Hazewinkel Handbook of Algebra

1. Auflage 2008
ISBN: 978-0-08-056499-9
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, Band Volume 5, 576 Seiten

Reihe: Handbook of Algebra

Algebra, as we know it today, consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a name or a convenient designation. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. If this happens, one should be able to find enough information in this Handbook to judge if it is worthwhile to pursue the quest.In addition to the primary information given in the Handbook, there are references to relevant articles, books or lecture notes to help the reader. An excellent index has been included which is extensive and not limited to definitions, theorems etc.The Handbook of Algebra will publish articles as they are received and thus the reader will find in this third volume articles from twelve different sections. The advantages of this scheme are two-fold: accepted articles will be published quickly and the outline of the Handbook can be allowed to evolve as the various volumes are published.A particularly important function of the Handbook is to provide professional mathematicians working in an area other than their own with sufficient information on the topic in question if and when it is needed.- Thorough and practical source of information - Provides in-depth coverage of new topics in algebra - Includes references to relevant articles, books and lecture notes

Autoren/Hrsg.

Hazewinkel, M.

Leseproben

Outline of the Series

Publisher Summary

This chapter presents the various concepts of algebra, such as linear algebra, linear (in)dependence, galois theory, and algebraic number theory.

(as of June 2007)

Philosophy and principles of the Handbook of Algebra

Compared to the outline in Volume 1 this version differs in several aspects.

First there is a major shift in emphasis away from completeness as far as the more elementary material is concerned and towards more emphasis on recent developments and active areas. Second the plan is now more dynamic in that there is no longer a fixed list of topics to be covered, determined long in advance. Instead there is a more flexible nonrigid list that can and does change in response to new developments and availability of authors.

The new policy, starting with Volume 2, is to work with a dynamic list of topics that should be covered, to arrange these in sections and larger groups according to the major divisions into which algebra falls, and to publish collections of contributions (i.e. chapters) as they become available from the invited authors.

The coding below is by style and is as follows.

– Author(s) in bold, followed by chapter title: articles (chapters) that have been received and are published or are being published in this volume.

– title in : chapters that are being written.

– Chapter title in plain text: topics that should be covered but for which no author has yet been definitely contracted.

Chapters that are included in Volumes 1–5 have a (x; yy pp.) after them, where ‘x’ is the volume number and ‘yy’ is the number of pages.

Compared to the plan that appeared in Volume 1 the section on “Representation and invariant theory” has been thoroughly revised from Volume 2 on.

Compared to the plan that appeared in Volume 4, Section 4H (Rings and algebras with additional structure) has been split into two parts: 4H (Hopf algebras and related structures) and 4I (Other rings and algebras with additional structure). The old Section 4I (Witt vectors) has been absorbed into the section on Hopf algebras.

There also a few more changes; mostly addition of some more topics.

Editorial set-up

Managing editor: M Hazewinkel.

Editorial board: M. Artin, M. Nagata, C. Procesi, O. Tausky-Todd†, R.G. Swan, P.M. Cohn, A. Dress, J. Tits, N.J.A. Sloane, C. Faith, S.I. A’dyan, Y. Ihara, L. Small, E. Manes, I.G. Macdonald, M. Marcus, L.A. Bokut', Eliezer (Louis Halle) Rowen, John S. Wilson, Vlastimil Dlab. Note that three editors have been added starting with Volume 5.

Planned publishing schedule (as of July 2007) 1996: Volume 1 (published)

2001: Volume 2 (published)

2003: Volume 3 (published)

2005: Volume 4 (published)

2007: Volume 5 (last quarter)

Further volumes at the rate of one every year.

Section 1. Linear algebra. Fields. Algebraic number theory

G.P. Egorychev, Van der Waerden conjecture and applications (1; 22 pp.)

V.L. Girko, Random matrices (1; 52 pp.)

A.N. Malyshev, Matrix equations. Factorization of matrices (1; 38 pp.)

L. Rodman, Matrix functions (1; 38 pp.)

Correction to the chapter by L. Rodman, Matrix functions (3; 1 p.)

J.A. Hermida-Alonso, Linear algebra over commutative rings (3; 49 pp.)

Linear algebra over skew fields

J.P.S. Kung, Matroids (1; 28 pp.)

Other algebras arising from vector spaces (working title only)

(There is also a chapter on ordered fields in Section 4)

J.K. Deveney, J.N. Mordeson, Higher derivation Galois theory of inseparable field extensions (1; 34 pp.)

I. Fesenko, Complete discrete valuation fields. Abelian local class field theories (1; 48 pp.)

M. Jarden, Infinite Galois theory (1; 52 pp.)

R. Lidl, H. Niederreiter, Finite fields and their applications (1; 44 pp.) W. Narkiewicz, Global class field theory (1; 30 pp.)

H. van Tilborg, Finite fields and error correcting codes (1; 28 pp.)

Constructive Galois theory (realizations of groups as Galois groups)

T. Albu, From field theoretic to abstract co-Galois theory (5; 81 pp.)

(to be arranged in several chapters by Y. Ihara)

U. Hebisch, H.J. Weinert, Semi-rings and semi-fields (1; 38 pp.)

G. Pilz, Near rings and near fields (1; 36 pp.)

Section 2. Category theory. Homological and homotopical algebra. Methods from logic

S. MacLane, I. Moerdijk, Topos theory (1; 28 pp.)

R. Street, Categorical structures (1; 50 pp.)

B.I. Plotkin, Algebra, categories and databases (2; 71 pp.)

P.S. Scott, Some aspects of categories in computer science (2; 71 pp.)

E. Manes, Monads of sets (3; 48 pp.)

M. Markl, Operads and PROPs (5; 54 pp.)

J.F. Carlson, The cohomology of groups (1; 30 pp.)

A. Generalov, Relative homological algebra. Cohomology of categories, posets, and coalgebras (1; 28 pp.)

J.F. Jardine, Homotopy and homotopical algebra (1; 32 pp.)

B. Keller, Derived categories and their uses (1; 32 pp.)

A.Ya. Helemskii, Homology for the algebras of analysis (2; 143 pp.))

Galois cohomology

Cohomology of commutative and associative algebras

Cohomology of Lie algebras

Cohomology of group schemes

V. Lyubashenko, O. Manzyuk, A$$-algebras, A$$-categories, and A$$-functors (5; 46 pp.)

B.V. Novikov, 0-Cohomology of semigroups (5; 21 pp.)

A. Kuku, Classical algebraic -theory: the functors 012 (3; 55 pp.)

A. Kuku, Algebraic -theory: the higher -functors (4; 122 pp.)

2

-theory and EXT

C*

-theory of noncommutative rings

Algebraic -theory

Cyclic cohomology

Asymptotic morphisms and -theory

Hirzebruch formulae

(See also P.C. Eklof, Whitehead modules, in Section 3B)

M. Prest, Model theory for algebra (3; 31 pp.)

M. Prest, Model theory and modules (3; 34 pp.)

F.O. Wagner, Stable groups (2; 36 pp.)

The Ax–Ershov–Kochen theorem and its relatives and applications

Rings up to homotopy

Simplicial algebras

Section 3. Commutative and associative rings and algebras

(See also C. Faith, Coherent rings and annihilator conditions in matrix and polynomial rings, in Section 3B)

(See also Freeness theorems for group rings and Lie algebras in Section 5A)

J.P. Lafon, Ideals and modules (1; 24 pp.)

General theory. Radicals, prime ideals, etc. Local rings (general). Finiteness and chain conditions

Homological algebra and commutative rings. Ext, Tor, etc. Special properties (p.i.d., factorial, Gorenstein,...