We provide a general procedure for computing the algebraic ktheory of finitely generated virtually free groups. Waldhausen started out as a 3manifold theorist, and he realized that certain phenomena in the topology of 3manifolds would be explained if the whitehead groups of. They also include some material from an earlier course on abelian categories, elaborating certain parts of gabriels thesis. Algebraic ktheory describes a branch of algebra that centers about two functors. Lectures on k theory pdf 95p this lecture note covers the following topics. Topological k theory, topological preliminaries on vector bundles, homotopy, bott periodicity and cohomological properties, chern character and chern classes, analytic k theory, applications of adams operations, higher algebraic k theory, algebraic preliminaries and the the grothendieck group, the whitehead and the steinberg groups.
The identity in this ring is the class rr1s of the free module r1. One motivation of cyclic homology was the need for an approximation of k theory that is defined, unlike k theory, as the homology of a chain complex. In 1988 i wrote out a brief outline, following quillens paper higher algebraic k theory i. Talks at the conference covered a wide range of current research activities in algebraic \k\theory. This book gives a superb overview of algebraic k theory, and could be read by anyone who has taken a course in commutative algebra or a course in the theory of rings. Every such extension can be represented as all polynomials in an algebraic number k q. Pdf on jan 1, 2007, aderemi kuku and others published representation theory and higher algebraic ktheory find, read and cite all the.
Introduction to algebraic ktheory john milnor download. Euler chaxacteristics and the wall finiteness obstruction 41 chapter 2. Ktheory itself, rather than on these applications of algebraic ktheory. Free k theory books download ebooks online textbooks tutorials. Higher algebraic ktheory of schemes and of derived categories. The procedure describes these groups in terms of the algebraic k theory of various finite subgroups and various farrell nil groups. Geometric, algebraic, and arithmetic objects are assigned objects called k groups. Algebraic ktheory and its applications graduate texts in. Other readers will always be interested in your opinion of the books youve read. We illustrate this process by carrying out the computation for several interesting classes of examples. I suggest looking at the introduction to waldhausens original paper on algebraic k theory algebraic k theory of generalized free products, part i, ann. This process is experimental and the keywords may be updated as the learning algorithm improves. And is there material lecture video or good pdf script where the algebraic ktheory is explained.
In 1985, i started hearing a persistent rumor that i was writing a book on algebraic ktheory. The procedure describes these groups in terms of 1 the algebraic ktheory of. Cyclic cohomology is in fact endowed with a pairing with k theory, and one hopes this pairing to be nondegenerate. Algebraic ktheory is a modern branch of algebra which has many important. Free k theory books download ebooks online textbooks.
The procedure describes these groups in terms of 1 the algebraic k theory of. Thomason and thomas trobaugh to alexander grothendieck on his 60th birthday in this paper we prove a localization theorem for the atheory of com mutative rings and of schemes, theorem 7. The first objective is to understand the relationships between products and the kernel of the hurewicz homomorphism. Springer have made a bunch of books available for free, here. Informally, \ k \ theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebraic and geometric questions. Algebraic k theory and quadratic forms springerlink. Algebraic ktheory is a tool from homological algebra that defines a sequence of functors from rings to abelian groups. Professor milnor sets out, in the present work, to define and study an analogous functor k2, also from associative rings to abelian groups. Algebraic k theory plays an important role in many areas of modern mathematics.
An algebraic introduction to k theory this book is an introduction to k theory and a text in algebra. Methods of algebraic ktheory are actively used in algebra and related fields, achieving interesting results. After a few years, i had heard the rumor from at least a dozen people. This is referred to as vbkt, its provisional title being vector bundles and ktheory. Similarly, the forgetful functor from kalgebras to kmodules k is a commutative ring has the tensor algebra functor as coadjoint. One class of spaces in rn we will be studying will be manifolds or k. Algebraic number theory involves using techniques from mostly commutative. Algebraic ktheory and its applications jonathan rosenberg. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the classical algebraic k theory. As a consequence, we get new groups with trivial whitehead groups. Algebraic ktheory is a modern branch of algebra which has many important applications in fundamental areas of mathematics connected with algebra, topology, algebraic geometry, functional analysis and algebraic number theory. Crediting the initial theory to serre rather than grothendieck is strange and just not true. Dominique arlettaz, algebraic ktheory of rings from a topological viewpoint.
Sep 15, 2006 these keywords were added by machine and not by the authors. Seminar on algebraic k theory and algebraic number theory 1987. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. The equivariant algebraic ktheory is an algebraic ktheory associated to the category. Algebraic topology m382c michael starbird fall 2007.
Algebraic k theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Pdf algebraic k theory and its applications download. Higher algebraic ktheory of schemes and of derived categories r. An introduction to algebraic k theory christian ausoni abstract. Algebraic k theory and its applications, jonathan rosenberg. In algebraic k theory, the k theory of a category c usually equipped with some kind of additional data is a sequence of abelian groups k i c associated to it. These notes are taken from a course on algebraic k theory given at the university of chicago in 1967. The serre spectral sequence and serre class theory 237 9. In mathematics, there are several theorems basic to algebraic k theory throughout, for simplicity, we assume when an exact category is a subcategory of another exact category, we mean it is strictly full subcategory i. The broad range of these topics has tended to give the subject an aura of inapproachability. Pdf representation theory and higher algebraic ktheory. The theoremdefinitionexercise numbers are the same. Daniel grayson, quillens work in algebraic ktheory, j. Download algebraic k theory pdf 45p download free online book chm pdf.
If c is an abelian category, there is no need for extra data, but in general it only makes sense to speak of k theory after specifying on c a structure of an exact category, or of a waldhausen category, or of a dgcategory, or. Algebraic groups and class fields, jeanpierre serre. This paper investigates the product structure in algebraic ktheory of rings. Algebraic l theory, also known as hermitian k theory, is important in surgery theory. Eastweat center algebraic k theory and algebraic number theory. An introduction to algebraic ktheory christian ausoni. This note will develop the ktheory of banach algebras, the theory of extensions of c algebras, and the operator ktheory of kasparov from scratch to its most advanced aspects. The reader will see a common theme throughout algebraic k theory, namely that of abelianization, which is very prevalent throughout modern mathematics. Suslins proof of the karoubi conjecture showing topological k is a special case of algebraic k should also be here, and the q construction should be mentioned. I talked to hy bass, the author of the classic book algebraic k theory, about what would be involved in writing such a book. It was meant as a leisurely exposition of classical aspects of algebraic k theory, with some of its applications to geometry and topology. In this paper, we extend waldhausens results on algebraic ktheory of generalized free produts in a more general setting and we give some properties of the nil functors. We provide a general procedure for computing the algebraic k theory of finitely generated virtually free groups.
Algebraic k theory pdf 45p download book freebookcentre. In 1985, i started hearing a persistent rumor that i was writing a book on algebraic k theory. These are the notes of an introductory lecture given at the 20th winter school for geometry and physics, at srni. W e now want to analyze the algebraic k theory of the virtually free group first of all, we need to look at the quotient graph of groups. Blackadars book for ktheory of operator algebras, and atiyahs book for topological ktheory as it stood in the 1960s, are readable without a lot of algebraic prerequisites. Algebraic ktheory of virtually free groups proceedings of. In mathematics, algebraic l theory is the k theory of quadratic forms. Some formulas in operator theory, involving determinants, are best understood in terms of algebraic k theory. In this paper, we extend waldhausens results on algebraic k theory of generalized free produts in a more general setting and we give some properties of the nil functors. Algebraic k theory plays an important role in many subjects, especially number theory, algebraic topology and algebraic geometry. For instance, the class group of a number eld kis essentially k0ok. This paper investigates the product structure in algebraic k theory of rings. There is also a third book in progress, on vector bundles, characteristic classes, and ktheory, which will be largely independent of ssat and also of much of the present book.
865 954 1484 122 158 1334 1504 312 597 196 1051 991 144 1523 942 1328 1565 1142 555 246 1349 338 854 1406 821 120 690 329 877 417 789 843 1044 1188 1200