Representation theory and higher algebraic K-theory. (©2007)

• Record Type: Book
• Further Information: Note: Aderemi Kuku.
• Other Names: Kuku, A. O (Aderemi O.)
• Contents: Introduction; REVIEW OF CLASSICAL ALGEBRAIC K-THEORY AND REPRESENTAION THEORY; Notes on Notations; ; Category of Representations and Constructions of Grothendieck Groups and Rings; Category of representations and G-equivariant categories; Grothendieck group associated with a semi-group; K0 of symmetric monoidal categories; K0 of exact categories - definitions and examples; Exercises; ; Some Fundamental Results on K0 of Exact and Abelian Categories with Applications to Orders and Group Rings; Some fundamental results on K0 of exact and Abelian categories; Some finiteness results on K0 and G0 of orders and groupings; Class groups of Dedekind domains, orders, and group rings plus some applications; Decomposition of G0 (RG) (G Abelian group) and extensions to some non-Abelian groups; Exercises; ; K1, K2 of Orders and Group Rings; Definitions and basic properties; K1, SK1 of orders and group-rings; Whitehead torsion; The functor K2; Exercises; ; Some Exact Sequences; Negative K-Theory; Mayer-Vietoris sequences; Localization sequences; Exact sequence associated to an ideal of a ring; Negative K-theory K-n, n positive integer; Lower K-theory of group rings of virtually infinite cyclic groups; ; HIGHER ALGEBRAIC K-THEORY AND INTEGRAL REPRESENTATIONS; Higher Algebraic K-Theory-Definitions, Constructions, and; Relevant Examples; The plus construction and higher K-theory of rings; Classifying spaces and higher K-theory of exact categories-constructions and examples; Higher K-theory of symmetric monoidal categories-definitions and examples; Higher K-theory of Waldhausen categories-definitions and examples; Exercises; ; Some Fundamental Results and Exact Sequences in Higher K-Theory; Some fundamental theorems; Localization; Fundamental theorem of higher K-theory; Some exact sequences in the K-theory of Waldhausen categories; Exact sequence associated to an ideal, excision, and Mayer-Vietoris sequences; Exercises; ; Some Results on Higher K-Theory of Orders, Group Rings and; Modules over "EI" Categories; Some finiteness results on Kn, Gn, SKn, SGn of orders and groupings; Ranks of Kn(?), Gn(?) of orders and group rings plus some consequences; Decomposition of Gn(RG) n = 0, G finite Abelian group; Extensions to some non-Abelian groups, e.g., quaternion and dihedral groups; Higher dimensional class groups of orders and group rings; Higher K-theory of group rings of virtually infinite cyclic groups; Higher K-theory of modules over "EI" -categories; Higher K-theory of P(A)G, A maximal orders in division algebras, G finite group; Exercises; ; Mod-m and Profinite Higher K-Theory of Exact Categories, Orders, and Groupings; Mod-m K-theory of exact categories, rings and orders; Profinite K-theory of exact categories, rings and orders; Profinite K-theory of p-adic orders and semi-simple algebras; Continuous K-theory of p-adic orders; ; MACKEY FUNCTORS, EQUIVARIANT HIGHER ALGEBRAIC K-THEORY, AND EQUIVARIANT HOMOLOGY THEORIES; Exercises; ; Mackey, Green, and Burnside Functors; Mackey functors; Cohomology of Mackey functors; Green functors, modules, algebras, and induction theorems; Based category and the Burnside functor; Induction theorems for Mackey and Green functors; Defect basis of Mackey and Green functors; Defect basis for KG0 -functors; Exercises; ; Equivariant Higher Algebraic K-Theory Together with Relative; Generalizations for Finite Group Actions; Equivariant higher algebraic K-theory; Relative equivariant higher algebraic K-theory; Interpretation in terms of group rings; Some applications; Exercises; ; Equivariant Higher K-Theory for Profinite Group Actions; Equivariant higher K-theory (absolute and relative); Cohomology of Mackey functors (for profinite groups); Exercises; ; Equivariant Higher K-Theory for Compact Lie Group Actions; Mackey and Green functors on the category A(G) of homogeneous spaces; An equivariant higher K-theory for G-actions; Induction theory for equivariant higher K-functors; Exercise; ; Equivariant Higher K-Theory for Waldhausen Categories; Equivariant Waldhausen categories; Equivariant higher K-theory constructions for Waldhausen categories; Applications to complicial bi-Waldhausen categories; Applications to higher K-theory of group rings; Exercise; ; Equivariant Homology Theories and Higher K-Theory of Group Rings; Classifying space for families and equivariant homology theory; Assembly maps and isomorphism conjectures; Farrell-Jones conjecture for algebraic K-theory; Baum-Connes conjecture; Davis-Lück assembly map for BC conjecture and its identification with analytic assembly map; Exercise; ; Appendices; A: Some computations; B: Some open problems; ; References; Index
• Publisher Details: Boca Raton : Chapman & Hall/CRC
• Publication Date: 2007
• Copyright Date: 2007
• Extent: 1 online resource (xxv, 442 pages)
• Subjects: 512/.66; K-theory; Representations of categories; Representations of groups; K-théorie; Représentations de catégories; Représentations de groupes; MATHEMATICS -- Algebra -- Intermediate; K-theory; Representations of categories; Representations of groups; Álgebra; K-teoria algébrica; Electronic books
• Languages: English
• ISBNs: 9781584886037; 9781420011128
• Related ISBNs: 158488603X; 142001112X
• Notes: Note: Includes bibliographical references (pages 423-436) and index.; Note: Print version record.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.