Representation Theory and Higher Algebraic K-Theory, by Aderemi Kuku
The definitive resource for algebraic K-theory
'Representation Theory and Higher Algebraic K-Theory' is the first book to present higher algebraic K-theory of orders and group rings as well as characterize higher algebraic K-theory as Mackey functors that lead to equivariant higher algebraic K-theory and their relative generalizations. Thus, this book makes computations of higher K-theory of group rings more accessible and provides novel techniques for the computations of higher K-theory of finite and some infinite groups.
Authored by a premier authority in the field, the book begins with a careful review of classical K-theory, including clear definitions, examples, and important classical results. Emphasizing the practical value of the usually abstract topological constructions, the author systematically discusses higher algebraic K-theory of exact, symmetric monoidal, and Waldhausen categories with applications to orders and group rings and proves numerous results. He also defines profinite higher K- and G-theory of exact categories, orders, and group rings. Providing new insights into classical results and opening avenues for further applications, the book then uses representation-theoretic techniques - especially induction theory - to examine equivariant higher algebraic K-theory, their relative generalizations, and equivariant homology theories for discrete group actions. The final chapter unifies Farrell and Baum-Connes isomorphism conjectures through Davis-Lück assembly maps.
Features
- Presents higher algebraic K-theory of orders and group rings for the first time in book form
- Explores connections between C_G and higher algebraic K-theory of C_G for suitable categories, such as exact, symmetric monoidal, and Waldhausen
- Collects methods that have been known to work for computations of higher K-theory of noncommutative rings, such as orders and group rings
- Describes all higher algebraic K-theory as Mackey functors that lead to equivariant higher algebraic K-theory and their relative generalizations for finite, profinite, and compact Lie group actions
- Obtains results on higher K-theory of orders lambda, and hence group rings, for all n ≥ 0
- Uses computations of higher K-theory of orders that automatically yield results on higher K-theory of RG(G finite) to produce results on higher K-theory of some infinite groups
- Provides appendices with many known computations and open problems in classical and higher algebraic K-theory of orders, group rings, and related structures
Contents
Category of Representations and Constructions of Grothendieck Groups and Rings
Category of representations and G-equivariant categories
Grothendieck group associated with a semi-group
K_{0} of symmetric monoidal categories
K_{0} of exact categories - definitions and examples
Exercises
Some finiteness results on K_{0} and G_{0} of orders and group rings
Class groups of Dedekind domains, orders, and group rings plus some applications
Decomposition of G_{0}(RG) (G Abelian group) and extensions to some non-Abelian groups
Exercises
K_{1}, SK_{1} of orders and group-rings; Whitehead torsion
The functor K_{2}
Exercises
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
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
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
Ranks of Kn(L), Gn(L) of orders and in rings plus some consequences
Decomposition of G_{n}(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
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
Exercises
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 (KG_{0})^{G} -functors
Exercises
Relative equivariant higher algebraic K-theory
Interpretation in terms of group rings
Some applications
Exercises
Cohomology of Mackey functors (for profinite groups)
Exercises
An equivariant higher K-theory for G-actions
Induction theory for equivariant higher K-functors
Exercise
Equivariant higher K-theory constructions for Waldhausen categories
Applications to complicial bi-Waldhausen categories
Applications to higher K-theory of group rings
Exercise
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
B: Some open problems
Index
JOC/EFR May 2019
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