be the mapping fiber of the map of K-theory spectra induced by the canonical projection from A to A/I. Hence, there is anatural exact triangle of spectra

$$ K(A,I) \rightarrow K(A) \rightarrow K(A/I) \xrightarrow{\partial} K(A,I)[-1] $$

and an induced natural long-exact sequence of K-groups

$$ ... \rightarrow K_{q}(A,I) \rightarrow K_{q}(A) \rightarrow K_{q}(A/I) \xrightarrow{\partial} K_{q-1}(A,I) \rightarrow ... $$

Lars Hesselholt

I.4. Bott Periodicity in Topological, Algebraic and Hermitian K-Theory

Abstract

This paper is devoted to classical Bott periodicity, its history and more recent extensions in algebraic and Hermitian K-theory. However, it does not aim at completeness. For instance, the variants of Bott periodicity related to bivariant K-theory are described by Cuntz in this handbook. As another example, we don’t emphasize here the relation between motivic homotopy theory and Bott periodicity since it is also described by other authors of this handbook (Grayson, Kahn,…).

Max Karoubi

I.5. Algebraic K-Theory of Rings of Integers in Local and Global Fields

Abstract

This survey describes the algebraic K-groups of local and global fields, and the K-groups of rings of integers in these fields. We have used the result of Rost and Voevodsky to determine the odd torsion in these groups.

Charles Weibel

K-Theory and Algebraic Geometry

Frontmatter

II.1. Motivic Cohomology, K-Theory and Topological Cyclic Homology

Abstract

We give a survey onmotivic cohomology, higher algebraic K-theory, and topological cyclic homology. We concentrate on results which are relevant for applications in arithmetic algebraic geometry (in particular, we do not discuss non-commutative rings), and our main focus lies on sheaf theoretic results for smooth schemes, which then lead to global results using local-to-global methods.

Thomas H. Geisser

II.2. K-Theory and Intersection Theory

Abstract

The problem of defining intersection products on the Chow groups of schemes has a long history. Perhaps the first example of a theorem in intersection theory is Bézout’s theorem, which tells us that two projective plane curves C and D, of degrees c and d and which have no components in common, meet in at most cd points. Furthermore if one counts the points of C ∩ D with multiplicity, there are exactly cd points. Bezout’s theorem can be extended to closed subvarieties Y and Z of projective space over a field k, ℙ _k ⁿ , with dim(Y) + dim(Z) = n and for which Y ∩ Z consists of a finite number of points.

Henri Gillet

II.3. Regulators

Abstract

The ζ-function is one of the most deep and mysterious objects in mathematics. During the last two centuries it has served as a key source of new ideas and concepts in arithmetic algebraic geometry. The ζ-function seems to be created to guide mathematicians into the right directions. To illustrate this, let me recall three themes in the 20th century mathematics which emerged from the study of the most basic properties of ζ-functions: their zeros, analytic properties and special values.

Alexander B. Goncharov

II.4. Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry

Abstract

Warning: This chapter is full of conjectures. If you are allergic to them it may be harmful to your health. Parts of them are proven, though.

In algebraic geometry, one encounters two important kinds of objects: vector bundles and algebraic cycles. The first lead to algebraic K-theory while the second lead to motivic cohomology. They are related via the Chern character and Atiyah–Hirzebruch-like spectral sequences.

Bruno Kahn

II.5. Mixed Motives

Abstract

During the early and mid-eighties, Beilinson [2] and Deligne [24] independently described a conjectural abelian tensor category of mixed motives over a given base field k, ℳℳ_k, which, in analogy to the category of mixed Hodge structures, should contain Grothendieck’s category of pure (homological) motives as the full subcategory of semi-simple objects, but should have a rich enough structure of extensions to allow one to recover the weight-graded pieces of algebraic K-theory. More specifically, one should have, for each smooth scheme X of finite type over a given field k, an object h(X) in the derived category D ^b(ℳℳ_k), as well as Tate twists h(X)(n), and natural isomorphisms

$$ \text{Hom}_{D^{b}(\mathcal{MM}_{k})}(1,h(X)(n)[m]) \otimes \mathbb{Q} \cong K_{2n-m}(X)^{(n)} $$

where K _p(X)⁽ⁿ⁾ is the weight n eigenspace for the Adams operations. The abelian groups

$$ H^{p}_{\mathcal{M}}(X,\mathbb{Z}(q)) := \text{Hom}_{D^{b}(\mathcal{MM}_{k})}(1,h(X)(q)[p]) $$

should form the universal Bloch–Ogus cohomology theory on smooth k-schemes of finite type; as this theory should arise from mixed motives, it is called motivic cohomology.

Marc Levine

K-Theory and Geometric Topology

Frontmatter

III.1. Witt Groups

Abstract

In his 1937 paper [86], Ernst Witt introduced a group structure – and even a ring structure – on the set of isometry classes of anisotropic quadratic forms, over an arbitrary field k. This object is now called the Witt group W(k) of k. Since then, Witt’s construction has been generalized from fields to rings with involution, to schemes, and to various types of categories with duality. For the sake of efficacy, we review these constructions in a non-chronological order. Indeed, in Sect. 1.2, we start with the now “classical framework” in its most general form, namely over exact categories with duality. This folklore material is a basically straightforward generalization of Knebusch’s scheme case [41], where the exact category was the one of vector bundles. Nevertheless, this level of generality is hard to find in the literature, like e.g. the “classical sublagrangian reduction” of Sect. 1.2.5. In Sect. 1.3, we specialize this classical material to the even more classical examples listed above: schemes, rings, fields. We include some motivations for the use of Witt groups.

Paul Balmer

III.2. K-Theory and Geometric Topology

Abstract

Historically, one of the earliest motivations for the development of K-theory was the need to put on a firm algebraic foundation a number of invariants or obstructions that appear in topology. The primary purpose of this chapter is to examine many of these K-theoretic invariants, not from a historical point of view, but rather a posteriori, now that K-theory is a mature subject.

Jonathan Rosenberg

III.3. Quadratic K-Theory and Geometric Topology

Abstract

Suppose R is a ring with an (anti)-involution −: R → R, and with choice of central unit ε such that $ \bar{\epsilon}\epsilon = 1 $. Then one can ask for a computation of $ \mathbb{K} Quad (R,-,\epsilon) $, the K-theory of quadratic forms. Let H: $ \mathbb{K}R \rightarrow \mathbb{K} Quad (R,-,\epsilon) $ be the hyperbolic map, and let F : $ \mathbb{K} Quad (R,-,\epsilon) \rightarrow \mathbb{K}R $ be the forget map. Then the Witt groups

$$ W_{0}(R,-,\epsilon) = coker (K_{0}R \xrightarrow{H} K_{0}Quad(R,-,\epsilon)) $$

$$ W_{1}(R,-,\epsilon) = ker (F: KQuad_{1}(R,-,\epsilon) \xrightarrow{F} K_{1}R) $$

have been highly studied. See [6], [29, 32, 42, 46, 68], and [86]–[89]. However, the higher dimensional quadratic K-theory has received considerably less attention, than the higher K-theory of f.g. projective modules. (See however, [34, 35, 39], and [36].)

Bruce Williams

K-Theory and Operator Algebras

Frontmatter

IV.1. Bivariant K- and Cyclic Theories

Abstract

Bivariant K-theories generalize K-theory and its dual, often called K-homology, at the same time. They are a powerful tool for the computation of K-theoretic invariants, for the formulation and proof of index theorems, for classification results and in many other instances. The bivariant K-theories are paralleled by different versions of cyclic theories which have similar formal properties. The two different kinds of theories are connected by characters that generalize the classical Chern character. We give a survey of such bivariant theories on different categories of algebras and sketch some of the applications.

Joachim Cuntz

IV.2. The Baum–Connes and the Farrell–Jones Conjectures in K- and L-Theory

Abstract

We give a survey of the meaning, status and applications of the Baum–Connes Conjecture about the topological K-theory of the reduced group C*-algebra and the Farrell–Jones Conjecture about the algebraic K- and L-theory of the group ring of a (discrete) group G.

Wolfgang Lück, Holger Reich

IV.3. Comparison Between Algebraic and Topological K-Theory for Banach Algebras and C*-Algebras

Abstract

For a Banach algebra, one can define two kinds of K-theory: topological K-theory, which satisfies Bott periodicity, and algebraic K-theory, which usually does not. It was discovered, starting in the early 80’s, that the “comparison map” from algebraic to topological K-theory is a surprisingly rich object. About the same time, it was also found that the algebraic (as opposed to topological) K-theory of operator algebras does have some direct applications in operator theory. This article will summarize what is known about these applications and the comparison map.

Jonathan Rosenberg

Other Forms of K-Theory

Frontmatter

V.1. Semi-topological K-Theory

Abstract

The semi-topological K-theory of a complex variety X, written K _* ^sst (X), interpolates between the algebraic K-theory, K _* ^alg (X), of X and the topological K-theory, K _top ^* (X ^an), of the analytic space (X ^an) associated to X. (The superscript “sst” stands for “singular semi-topological”.) In a similar vein, the real semi-topological K-theory, written Kℝ _* ^sst (Y), of a real variety Y interpolates between the algebraic K-theory of Y and the Atiyah Real K-theory of the associated space with involution Y _ℝ(ℂ). We intend this survey to provide both motivation and coherence to the field of semi-topological K-theory. We explain the many foundational results contained in the series of papers by the authors [27, 31, 32], as well as in the recent paper by the authors and Christian Haesemeyer [21]. We shall alsomention various conjectures that involve challenging problems concerning both algebraic cycles and algebraic K-theory.

Eric M. Friedlander, Mark E. Walker

V.2. Equivariant K-Theory

Abstract

The equivariant K-theory was developed by R. Thomason in [21]. Let an algebraic group G act on a variety X over a field F. We consider G-modules, i.e., $ \mathcal{O}_{X} $-modules over X that are equipped with an G-action compatible with one on X. As in the non-equivariant case there are two categories: the abelian category ℳ(G; X) of coherent G-modules and the full subcategory $ \mathcal{P}(G;X) $ consisting of locally free $ \mathcal{O}_{X} $-modules. The groups K' _n(G; X) and K _n(G; X) are defined as the K-groups of these two categories respectively.

Alexander S. Merkurjev

V.3. K(1)-Local Homotopy, Iwasawa Theory and Algebraic K-Theory

Abstract

The Iwasawa algebra Λ is a power series ring Z _ℓ[[T]], ℓ a fixed prime. It arises in number theory as the pro-group ring of a certain Galois group, and in homotopy theory as a ring of operations in ℓ-adic complex K-theory. Furthermore, these two incarnations of Λ are connected in an interesting way by algebraic K-theory. The main goal of this paper is to explore this connection, concentrating on the ideas and omitting most proofs.

Stephen A. Mitchell

V.4. The K-Theory of Triangulated Categories

Abstract

The purpose of this survey is to explain the open problems in the K-theory of triangulated categories. The survey is intended to be very easy for non-experts to read; I gave it to a couple of fourth-year undergraduates, who had little trouble with it. Perhaps the hardest part is the first section, which discusses the history of the subject. It is hard to give a brief historical account without assuming prior knowledge. The students are advised to skip directly to Sect. 4.2.

Amnon Neeman

Backmatter

Titel: Handbook of K-Theory
herausgegeben von: Eric M. Friedlander
Daniel R. Grayson
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-27855-9
Print ISBN: 978-3-540-23019-9
DOI: https://doi.org/10.1007/978-3-540-27855-9