The conference concluded the activity of the algebraic \ k \ theory seminar held at the adam mickiewicz university in the academic year 19941995. It is also often considered, for this reason, as a sub. Algebraic number theory distinguishes itself within number theory by its use of techniques from abstract algebra to approach problems of a numbertheoretic nature. For instance, the algebraic lgroups are the recipients for various surgery obstructions and hence highly relevant for the classi cation of manifolds. Algebraic ktheory of 1operads 616 k theory groups k n. Algebraic ktheory, assembly maps, controlled algebra, and. 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. This volume contains previously unpublished papers on algebraic \ k \ theory written by leningrad mathematicians over the last few years. These lectures notes follow the structure of the lectures given by c. Some formulas in operator theory, involving determinants, are best understood in terms of algebraic k theory. Geometric, algebraic, and arithmetic objects are assigned objects called k groups. Algebraic topology this book, published in 2002, is a beginning graduatelevel textbook on algebraic topology from a fairly classical point of view.
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. 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. F will be described in four interesting special cases. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the classical algebraic k theory. It is due to friedhelm waldhausen and concerns a category with cofibrations and weak equivalences. Algebraic number theory was born when euler used algebraic num bers to solve diophantine equations suc h as y 2 x 3.
Friedlander some remarks on the ktheory of fields, in applications of algebraic ktheory to algebraic geometry and number theory, part i, ii boulder, 1983, contemp. I suggest looking at the introduction to waldhausens original paper on algebraic k theory algebraic k theory of generalized free products, part i, ann. Scholl in this paper, which follows closely the talk given at the conference, i will sketch an example of a nontrivial element of k2 of a certain threefold, whose existence is related to the vanishing of. Quillens work in algebraic ktheory 5 for a group g, there is a classifying space bg, whose only nonvanishing homotopy group is its fundamental group, with. To find out more or to download it in electronic form, follow this link to the download page. Introductory algebraic number theory saban alaca, kenneth s. The whole theory was initiated by quillen qui73, but see also tho82. Algebraic ktheory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Weibel received 10 july 1984 revised 1 march 1985 dedicated to professor nobuo shimada on his 60th birthday introduction.
This paper gives a contribution to the computation of algebraic ktheory. Algebraic k theory is a tool from homological algebra that defines a sequence of functors from rings to abelian groups. Quillens work in algebraic ktheory semantic scholar. Combining this with the second corollary of the stability theorem. Di erential function spectra, the di erential beckergottlieb. Algebraic k theory plays an important role in many areas of modern mathematics. All these constructions give the same ktheory of a ring, but are useful in. Algebraic ktheory and its applications springerlink. Lectures on topics in algebraic ktheory by hyman bass note by amit roy no part of this book may be reproduced in any form by print, micro. We the study the algebraic ktheory of c algebras, forgetting the.
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. An algebraic introduction to k theory this book is an introduction to k theory and a text in algebra. The prerequisites for a course based on this book include a working. Applications of algebraic k theory to algebraic geometry and number theory, part 1 about this title. Simple knots surgery theory algebraic transversality the topological invariance of rational pontrjagin classes localization algebraic k and l theory invariants of knots codimension qsurgery fredholm localization structure part one algebraic k theory 1. Familiarity with these topics is important not just for a topology student but any student of pure mathematics, including the student moving towards research in geometry, algebra, or analysis. Beckergottlieb transfer, and applications to di erential algebraic ktheory ulrich bunke and david gepnery february 9, 2016 abstract we develop di erential algebraic ktheory for rings of integers in number elds and we construct a cycle map from geometrized bundles of modules over such a ring to the di erential algebraic ktheory. Article pdf available in ktheory 121 september 1997 with 91 reads. Hausdorff \ etale zappasz\ep product groupoid is a cblend, in the. Methods of algebraic ktheory are actively used in algebra and related fields, achieving interesting results. On one hand the resulting space k c should be betterbehaved, more strongly structured and possibly more easily analyzed than c. 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 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. Algebraic ktheory of 1operads 616 ktheory groups k n. Organizers thomas geisser, tokyo lars hesselholt, copenhagen annette huberklawitter, freiburg moritz kerz, regensburg. Newest algebraicktheory questions mathematics stack. Algebraic ktheory describes a branch of algebra that centers about two functors. Ktheory 11 20, 527547 pdf an introductory textbook account is in charles weibel, the kbook. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.
Higher algebraic k theory of schemes 249 theory of perfect complexes on schemes from sga 6. In the classical theory of quadratic forms the ground ring is a eld, or a ring of integers in an algebraic number eld, and quadratic forms are. In chapter 2 we will see that the converse of exercise 1. The term algebraic l theory was coined by wall, to mean the algebraic k theory of quadratic forms, alias hermitian k theory. Lectures on the theory of algebraic functions of one variable. Algebraic graph theory without orientation pdf free download. Daniel quillens seminal paper higher algebraic ktheory.
Quillens localisation theorem in algebraic ktheory is considerably easier to prove. Multiple categories and algebraic k theory kazuhisa shimakawa research institute for mathematical sciences, kyoto university, sakyoku, kyoto 606, japan communicated by h. Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. As such, algebraic ktheory is a metatheme for mathematics, but the successful. Algebraic ktheory, algebraic cycles and arithmetic geometry. Numbertheoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields.
On the other hand, k 1p o fq o f is a nontrivial theorem. 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. In section 1 the mod two calculations are re viewed. Comparison between algebraic and topological ktheory for.
Geometric, algebraic, and arithmetic objects are assigned objects called kgroups. The kbook an introduction to algebraic ktheory citeseerx. Quillen, higher k theory for categories with exact sequences, to appear in the proceedings of the june 1972 oxford symposium, new. It is a perfect source book for seasoned graduate students and working researchers who are willing and eager to follow the authors expository path and who are ready for a lot of additional reading and selfreliant work. Algebraic ktheory of group rings and the cyclotomic trace map. Professor milnor sets out, in the present work, to define and study an analogous functor k2, also from associative rings to abelian groups. Key to quillens proof 56 of the adams conjecture 5 see also 24 was his discovery that for each power q of a prime number p there is a homology equivalence bgl 1f.
The broad range of these topics has tended to give the subject an aura of inapproachability. Download pdf k theory and algebraic geometry free online. Many areas of mathematics are used in coding theory, and we focus on the interplay between algebra and coding theory. Talks at the conference covered a wide range of current research activities in algebraic \ k \ theory. Grotherdieck who first studied k0c then written k c where for a scheme x, c is the category px of locally free sheaves of oxmodules. Charles weibels k book offers a plethora of material from both classical and more recent algebraic k theory. Daniel grayson, quillens work in algebraic ktheory, j. It was discovered, starting in the early 80s, that the \comparison map from algebraic to topological k theory is a surprisingly rich object. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. Ktheory itself, rather than on these applications of algebraic ktheory.
An important feature of algebraic ktheory, and one which has led to genuinely new. The basic idea of algebraic ktheory is to consider a modi. Lectures on k theory pdf 95p this lecture note covers the following topics. Under the relation between modules and vector bundles. The seminar on algebraic k theory and algebraic number theory was held at the east west center in honolulu, hawaii on january 1216, 1987. Algebraic ktheory by olivier isely epfl algebraic ktheory is a branch of algebra dealing with linear algebra over a general ring a instead of over a field.
In algebra, the sconstruction is a construction in algebraic k theory that produces a model that can be used to define higher k groups. Pdf the algebraic ktheory of operator algebras researchgate. Section 4 provides the ab ov e exact sequences involving the k. An algebraic number is an algebraic integer if it is a root of some monic polynomial fx 2 zx i. Pdf algebraic k theory and its applications download. The overriding concern of algebraic number theory is the study. Waldhausen k theory of the derived category and local chern classes.
Thus the task of the present paper is to erect the twoadic type upon the mod two type. Every such extension can be represented as all polynomials in an algebraic number k q. Fermat had claimed that x, y 3, 5 is the only solution in. Algebraic coding theory is an area of discrete applied mathematics that is concerned in part with developing errorcontrol codes and encodingdecoding procedures. Algebraic ktheory of generalized free products, part 1 authors. Algebraic k theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Algebraic ktheory and quadratic forms reed college. So gauss, dirichlet, kummer, and dedekind were all actually studying algebraic ktheory of number. The expert might skip this, but should look at the characterization of perfect complexes as finitely presented objects 2. Algebraic ktheory plays an important role in many subjects, especially number theory, algebraic topology and algebraic geometry.
Methods of algebraic k theory are actively used in algebra and related fields, achieving interesting results. If is a rational number which is also an algebraic integer, then 2 z. Higher algebraic ktheory of schemes and of derived categories. Algebraic ktheory and quadratic forms 321 to conclude this section, the ring k. The main topic of the first part is the computation of \ k \ theory and \ k \cohomology for special varieties, such as group varieties and their principal homogeneous spaces, flag fiber bundles and their twisted forms, \\lambda\operations in higher \ k. As in the free product case, the functions so and vo combine to give a. Calculations of algebraic ktheory are very rare and hard to come by. 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. A few words these are lecture notes for the class on introduction to algebraic number theory, given at ntu from january to april 2009 and 2010.
1452 1258 1252 1137 1219 1193 752 420 845 1386 1451 762 1549 826 1318 1369 1540 27 1091 1426 673 4 125 1352 721 1407 1183 80 1120 1561 915 102 1 1328 1013 222 296 719 967 471