Seminar on algebraic ktheory and algebraic number theory 1987. The higher kgroups, as defined by quillen, will be the subject of the course higher algebraic ktheory in the fall semester of 2011. Lind, hisham sati, and craig westerland june 8, 2017 abstract we introduce a periodic form of the iterated algebraic ktheory of ku, the connective complex ktheory spectrum, as well as a natural twisting of this cohomology theory by higher gerbes. Comparison between algebraic and topological ktheory for banach algebras and c algebras jonathan rosenberg. If is a rational number which is also an algebraic integer, then 2 z. The main reference used in the abstracts is ros94, but further text books are mentioned at the end sri96,wei. A minicourse on recent progress in algebraic ktheory and its. Eastweat center algebraic ktheory and algebraic number theory. Selected titles in this series american mathematical society. The semitopological ktheory of a complex variety was defined in fw2 with the expectation that it would prove to be a theory lying part way between the algebraic ktheory of the variety. Quillens constructions of higher algebraic ktheory.
Algebraic theory is a sequence of functors which, roughly speaking, extracts abelian invariants from linear algebra construction of. Ktheory itself, rather than on these applications of algebraic ktheory. Algebraic ktheory has become an increasingly active area of research. The euler operator defines via kasparov theory an element, called the equivariant euler class, in the equivariant k o homology of m. This was before the general definition of algebraic ktheory by quillen. About the same time, it was also found that the algebraic as opposed to topological ktheory of operator algebras does have some direct applications in operator theory. Algebraic ktheory and its applications errata to the.
Download pdf k theory and algebraic geometry free online. Algebraic ktheory and its applications graduate texts in mathematics, no. They contain detailed information about the original object but are notoriously difficult to compute. The question, i think, is whether basic theorems of algebraic ktheory is an appropriate new topic to draft, or whether it. Twisted iterated algebraic ktheory and topological tduality for sphere bundles john a. The whole theory was initiated by quillen qui73, but see also tho82. Driven by applications to number theory, and then algebraic geometry, higher algebraic ktheory was defined around 1970 by quillen and others. Algebraic ktheory and its applications by jonathan rosenberg. The broad range of these topics has tended to give the subject an aura of inapproachability.
Part c category theory or homological algebra part c algebraic geometry or introduction to schemes part b galois theory or algebraic number theory useful reading. Lectures on k theory pdf 95p this lecture note covers the following topics. An introduction to algebraic ktheory christian ausoni. This text is designed to help graduate students in other areas learn the basics of ktheory and get a feel for its many applications. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and students in mathematics. Comparison between algebraic and topological ktheory for. Lectures on topics in algebraic k theory pdf 8p this note covers the following topics.
The exact sequence of algebraic ktheory, categories of modules and their equivalences, brauer group of a commutative ring, brauerwall group of graded azumaya algebras and the structure of the clifford functor. Lecture notes on algebraic k theory pdf 252p download book. Every such extension can be represented as all polynomials in an algebraic number k q. Algebraic ktheory we now briefly explain the basics of algebraic theory see 7 and 8 and state borels theorem on these rational groups of. We the study the algebraic ktheory of c algebras, forgetting the.
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. Let g be a countable discrete group and let m be a smooth proper cocompact gmanifold without boundary. Algebraic ktheory is crucial in many areas of modern mathematics, especially algebraic topology, number. I am quite grateful to those who have sent me their. The universal equivariant euler characteristic of m, which lives in a group u g m, counts the equivariant cells of m, taking the component structure of the. Algebraic ktheory and its applications springerlink. This book is based on lectures given by the author at the tata institute in bombay. I suggest looking at the introduction to waldhausens original paper on algebraic ktheory algebraic ktheory of generalized free products, part i, ann. Algebraic ktheory and its applications ebok jonathan. Bott periodicity in topological, algebraic and hermitian k. This book, based on a course at the university of maryland in the fall of 1990, is intended to enable graduate. Pdf algebraic k theory and its applications download.
Algebraic ktheory, which to any ring r associates a sequence of groups k 0 r, k 1 r, k 2 r, etc. Methods of algebraic ktheory are actively used in algebra and related fields, achieving interesting results. Historically, the algebraic ktheory of a commutative ring what today is the 0th algebraic ktheory group was originally defined to be the grothendieck group of its symmetric monoidal category of projective modules under tensor product of modules. Please join the simons foundation and our generous member organizations in supporting arxiv during our giving campaign september 2327. I especially thank paul arne ostvaer, ioannis emmanouil, desmond sheiham, efton park, jon berrick, henrik holm, mike boyle, and hanfeng li for their corrections. Algebraic number theory involves using techniques from mostly commutative algebra and nite group theory to gain a deeper understanding of the arithmetic of number elds and related objects e. We will study in detail the first three of these groups. Euler chaxacteristics and the wall finiteness obstruction 41 chapter 2. Algebraic ktheory and its applications, jonathan rosenberg. 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. Algebraic ktheory and its applications jonathan rosenberg. Algebraic ktheory and its applications by jonathan rosenberg errata to the second printing, 1996 i am quite grateful to those who have sent me their comments on this book. When people write algebraic ktheory today they mean the quillen version or other later developments of higher algebraic ktheory, and specify milnor ktheory when they mean that. Jonathan rosenberg algebraic ktheory plays an important role in many areas of modern mathematics.
After a few years, i had heard the rumor from at least a dozen people. Topological ktheory of complex noncommutative spaces volume 152 issue 3 anthony blanc. Surveys with accounts of the historical development include. Topological and bivariant ktheory joachim cuntz springer. Rosenberg, \on a combinatorial identity of djakov and mityagin, utilitas mathematica 100 2016, pp. Algebraic ktheory plays an important role in many areas of modern mathematics. Algebraic ktheory of 1operads 616 ktheory groups k n. Pdf algebraic ktheory and its applications semantic scholar. Click download or read online button to k theory and algebraic geometry book pdf for free now. The subject developed steadily in the 1970s, balanced between its intrinsic structure and its applications. The reader will see a common theme throughout algebraic ktheory, namely that of abelianization, which. Daniel grayson, quillens work in algebraic ktheory, j. For applications of algebraic ktheory in geometric topology, the most. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic ktheory.
Under the relation between modules and vector bundles. Algebraic ktheory plays an important role in many areas of modern. Jonathan micah rosenberg born december 30, 1951 in chicago, illinois is an american mathematician, working in algebraic topology, operator algebras, ktheory and representation theory, with applications to string theory especially dualities in physics rosenberg received his ph. Topological ktheory is one of the most important invariants for noncommutative algebras equipped with a suitable topology or bornology. Classically, algebraic ktheory of rings is the study of the family of ktheory. This book gives a superb overview of algebraic ktheory, and could be read by anyone who has taken a course in commutative algebra or a course in the theory of rings.
Mixed tate motives, algebraic ktheory and multiple zeta. Ros94 jonathan rosenberg, algebraic ktheory and its applications, graduate texts in mathematics, vol. Springer have made a bunch of books available for free. Suppose fab 0 where fx p n j0 a jx j with a n 1 and where a and b are relatively prime integers with b0. Gennady kasparov, the operator k kfunctor and extensions of c c\astalgebras, izv.
Algebraic ktheory and its applications graduate texts in. Jonathan rosenberg at university of maryland, college park. In 1985, i started hearing a persistent rumor that i was writing a book on algebraic ktheory. Topological ktheory of complex noncommutative spaces. Algebraic ktheory and its applications 1995, graduate texts in mathematics 147, springer. Recent progress in algebraic ktheory and its relationship with.
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