Prerequisites in Module Theory
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Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
Lecture Notes in Mathematics 1081
Lecture Notes in Mathematics 1081 Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris David J. Benson Modular Representation Theory New Trends and Methods Second printing Sprin ger Author David J. Benson Department of Mathematical Sciences University of Aberdeen Meston Building King's College Aberdeen AB24 SUE Scotland UK Modular Representation Theory Library of Congress Cataloging in Publication Data. Benson, David, 1955-. Modular representation theory. (Lecture notes in mathematics; 1081) Bibliography: p. Includes index. 1. Modular representations of groups. 2. Rings (Algebra) I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1081. QA3.L28 no. 1081 [QA171] 510s [512'.2] 84-20207 ISBN 0-387-13389-5 (U.S.) Mathematics Subject Classification (1980): 20C20 Second printing 2006 ISSN 0075-8434 ISBN-10 3-540-13389-5 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-13389-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science-i-Business Media springer.com © Springer-Verlag Berlin Heidelberg 1984 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. -
Arxiv:Math/0504051V2 [Math.AT] 12 Jul 2005 A:4 5 8338370 251 49 FAX: Www: Aebe Endadivsiae Ndti O Nt Rus H P the Groups
The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups Wolfgang L¨uck∗ Fachbereich Mathematik Universit¨at M¨unster Einsteinstr. 62 48149 M¨unster Germany June 5, 2018 Abstract After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (dis- crete) groups. The first three are the finite-G-set-version, the inverse- limit-version and the covariant Burnside group. The most sophisticated one is the fourth definition as the zero-th equivariant stable cohomotopy of the classifying space for proper actions. In order to make sense of this definition we define equivariant stable cohomotopy groups of finite proper equivariant CW -complexes in terms of maps between the sphere bundles associated to equivariant vector bundles. We show that this yields an equivariant cohomology theory with a multiplicative structure. We for- mulate a version of the Segal Conjecture for infinite groups. All this is analogous and related to the question what are the possible extensions of the notion of the representation ring of a finite group to an infinite group. Here possible candidates are projective class groups, Swan groups and the equivariant topological K-theory of the classifying space for proper actions. arXiv:math/0504051v2 [math.AT] 12 Jul 2005 Key words: Burnside ring, equivariant stable cohomotopy, infinite groups. Mathematics Subject Classification 2000: 55P91, 19A22. 0 Introduction The basic notions of the Burnside ring and of equivariant stable cohomotopy have been defined and investigated in detail for finite groups. The purpose of ∗email: [email protected] www: http://www.math.uni-muenster.de/u/lueck/ FAX: 49 251 8338370 1 this article is to discuss how these can be generalized to infinite (discrete) groups. -
The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups
Pure and Applied Mathematics Quarterly Volume 1, Number 3 (Special Issue: In Memory of Armand Borel, Part 2 of 3 ) 479|541, 2005 The Burnside Ring and Equivariant Stable Cohomotopy for In¯nite Groups Wolfgang LÄuck 0. Introduction The basic notions of the Burnside ring and of equivariant stable cohomotopy have been de¯ned and investigated in detail for ¯nite groups. The purpose of this article is to discuss how these can be generalized to in¯nite (discrete) groups. The guideline will be the related notion of the representation ring which allows sev- eral generalizations to in¯nite groups, each of which reflects one aspect of the original notion for ¯nite groups. Analogously we will present several possible generalizations of the Burnside ring for ¯nite groups to in¯nite (discrete) groups. There seems to be no general answer to the question which generalization is the right one. The answer depends on the choice of the background problem such as universal additive properties, induction theory, equivariant stable homotopy theory, representation theory, completion theorems and so on. For ¯nite groups the representation ring and the Burnside ring are related to all these topics si- multaneously and for in¯nite groups the notion seems to split up into di®erent ones which fall together for ¯nite groups but not in general. The following table summarizes in the ¯rst column the possible generalizations to in¯nite groups of the representation ring RF (G) with coe±cients in a ¯eld F of characteristic zero. In the second column we list the analogous generalizations for the Burnside ring. -
Ldempotents of Burnside Rings and Dress Induction Theorem
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 80, 90-105 (1983) ldempotents of Burnside Rings and Dress induction Theorem TOMOYUKI YOSHIDA Department of Mathematics, Hokkaido University, Sapporo, Japan Communicated by W. Feit Received July 5, 1980 1. INTRODUCTION The purpose of this paper is to state the connection between an explicit formula of idempotents of the Burnside rings of a finite group and some induction theorems in representation theory of finite groups. The study in this direction has its origin in Solomon’s paper [ 131, in which he found that primitive idempotents in the Burnside ring Q @ Q(G) of a finite group G could be presented by the Mobius function of the poset of conjugate classes of subgroups of G and that the formula implies Artin’s induction theorem in the explicit form by Brauer [3]. Following him, Conlon gave another kind of idempotent formula and show that many induction theorems, similar to Artin’s one, result for representation rings [ 151. In this paper, we first give an explicit formula of idempotents of the Burnside ring by using the Mobius function of the subgroup lattice instead of the poset of conjugate classesof subgroups (Theorem 3.1). This improvement makes calculations of Burnside rings simpler and produces some applications. As the first application, we obtain an alternate proof of a generalization of Brown’s theorem [4, lo] for the Euler characteristic of the poset of nontrivial p-subgroups of a finite group (Corollaries 3.2 and 3.3). -
On the Exponential Map of the Burnside Ring
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Bilkent University Institutional Repository ON THE EXPONENTIAL MAP OF THE BURNSIDE RING a thesis submitted to the department of mathematics and the institute of engineering and sciences of bilkent university in partial fulfillment of the requirements for the degree of master of science By Ay¸seYaman July, 2002 I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Assoc. Prof. Dr. Laurence J. Barker(Principal Advisor) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Assoc. Prof. Dr. Ali Sinan Sertoz I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Asst. Prof. Dr. Semra Kaptano˜glu Approved for the Institute of Engineering and Sciences: Prof. Dr. Mehmet Baray Director of Institute of Engineering and Sciences ii ABSTRACT ON THE EXPONENTIAL MAP OF THE BURNSIDE RING Ay¸seYaman M.S. in Mathematics Supervisor: Assoc. Prof. Dr. Laurence J. Barker July, 2002 We study the exponential map of the Burnside ring. We prove the equiv- alence of the three different characterizations of this map and examine the surjectivity in order to describe the elements of the unit group of the Burnside ring more explicitly. -
Cohomology of Burnside Rings
Cohomology of Burnside Rings Benen Harrington PhD University of York Mathematics November 2018 Abstract l We study the groups ExtA(G)(ZH ; ZJ ) where A(G) is the Burnside ring of a finite group G and for a subgroup H ⊂ G, the A(G)-module ZH is defined by the mark homomorphism corresponding to H. If jGj is square-free we give a complete description of these groups. If jGj is not square-free we show that l for certain H; J ⊂ G the groups ExtA(G)(ZH ; ZJ ) have unbounded rank. We also extend some of these results to the rational and complex rep- resentation rings of a finite group, and describe a new generalisation of the Burnside ring for infinite groups. 2 Contents Abstract 2 Contents 4 Acknowledgements 7 Declaration 9 Introduction 11 1 B-rings 14 1.1 The spectrum of a B-ring . 19 1.2 The Burnside ring . 20 1.2.1 The Grothendieck group associated to a commutative monoid 20 1.2.2 The Burnside ring as a B-ring . 21 1.3 Higher Ext groups and Tor . 24 1.4 Reducing to the modular case . 28 1.5 B-rings modulo a prime . 32 2 Cohomology of commutative local k-algebras 38 l S 2.1 Relating ExtS(k; k) and Torl (k; k)................... 38 2.2 Indecomposable summands of the modular Burnside ring . 40 2.3 The case dim M2 =1........................... 41 2.3.1 Degenerate cases . 42 2.3.2 Constructing a resolution . 42 2.3.3 Determining Al .......................... 44 2.3.4 Gr¨obnerbases . -
Segal's Burnside Ring Conjecture
Segal's Burnside Ring Conjecture Outline: I Atiyah's Theorem I Classifying spaces I Weak form of Segal's Conjecture I Strong form of Segal's Conjecture I Relation to algebraic K-theory I The Segal conjecture for tori I Towards motivic Segal conjectures. Topology Seminar, Bergen, January 24, 2013 Morten Brun, Universitetet i Bergen Atiyah's Theorem G finite group BG its classifying space KU∗ complex periodic K-theory R(G) complex representation ring of G I kernel of augmentation ": R(G) ! Z given by dimension. Theorem (Atiyah, 1961) There exists an isomorphism ^ 0 R(G)I ! KU (BG) 0 KU (X ) = [X ; BU × Z], where U = [nU(n) infinite unitary group and BU its classifying space. The Representation Ring Consider the set of isomorphism classes of finite dimensional complex representations of G. Direct sum of representations makes this set into an abelian monoid. The underlying additive group of R(G) is the group-completion of this abelian monoid. Multiplication in R(G) is induced by the tensor product over C. The representation ring of the trivial group is isomorphic to Z. Forgetting the action of the group we obtain the augmentation ∼ ": R(G) ! R(e) = Z Fiber Bundles A fiber bundle consists of a surjective continous map π : E ! X together with a topological space F . The map π is required to satisfy the following local triviality condition: for every point in X , there is an open neighborhood U of that point and a homeomorphism of the form ': U × F ! π−1(U) such that for each x 2 U, (π ◦ ')(x; v) = x for every v in F The Hopf Fibration 1 1 2 Recall that RP is obtained from the 1-sphere S ⊆ R via the 0 identification x ∼ λx for λ 2 S = f1; −1g ⊆ R. -
UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations
UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations Title Fusion systems and biset functors via ghost algebras Permalink https://escholarship.org/uc/item/0ts629vq Author O'Hare, Shawn Michael Publication Date 2013 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA SANTA CRUZ FUSION SYSTEMS AND BISET FUNCTORS VIA GHOST ALGEBRAS A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS by Shawn Michael O’Hare June 2013 The Dissertation of Shawn Michael O’Hare is approved: Professor Robert Boltje, Chair Professor Geoff Mason Professor Martin Weissman Tyrus Miller Vice Provost and Dean of Graduate Studies Copyright c by Shawn Michael O’Hare 2013 Table of Contents Abstract v Dedication vi Acknowledgments vii Introduction1 Some Notation . .4 1 Background7 1.1 Fusion System Basics . .7 1.2 Bisets . 10 1.3 Double Burnside Groups . 15 1.4 Two Ghost Groups . 17 1.5 Subgroups of Burnside Groups . 21 1.6 Biset Categories . 24 2 Biset Categories 27 2.1 A Special Class of Groups . 28 2.2 Fusion Preserving Isomorphisms . 31 3 Characteristic Idempotents 36 3.1 Calculating the Characteristic Idempotent . 36 3.2 Bideflation of Characteristic Idempotents . 42 4 A Generalized Burnside Functor 47 4.1 Pseudo-rings . 47 Condensation and Decondensation . 49 4.2 Decondensation of the Burnside Functor . 51 4.3 The Action of Elementary Subgroups . 53 4.4 Subfunctors . 57 References 67 iii A Single Burnside Rings 70 A.1 Sets with a Group Action . 70 A.2 Operations on Sets with a Group Action . -
The Burnside-Ring of a Compact Lie Group
5. The Burnside-Ring of a Compact Lie Group. 5.1. Euler Characteristic. We collect the properties of the Euler-CharacteristJc that we shall need in the sequel and indicate proofs when appropriate references cannot be given. Let R be a commutative ring and let A be an associative R-algebra with identity (e.g. A = R; A = RIG], G a finite group). In general, an Euler-Poincar~ map is a map from a certain category of A-modules to an abeiian group which is additive on certain exact sequences. We consider the following sufficiently general situation: Let GrR(A) be the abelian group (Grothendieck group) with generators M] where M is a left A-module which is finitely generated and projective as an R-module, with relations [M] = [M'] + [M'~ for each exact sequence O--~ M'---%M--) M"--90 of such modules. Let Gr(A) be the Grothendieck group of finitely generated left A-modules and the ana- logous relations for exact sequences. A ring R is called regular if it is noetherian and every finitely generated R-module has a finite resolution by finitely generated projective R-modules. Proposition 5.1.1. Let R be a regular ring and A a__nn R-algebra which is finitel~ generated and projective as an R-module. Then the forgetful map GrR(A) ---~ Gr(A) is an isomorphism. Proof. Swan-Evans [458] , p. 2. (The symbol G o is used in ~$8] where we use Gr. Since we do not need G I and use G to denote groups we have chosen this non-standard notation.) 83 Remark. -
Polynomial Operations from Burnside Rings to Representation Functors
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 65 (1990) 163-190 163 North-Holland POLYNOMIAL OPERATIONS FROM BURNSIDE RINGS TO REPRESENTATION FUNCTORS Ernest0 VALLEJO* Matematisches Institut der Universittit, Im Neuenheimer Feld 288 D-6900 Heidelberg, Fed. Rep. Germany Communicated by A. Heller Received 13 July 1988 Revised 8 June 1989 We introduce the concept of polynomial operation from the Burnside ring functor A to other representation functors R, which includes operations such as symmetric powers, I-operations and Adams operations. The set of polynomial operations from A to R, Pol(A, R), has a canonical ring structure. We give a complete description of the additive structure of this ring as well as a family of generating operations. Introduction In this paper we study polynomial operations q : A + R where A denotes the Burn- side ring functor, and R is a representation functor, see Definition 2.1. Polynomial operations (1.6) are natural transformations between contravariant functors such that vG : A(G) + R(G) is a polynomial map for each finite group G. The set of all polynomial operations Pol(A,R) is closed under the addition and multiplication defined from the ring structures on the R(G)‘s, thus having a ring structure. Our purpose is to give a description of the additive structure of Pol(A, R). In order to do that we construct a family of polynomial operations F(a):A +R, one for each a~ RS(m), mr0, where S(m) is the symmetric group of degree m. -
University of California Santa Cruz the Unit Group of The
UNIVERSITY OF CALIFORNIA SANTA CRUZ THE UNIT GROUP OF THE BURNSIDE RING AS A BISET FUNCTOR FOR SOME SOLVABLE GROUPS A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS by Jamison Blair Barsotti June 2018 The Dissertation of Jamison Blair Barsotti is approved: Professor Robert Boltje, Chair Professor Samit Dasgupta Professor Junecue Suh Tyrus Miller Vice Provost and Dean of Graduate Studies Copyright c by Jamison Blair Barsotti 2018 Table of Contents Abstract v Acknowledgments vi 1 Introduction 1 2 Background 4 2.1 Grothendieck Groups . .4 2.2 Burnside Rings . .5 2.3 Bisets . .9 2.4 Elementary Bisets . 10 2.5 Biset Functors . 14 3 The Double Burnside Ring 19 3.1 Idempotents of RB(G,G) . 19 4 The Unit Group of the Burnside Ring 22 4.1 The Biset Functor B× ............................... 22 4.2 The Unit Group of the Burnside Ring for p-groups . 24 5 The Unit Group of the Burnside Ring for Some Solvable Groups 26 5.1 Groups With Abelian Subgroups of Index 1 or 2 . 26 5.2 Extending the Main Theorem . 35 6 Residual groups for B× 40 6.1 Residual Groups . 40 6.2 Residual Objects of C 0 with Respect to B× .................... 42 7 Subfunctors of B× 54 7.1 The Subfunctor Lattice of B× over C 0 ...................... 54 7.2 Composition Factors . 64 iii 8 Applications to Simple Biset Functors 66 × 8.1 Computing SG;F2 (H) for G residual with respect to B ............. 66 8.2 Surjectivity of the Exponential Map B ! B× ..................