Higher algebraic ktheory after quillen, thomason and others marco schlichting mathematics institute, zeeman building, university of warwick, coventry cv4 7al uk, m. Other readers will always be interested in your opinion of the books youve read. Higher algebraic ktheory after quillen, thomason and others 3 b let abe an exact category, and let babe a full additive subcategory closed under extensions in a. After surveying higher \ k\ theory of toric varieties, we present totaros old c. Some applications 350 nonvanishing of class groups and whitehead groups 350 idempotents in calgebras 355 group rings and assembly maps 362 references 369 books and monographs on related areas of algebra, analysis, number theory, and topology 369 books and monographs on algebraic iftheory 371. Characteristic classes of higher rank bundles 58 4. An algebraic introduction to ktheory this book is an introduction to ktheory 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 ktheory is about natural constructions of cohomology theoriesspectra from. Abstract we present an introduction with a few proofs to higher algebraic k theory of schemes based on the work of quillen, waldhausen. Thomason and thomas trobaugh to alexander grothendieck. But thats the original motivation and not the way the work in the field is currently going from your question it seems like youre asking about a motivation for higher algebraic k theory. Algebraic ktheory of brave new rings provides a very interesting link between algebraic. Topological ktheory of complex noncommutative spaces volume 152 issue 3 anthony blanc.
Lecture notes on the ktheory of operator algebras based primarily on m. Algebraic ktheory and quadratic forms reed college. Pdf higher algebraic ktheory after quillen, thomason and. Algebraic ktheory and its applications graduate texts in. They contain detailed information about the original object but are notoriously difficult to compute. Professor milnor sets out, in the present work, to define and study an analogous functor k2, also from associative rings to abelian groups. On the other hand, k 1p o fq o f is a nontrivial theorem due to bass, milnor, and serre see. What links here related changes upload file special pages permanent. A variant of algebraic ktheory for operator algebras banach algebras and in particular cstar algebras is operator ktheory. We identify the milnor ktheory of a field with a certain higher chow group. Abstract we present an introduction with a few proofs to higher algebraic ktheory of schemes based on the work of quillen, waldhausen. Making use of the recent theory of noncommutative motives, we prove that every additive invariant satisfies galois descent.
Using this, we give new, higher categorical proofs of the approximation, additivity, and fibration theorems of waldhausen in this context. Nigel higson, categories of fractions and excision in kktheory j. Higher algebraic ktheory of schemes and of derived categories. 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.
A variant of algebraic k theory for operator algebras banach algebras and in particular cstar algebras is operator k theory. Milnor ktheory is the simplest part of algebraic ktheory. The ktheory of z 579 bibliography 589 indexofnotation 599 index 605. The algebraic k theory of the whole subcategory of cofibrant objects of a model category is trivial by a version of the eilenberg swindle. Ehrlich fibrations and a transfer map in algebraic k theory cells and moreover b v vs will dominate a pp.
Quillens localisation theorem in algebraic ktheory is considerably easier to prove. For quillens ktheory, quillen showed in q that applying the functor. Textbook or lecture notes in topological ktheory mathoverflow. Higher algebraic ktheory of schemes and of derived categories r. Algebraic k theory plays an important role in many areas of modern mathematics. 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. F will be described in four interesting special cases. K theory, kkdualities and applications to topological \mathbbtduality pdf. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the classical algebraic k theory.
From the perspective of homotopy theory, algebraic k theory has a certain universality. Tabuada, higher ktheory via universal invariants tabuada, noncommutative motives. Waldhausen to a certain type of rings up to homotopy, called brave new rings. In algebraic topology, it is a cohomology theory known as topological k theory. The reader will see a common theme throughout algebraic k theory, namely that of abelianization, which is very prevalent throughout modern mathematics. Lectures on k theory pdf 95p this lecture note covers the following topics. The author shows that this definition of ktheory does coincide with that of k0 and k1 done earlier in the book. Galois descent of additive invariants bulletin of the. 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. 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. The first part pf the paper is concerned with the general theory of these xgroups. As applications of this technology, we study the algebraic ktheory of associative rings in a wide range of homotopical contexts and of.
The meaning of theory 175 question, it does not force us to think about metaphysics, reality, essences, being, and the like. Munkholm mathematics department, university of odense, dk5230 odense m, denmark communicated by h. Professor milnor sets out, in the present work, to define and study an analogous functor. Call a sequence 1 in ba conation if it is a conation in a. Lectures on operator ktheory and the atiyahsinger index. Algebraic ktheory and its applications by jonathan rosenberg. Algebraic ktheory of brave new rings provides a very interesting link between algebraic ktheory of rings and geometry. Milnor k theory is the simplest part of algebraic k theory burt totaro department of mathematics, university, of chicago, 5734 s. We present an introduction with a few proofs to higher algebraic ktheory of schemes based on the work of quillen, waldhausen, thomason and others. On the algebraic ktheory of model categories sciencedirect.
I am looking for a good introductory level textbook or set of lecture notes on classical topological k theory that would be suitable for a onesemester graduate course. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the classical algebraic ktheory. Dominique arlettaz, algebraic ktheory of rings from a topological viewpoint pdf. Algebraic ktheory and quadratic forms 321 to conclude this section, the ring k. Higher algebraic ktheory after quillen, thomason and others. As applications of this technology, we study the algebraic ktheory of associative rings in a wide range of homotopical contexts and of spectral delignemumford stacks. Robert wayne thomason 5 november 1952 tulsa, oklahoma, u. A central part of the book is a detailed exposition of the ideas of quillen as contained in his classic papers higher algebraic ktheory, i, ii. Wall shows that any element x e ko7rib can be realized by a space a build up from b v vs by attaching higher dimensional 4 k. A geometric introduction to ktheory daniel dugger contents preface 3 introduction 4 1. I it is ths goal of the present paper to show that this definition leads to an interesting theory. So gauss, dirichlet, kummer, and dedekind were all actually studying algebraic ktheory of number. Waldhausen gave a more general definition of algebraic ktheory that accepts a more flexible input and can be applied to a broader class of situations the language of model categories, also introduced by quillen, provides an abstract setup for homotopy theory.
Algebraic k theory and configuration spaces 411 using 0. Algebraic k theory and quadratic forms 321 to conclude this section, the ring k. Characteristic classes for complex line bundles 54 3. In algebra and algebraic geometry, it is referred to as algebraic k theory. An introduction to algebraic k theory christian ausoni abstract. One checks that bequipped with this family of conations is an exact category making binto a fully exact. It is a higher analog of the rational chern character isomorphism for general toric schemes.
Higher chow groups form, not a cohomology theory, but a borelmoore. Lecture notes on algebraic k theory pdf 252p download book. Our decision to begin this lecture series on modern social theory with the question what is theory. From the perspective of homotopy theory, algebraic ktheory has a certain universality. For the topological equivariant ktheory, see topological ktheory. Algebraic ktheory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. Higher algebraic ktheory of schemes and of derived. Algebraic ktheory describes a branch of algebra that centers about two functors.
Our theorem utilizes this sheaftheoretic description to reduce the. This paper is concerned with higher algebraic ktheory and actions of algebraic groups g on such nice categories as the category of algebraic vector bundles on a scheme x. In particular, the higher ktheory of a ring r is defined as the product of the group of units of the classifying space of r and k0r. We identify the milnor k theory of a field with a certain higher chow group. Blumberg, david gepner, goncalo tabuada download pdf. Marius dadarlat, shape theory and asymptotic morphisms for c c\astalgebras, duke math.
Geometric, algebraic, and arithmetic objects are assigned objects called kgroups. A more elementary proof of the theorem of merkujevsuslin is given in this edition. A universal characterization of higher algebraic ktheory. In this paper we prove a localization theorem for the ktheory of commutative rings and of schemes, theorem 7. Fibrations and a transfer map in algebraic ktheory. Introduction to kktheory and etheory, lecture notes lisbon 2009 a standard textbook account is in section 25 of. An algebraic introduction to k theory this book is an introduction to k theory and a text in algebra. 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. We establish some useful properties like invariance under the appropriate equivalences and long exact sequences which allow us to compute these groups in some examples. Topological ktheory of complex noncommutative spaces.
Specifically, we prove that connective algebraic k theory is the universal additive invariant, ie the universal functor with values in spectra which inverts morita equivalences, preserves filtered colimits and satisfies waldhausens additivity theorem. Higher algebraic ktheory was introduced by quillen to extend the existing theory of the lower kgroups of rings. Buy algebraic ktheory and its applications graduate texts in mathematics v. It was meant as a leisurely exposition of classical aspects of algebraic k theory, with some of its applications to geometry and topology. Algebraic ktheory, algebraic cycles and arithmetic geometry. Our emphasis is on the application of triangulated category methods in algebraic ktheory. In this paper we prove a localization theorem for the k theory of commutative rings and of schemes, theorem 7. Our emphasis is on the application of triangulated category methods in algebraic k theory. Transfer on algebraic ktheory and whitehead torsion for. Unlike the teleological question, it does not appear to require that we are clear about the nature and aims of sociology. We present an introduction with a few proofs to higher algebraic k theory of schemes based on the work of quillen, waldhausen, thomason and others. Ktheory itself, rather than on these applications of algebraic ktheory.
Algebraic ktheory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Unlike the evaluative question, it does not involve the tricky predicate be good. The broad range of these topics has tended to give the subject an aura of inapproachability. His results include a proof that all infinite loop space machines are in some sense equivalent, and progress on. We use cookies to make interactions with our website easy and meaningful, to better understand the use of our services, and to tailor advertising.
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