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Thursday, April 30, 2020 | History

5 edition of Algebraic K-theory and localised stable homotopy theory found in the catalog.

Algebraic K-theory and localised stable homotopy theory

  • 193 Want to read
  • 17 Currently reading

Published by American Mathematical Society in Providence, R.I., USA .
Written in English

    Subjects:
  • K-theory.,
  • Homotopy theory.

  • Edition Notes

    Bibliography: p. 98-102.

    StatementVictor Snaith.
    SeriesMemoirs of the American Mathematical Society,, no. 280
    Classifications
    LC ClassificationsQA3 .A57 no. 280, QA612.33 .A57 no. 280
    The Physical Object
    Paginationxi, 102 p. ;
    Number of Pages102
    ID Numbers
    Open LibraryOL3161924M
    ISBN 100821822802
    LC Control Number83003726

    The theory of Algebraic Cycles, Higher Algebraic K-theory, and Motivic Homotopy Theory are modern versions of Grothendieck's legacy. In recent years it has seen some spectacular developments, on which we want to build further. The programme will also specifically explore the connections between the following areas. Algebraic K-theory of spaces. A¹-homotopy invariance of algebraic K-theory with coefficients and du Val singularities $ of representations of a quiver with the structure of an algebra Author: Friedhelm Waldhausen.   Algebraic geometry and homotopy theory enjoy rich interaction. Their relationship can be seen in part in two exciting fields of mathematics, both of which emerged only recently. There exists a homotopy theory of smooth schemes: motivic homotopy th. This workshop, sponsored by AIM and the NSF, will explore the applications to chromatic homotopy and algebraic K-theory of the new computational techniques in equivariant stable homotopy. The recent renaissance in equivariant stable homotopy theory has provided several new tools which allow us to better understand classical problems and carry.


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Algebraic K-theory and localised stable homotopy theory by V. P. Snaith Download PDF EPUB FB2

Algebraic K-theory, which is the main character of this book, deals mainly with studying the structure of rings. However, it turns out that even working in a purely algebraic context, one requires techniques from homotopy theory to construct the Cited by: Algebraic K-theory and localised stable homotopy theory.

[Victor P Snaith] -- There is a homomorphism from the stable homotopy of the classifying space of the group of units in a ring to its algebraic [italic]K-theory.

Topological K-theory has become an important tool in topology. Using K- theory, Adams and Atiyah were able to give a simple proof that the only spheres which can be provided with H-space structures are S1, S3 and S7. Moreover, it is possible to derive a substantial part of stable homotopy theory from by: Topological K-theory has become an important tool in K- theory, Adams and Atiyah were able to give a simple proof that the only spheres which can be provided with H-space structures are S1, S3 and er, it is possible to derive a substantial part of stable homotopy theory from K-theory.

Beginning with an introduction to the homotopy theory of simplicial sets and topos theory, the book covers core topics such as the unstable homotopy theory of simplicial presheaves and sheaves, localized theories, cocycles, descent theory, non-abelian cohomology, stacks, and local stable homotopy : Springer-Verlag New York.

Kup książkę Algebraic K-Theory and Localised Stable Homotopy Theory (Victor P. Snaith) u sprzedawcy godnego zaufania.

Przeczytaj fragment, zapoznaj się z opiniami innych czytelników, przejrzyj książki o podobnej tematyce, które wybraliśmy dla Ciebie z naszej milionowej kolekcji. from our sellection of 20 million titles.

Algebraic K-Theory I. Proceedings of the Conference Held at the Seattle Research Center of Battelle Memorial Institute, August 28 - September 8, The result, An introduction to homological algebra, took over five years to write.

By this time (), the K-theory landscape had changed, and with it my vision of what my K-theory book should be. Was it an obsolete idea. After all, the new developments in Motivic Cohomology were affecting our knowledge of the K-theory of fields and varieties.

This unfinished book is intended to be a fairly short introduction to topological K-theory, starting with the necessary background material on vector bundles and including also basic material on characteristic classes. For further information or to download the part of the book that is written, go to the download page.

The most modern and thorough treatment of unstable homotopy theory available. The focus is on those methods from algebraic topology which are needed in the presentation of results, proven by Cohen, Moore, and the author, on the exponents of homotopy : Joseph Neisendorfer.

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'^. Moreover, it is possible to derive a substantial part of stable homotopy theory from A^-theory (cf.

Adams [2]). Further applications to analysis and algebra are found in the work of 5/5(2). Stable homotopy of algebraic theories Stefan Schwede Fakulta(tfu(r Mathematik, Universita(t Bielefeld, Bielefeld, Germany Received 1 December ; received in revised form 25 October ; accepted 19 April Abstract The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model categorystructure.

Axiomatic stable homotopy theory About this Title. Mark Hovey, John H. Palmieri and Neil P. Strickland. Publication: Memoirs of the American Mathematical Society Publication Year VolumeNumber ISBNs: (print); (online)Cited by: Book Description: This book contains accounts of talks held at a symposium in honor of John C.

Moore in October at Princeton University, The work includes papers in classical homotopy theory, homological algebra, rational homotopy theory, algebraic K-theory of. Connective stable homotopy theory 76 8. Semisimple stable homotopy theory 78 9.

Examples of stable homotopy categories 80 A general method 80 Chain complexes 82 The derived category of a ring 83 Homotopy categories of equivariant spectra 86 Cochain complexes of B-comodules 89 The stable category of B-modules 95 Cited by: With algebraic K-theory as an intermediary, there has been a growing volume of work that relates algebraic geometry to stable homotopy theory.

With Waldhausen’s introduction of the algebraic K-theory of spaces in the late ’s, stable homotopy became a bridge between algebraic K-theory and the study of diffeomorphisms of manifolds. Algebraic Methods in Unstable Homotopy Theory This is a comprehensive up-to-date treatment of unstable homotopy.

The focus is on those methods from algebraic topology which are needed in the presentation of results, proven by Cohen, Moore, and the author, on the exponents of homotopy groups.

K-theory 38 Bordism 40 Cohomotopy 42 The cohomology of groups 45 References 46 Introduction The study of symmetries on spaces has always been a major part of algebraic and geometric topology, but the systematic homotopical study of group actions is relatively recent.

The last decade has seen a great deal of activity in this. 10 Answers Algebraic K-theory originated in classical materials that connected class groups, unit groups and determinants, Brauer groups, and related things for rings of integers, fields, etc, and includes a lot of local-to-global principles.

Historically, the algebraic K-theory of a commutative ring R (what today is the “0th” algebraic K-theory group) was originally defined to be the Grothendieck group of its symmetric monoidal category of projective modules (under tensor product of modules).

Under the relation between modules and vector bundles. concocted from the given setup, and one defines a K-theory space associated to the geometric realization BCof this category. The K-theory groups are then the homotopy groups of the K-theory space.

In the first chapter, we introduce the basic cast of characters: projective modules and vector bundles (over a topological space, and over a scheme). Algebraic K-theory and abstract homotopy theory Article in Advances in Mathematics (4) August with 24 Reads How we measure 'reads'.

Another great reference is Hovey-Shipley-Smith Symmetric Spectra. On the more modern side, there's Stefan Schwede's Symmetric Spectra Book Project. All these references contain phrasing in terms of model categories, which seem indispensible to modern homotopy theory.

Good references are Hovey's book and Hirschhorn's book. This book provides an introduction to the basic concepts and methods of algebraic topology for the beginner. It presents elements of both homology theory and homotopy theory, and includes various applications.

The author's intention is to rely on the geometric approach by appealing to the reader's own intuition to help understanding. This book should be of interest to all researchers working in fields related to algebraic K-theory.

The techniques presented here are essentially combinatorial, and hence algebraic. An extensive background in traditional stable homotopy theory is not assumed. In mathematics, stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor.

A founding result was the Freudenthal suspension theorem, which states. Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups.

These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups.

Structured Stable Homotopy Theory and the Descent Problem for the Algebraic K-theory of Fields Gunnar Carlsson1 Department of Mathematics Stanford University Stanford, California Contents 1 Introduction 3 2 Preliminaries 11 3 Completions 15 4 Endomorphism algebras for K-theory spectra Algebraic K-theory is a tool from homological algebra that defines a sequence of functors from rings to abelian groups.

algebraic-geometry homotopy-theory higher-category-theory algebraic-k-theory stable-homotopy-theory. asked Oct 7 '19 at user algebraic-geometry commutative-algebra homological-algebra algebraic-k-theory.

Algebraic K-Theory and Localised Stable Homotopy Theory. 点击放大图片 出版社: American Mathematical Society. 作者: Snaith, Victor P. 出版时间: 年12月15 日. 10位国际标准书号: 13位国际标准 Algebraic K-Theory and Localised Stable Homotopy Theory. P.

Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, B Springer (). Google Scholar by: Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry.

As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few. The term ‘algebraic L-theory’ was coined by Wall, to mean the algebraic K-theory of quadratic forms, alias hermitian K-theory.

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 classi ed up to. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory.

It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.

Algebraic K-theory of rings of integers in local and global fields, pp in Handbook of K-theory, Springer-Verlag, Two-primary algebraic K-theory of rings of integers in number fields (J. Rognes and C. Weibel), J. AMS 13 (), Etale descent for two-primary algebraic K-theory of totally imaginary number fields.

Algebraic K-theory and localised stable homotopy theory by Victor P Snaith (Book) 6 editions published in in English and held by WorldCat member libraries worldwide. See for examples of i), ii) and iv), of algebraic models of homotopy types, and of many modes of study, for example that of rational homotopy theory.

A further aspect is that "deformation" methods are essential in a variety of subjects for the purposes of classification, and the comparison of these deformations, or homotopies, in different. The Mathematical Sciences Research Institute (MSRI), founded inis an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions.

The Institute is located at 17 Gauss Way, on the University of California, Berkeley campus, close to. K-theory itself, rather than on these applications of algebraic K-theory. In particular, we will focus directly on “higher algebraic K-theory”, the definition of which requires more categorical and homotopy theoretic subtlety than the simpler algebraic group completion process that is most immediately needed for some of the applications.

From a very modern perspective, higher algebraic K-theory is a functor from the category of stable -categories to the category of spectra.

We recover the usual lower K-groups by taking homotopy groups of the output spectrum — for example, where is incarnated as a stable category by its category. How to Cite This Entry: Algebraic K-theory. Encyclopedia of Mathematics. URL: ?title=Algebraic_K-theory&oldid=Beginning with an introduction to the homotopy theory of simplicial sets and topos theory, the book covers core topics such as the unstable homotopy theory of simplicial presheaves and sheaves, localized theories, cocycles, descent theory, non-abelian cohomology, stacks, and local stable homotopy theory.HOMOTOPY THEORY- Algebraic Topology(May ) (Copy) GalaxyMessier31; Jacob Lurie: Finiteness and Ambidexterity in K(n)-local stable homotopy theory .