In chapter 6 we describe the structure of the ktheory of. The point is that the hermitian ktheory of symmetric forms over r is also the hermitian ktheory of the ring m 2 r with involution which are homotopy equivalences forgetting the c 2 action. For each projective algebraic variety, grothendieck constructed a group from the category of coherent algebraic sheaves, and showed that it had many nice properties. Here is the download access for ebook introduction algebraic k theory. In this article, we further the study of higher ktheory of differential graded dg categories via universal invariants, initiated in g.
Fillmore, extensions of calgebras, operators with compact selfcommutators, and khomology, bull. An elementary introduction by max karoubi conference at the clay mathematics research academy the purpose of these notes is to give a feeling of ktheory, a new interdisciplinary subject within mathematics. The complex case was proved in 199091 by the second author and andrei suslin. A partial solution to the left is given by the theorem a few lines below. Sep 18, 2008 topological theory has become an important tool in topology. This theory was invented by alexander grothendieck1 bs in the 50s in order to solve some difficult problems in algebraic geometry the letter. This idea of ktheory has invaded other parts of mathematics, for example number theory ga, topology bott. It is also a fundamental tool in the field of operator algebras. This essay aims to provide a quick and accessible introduction to k theory, including how to calculate with it, and some of its additional features such as characteristic. The main purpose of the present article is to establish the real case of karoubis conjecture in algebraic ktheory. An elementary introduction by max karoubi conference at the clay mathematics research academy the purpose of these notes is to give a feeling of k theory, a new interdisciplinary subject within mathematics. For commutative unital calgebras, alias continuous functions on compact spaces, there are two equivalent descriptions of the k groups, each with its own advantages.
In algebraic topology, it is a cohomology theory known as topological k theory. Periodicity of hermitian ktheory and milnors k groups. This survey paper is an expanded version of lectures given at the clay mathematics academy. A much more complete introduction to ktheory may be found in the. Bott periodicity in topological, algebraic and hermitian ktheory. In closing, then, ktheory, an introduction is a phenomenally attractive book.
Topological k theory, k theory of c algebras, geometric and topological invarients, the functors k1 k2, k1, sk1 of orders and grouprings, higher algebraic k theory, higher dimensional class groups of orders and group rings, higher k theory of schemes, modm higher k theory of exact categories, schemes. Here is a preliminary discussion, intended to lead the way into the subject and to motivate some of the constructions involved. Some k theory of calgebras books also mention a little topological k theory as a background, you can see this book. 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. 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. Moreover, it is possible to derive a substantial part of stable homotopy theory from a theory cf. Grothendieck in his formulation of the riemann roch theorem cf. Therefore, they are accessible to a mathematician of any speciality willing to understand the subject. An introduction classics in mathematics series by max karoubi. An introduction mathematical association of america. I am making some videos of k theory from topological to operator in my language chinese, if you can read chinese or have some friend help to. Using theory, adams and atiyah were able to give a simple proof that the only spheres which can be provided with space structures are s, s and s. The theoremdefinitionexercise numbers are the same.
Universal karoubi characteristic classes of nuclear c. Ktheory, an introduction is a phenomenally attractive book. In algebra and algebraic geometry, it is referred to as algebraic k theory. Download k theory and algebraic geometry ebook pdf or read. Algebraic topology hatcher solution free pdf file sharing. Chang 115 an introduction to independence for analysts, h. Lectures on operator ktheory and the atiyahsinger index. For each projective algebraic variety, grothendieck constructed a group from the category. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. A more general version of the elliptic package 30 chapter 3.
An elementary introduction by max karoubi clay mathematics academy the purpose of these notes is to give a feeling of k theory, a new interdisciplinary subject within mathematics. The proof of the periodicity is that of atiyah and bott 11 with the modi. Textbook or lecture notes in topological ktheory mathoverflow. This theory was invented by alexander grothendieck1 bs in the 50s. Only a master like karoubi could have written the book, and it will continue to be responsible for many seductions of fledglings to the ranks of topological k theorists as well as serve as a fundamental reference and source of. For each projective algebraic variety, grothendieck constructed a group from the category of coherent algebraic sheaves, and showed that it had many. These considerations lead to the following definition of the ktheory of a ring with unit. An elementary introduction by max karoubi k theory. 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.
These considerations lead to the following definition of the ktheory of a ring. This handbook offers a compilation of techniques and results in k theory. In fact, rational cohomology is defined in terms of k theory. Algebraic and hermitian ktheory of krings request pdf. Grothendieck in his formulation of the riemann roch theorem. Get a printable copy pdf file of the complete article 296k, or click on a page image below to browse page by page. 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. Grotherdieck who first studied k0c then written kc where for a scheme x, c is the category px of locally free sheaves of oxmodules. Using k theory, adams and atiyah were able to give a simple proof that the only spheres which can be provided with hspace structures are s1, s3 and s7.
Geometric, algebraic, and arithmetic objects are assigned objects called k groups. K theory is one of the most important constructions in operator algebras, noncommutative geometry and in topology with a host of applications in mathematics and in physics. Such a preamble may be useful, since modern algebraic k theory relies on quite a large body of technical foundations, and it is easily possible to get sidetracked by developing. Karoubis classic ktheory, an introduction is to provide advanced students and mathematicians in other fields with the fundamental material in this subject. Pdf periodicity of hermitian ktheory and milnors k groups. Algebraic k theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Our projection bundle is a vector bundle in the sense of anderson. These two volumes consist of chapters, each of which is dedicated to a specific topic and is written by a leading expert. Topological ktheory has become an important tool in topology. This was before the general definition of algebraic ktheory by quillen. Lecture 3 presents some of machinery of topological ktheory. A much more complete introduction to ktheory may be found in the handbook of k. An elementary introduction by max karoubi conference at the clay mathematics research academy algebraic geometry, number theory, source. The theory is taken as far as the solution of the hopf invariant problem and a start is mode on the jhomomorphism.
Cartan, karoubi points out that ktheory was introduced by grothendieck in his formulation. Over the point this is the atiyahbottshapiro isomorphism. Given a differential k theory class on x, we define its analytic and topological indices as differential k theory classes on b. Moreover, it is possible to derive a substantial part of stable homotopy theory from ktheory. Download pdf k theory and algebraic geometry free online. Gonshor 1 lectures on the asymptotic theory of ideals, d.
866 1555 621 1567 1435 1061 75 1285 1553 191 806 76 759 773 410 1258 1500 346 298 1387 1420 162 1263 821 1205 307 21 125 1159 497 1106 261