I am reading algebraic ktheory but i have doubt not in the subject but in the name. An algebraic introduction to k theory this book is an introduction to k theory and a text in algebra. These notes evolved from the lecture notes of a minicourse given in swisk, the sedano winter school on ktheory held in sedano, spain, during the week january 2227 of 2007, and from those of a longer course given in the university of buenos aires, during the second half of 2006. Algebraic k theory 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. I suggest looking at the introduction to waldhausens original paper on algebraic k theory algebraic k theory of generalized free products, part i, ann. More significantly still, in the 1960s atiyah, the american isadore singer, and others found ways of connecting this work to the study of a wide variety of questions involving partial differentiation, culminating in the celebrated atiyah. An algebraic introduction to ktheory this book is an introduction to ktheory and a text in algebra.
Hirzebrch introduced the analogous theory for topological spaces 2 based on topological vector bundles. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the classical algebraic ktheory. Newest algebraicktheory questions mathematics stack. An introduction to ktheory for calgebras and secondarily on b. This book is an accessible introduction to the basics and provides detailed explanations of the various concepts required for a deeper understanding of the subject. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the classical algebraic k theory. Topological k theory robert hines december 15, 2016 the idea of topological k theory is that spaces can be distinguished by the vector bundles they support. Some familiarity with basic calgebra theory is assumed. Theory x management style generally get poor results compare with managers use theory y, which produces better performance and results, and allows people to grow and develop. What we have in mind here is that the square diagram ka k ka p. Using this, we give new, higher categorical proofs of the approximation, additivity, and fibration theorems of waldhausen in this context.
Algebraic k theory, which is the main character of this book, deals mainly with studying the structure of rings. Algebraic ktheory, which is the main character of this book, deals mainly with studying the structure of rings. K theory is often considered a complicated mathematical theory for specialists only. The material on ktheory of schemes is isolated in a separate section, so it may be skipped by those not interested in algebraic geometry. Lectures on topics in algebraic k theory pdf 8p this note covers the following topics. The author also discusses to what extent ktheory can be viewed as a homology for rings.
A much more complete introduction to k theory may be found in the handbook of k theory, recently edited by springer. Higher algebraic ktheory of schemes and of derived categories. Ktheory and calgebras rainer matthes wojciech szyma nski. However, it turns out that even working in a purely algebraic context, one requires techniques from homotopy theory to construct the higher k groups and to perform computations.
Because k0cclassifies the isomorphism classes in c and he wanted the name of the theory to reflect class, he used the first letter k in. In algebra and algebraic geometry, it is referred to as algebraic ktheory. However, it turns out that even working in a purely algebraic context, one requires techniques from homotopy theory to construct the higher kgroups and to perform computations. While some emphasized that sociological theory or social theory is a i w hati st he or y.
Higher algebraic ktheory of schemes 249 theory of perfect complexes on schemes from sga 6. Given the recent developments between the journal of k theory editorial board and bak, im a whole lot less hopeful that issues regarding k theory will be sorted out anytime soon. The question, i think, is whether basic theorems of algebraic ktheory is an appropriate new topic to draft, or whether it. Let k r be the field with r qa elements, where q is a prime which does not divide the order of g. Every such extension can be represented as all polynomials in an algebraic number k q. It is also a fundamental tool in the field of operator algebras. Introduction to algebraic ktheory mona merling due to its historical development, algebraic ktheory has two main components. Algebraic ktheory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.
Our decision to begin this lecture series on modern social theory with the question what is theory. The other historical origin of algebraic k theory was the work of j. Topological ktheory robert hines december 15, 2016 the idea of topological ktheory is that spaces can be distinguished by the vector bundles they support. Therefore, they are accessible to a mathematician of any speciality willing to understand the subject. I want to ask what k stands in the word algebraic ktheory as well as in topological ktheory. An elementary introduction by max karoubi clay mathematics academy the purpose of these notes is to give a feeling of ktheory, a new interdisciplinary subject within mathematics. A much more complete introduction to ktheory may be found in the. Higher algebraic ktheory of schemes and of derived. I suggest looking at the introduction to waldhausens original paper on algebraic ktheory algebraic ktheory of generalized free products, part i, ann.
Daniel grayson, quillens work in algebraic ktheory, j. 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. They contain detailed information about the original object but are notoriously difficult to compute. 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. Algebraic ktheory and its applications jonathan rosenberg. Friedlander received 24 april 1986 let gl7l respectively slzl be the infinite general respectively special linear group and stzl the infinite steinberg group of 7l. Ktheory and motivic cohomology of schemes, i 3 an outline of the paper is as follows. Lectures on k theory pdf 95p this lecture note covers the following topics. Higher algebraic k theory of schemes 249 theory of perfect complexes on schemes from sga 6. In 1988 i wrote out a brief outline, following quillens paper higher algebraic ktheory i. Topological ktheory shortly after grothendieck introduced k 0x for an algebraic variety x, m. This is relative k theory, which is defined for twosided ideals in r. Lecture notes on algebraic k theory pdf 252p download book.
Algebraic k theory pdf 45p this note introduces the definitions of the ktheory groups such as the group k0, the group k1, the group k2 and higher ktheory groups. For instance, the class group of a number eld kis essentially k0ok. Crediting the initial theory to serre rather than grothendieck is strange and just not true. The expert might skip this, but should look at the characterization of perfect complexes as finitely presented objects 2.
Geometric, algebraic, and arithmetic objects are assigned objects called k groups. So gauss, dirichlet, kummer, and dedekind were all actually studying algebraic ktheory of number. Grotherdieck who first studied k0c then written kc where for a scheme x, c is the category px of locally free sheaves of oxmodules. Also, he gives the reader a taste of topological ktheory. Algebraic ktheory and its applications springerlink. Algebraic number theory involves using techniques from mostly commutative algebra and. Several examples of k0 are discussed, such as when r is a principal ideal domain, a local ring, and a dedekind domain. This is relative ktheory, which is defined for twosided ideals in r. In the third chapter we give a brief overview of the classical ktheory for k1 and k2 of a ring. Algebraic ktheory plays an important role in many areas of modern mathematics.
Geometric, algebraic, and arithmetic objects are assigned objects called kgroups. Also, he gives the reader a taste of topological k theory. Via the fundamental theorem, this leads to bass negative. I talked to hy bass, the author of the classic book algebraic ktheory, about what would be involved in writing such a book.
Dominique arlettaz, algebraic ktheory of rings from a topological viewpoint. Pdf holomorphic ktheory, algebraic cocycles, and loop. Some formulas in operator theory, involving determinants, are best understood in terms of algebraic k theory. On the other hand, k 1p o fq o f is a nontrivial theorem.
The main objects that we study in algebraic number theory are number. Algebraic k theory plays an important role in many subjects, especially number theory, algebraic topology and algebraic geometry. Mcgregors xy theory is a natural rule for managing people. Pdf algebraic k theory and its applications download. In mathematics, ktheory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. D epartm en t of m athem atics, n orthw estern u n iversity, e van ston, u s a l ectu res given at the s chool on a lgebraic k theory an d its a pplication s t rieste, 14 25 m ay 2007 l n s 0823001.
Ps if one is particularly persistent and know the right people, i believe there is a complete collection of k theory articles out there on the internet. This theory was invented by alexander grothendieck1 bs in the 50s in order to solve some difficult problems in algebraic geometry the letter. The relationship between mcgregors xy theory management. Methods of algebraic k theory are actively used in algebra and related fields, achieving interesting results. What happened to online articles published in ktheory. Daniel grayson, quillens work in algebraic k theory, j.
Ktheory is often considered a complicated mathematical theory for specialists only. Given the recent developments between the journal of ktheory editorial board and bak, im a whole lot less hopeful that issues regarding ktheory will be sorted out anytime soon. As applications of this technology, we study the algebraic ktheory of associative rings in a wide range of homotopical contexts and of. The book then follows a careful construction and analysis of the operator ktheory groups. What we have in mind here is that the square diagram ka kap k7. The broad range of these topics has tended to give the subject an aura of inapproachability. Algebraic k theory plays an important role in many areas of modern mathematics. The book then follows a careful construction and analysis of the operator k theory groups. Some formulas in operator theory, involving determinants, are best understood in terms of algebraic ktheory. Pdf holomorphic ktheory, algebraic cocycles, and loop groups. Surveys with accounts of the historical development include. Whitehead and others on what later became known as whitehead torsion there followed a period in which there were various partial definitions of higher k theory functors.
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. Mcgregors ideas suggest that there are two fundamental approaches to managing people. 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. This survey paper is an expanded version of lectures given at the clay mathematics academy.
These notes evolved from the lecture notes of a minicourse given in swisk, the sedano winter school on ktheory held in sedano, spain, during the week january 2227 of 2007, and from those of a longer course given in the. Algebraic ktheory and its applications graduate texts in. In doing so we need the following two equivalence relations. In algebraic topology, it is a cohomology theory known as topological ktheory. Algebraic k theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.
K theory itself, rather than on these applications of algebraic ktheory. Ps if one is particularly persistent and know the right people, i believe there is a complete collection of ktheory articles out there on the internet. Algebraic ktheory plays an important role in many subjects, especially number theory, algebraic topology and algebraic geometry. The author also discusses to what extent k theory can be viewed as a homology for rings.
768 189 1174 373 702 673 1449 1062 170 1407 430 566 972 204 1126 992 468 707 1296 965 485 1094 1023 392 1432 543 138 304 685 1364 565 476 205 975 24 807 1303 448 767 882 337