Characters of Groups and Lattices over Orders: From Ordinary to Integral Representation Theory (De Gruyter Textbook)

Preface
Contents
1 Ring theoretical foundations
	1.1 Algebras, actions and modules: definition of the basic objects
		1.1.1 Algebras
		1.1.2 Modules
		1.1.3 Direct sums and products of modules
		1.1.4 Group algebras and modules over a group
	1.2 Maschke, Wedderburn and Krull-Schmidt
		1.2.1 Maschke’s theorem
		1.2.2 The Krull-Schmidt theorem
		1.2.3 Wedderburn’s structure theorem on semisimple artinian rings
	1.3 Group rings as semisimple algebras
	1.4 Exercises
2 Characters
	2.1 Definitions and basic properties
	2.2 Class functions, linking characters to the semisimple group algebra
	2.3 Multiplicities of simple submodules; orthogonality relations
	2.4 Character degrees, Burnside’s theorem and the Dixon-Schneider algorithm
		2.4.1 Integral elements in rings, rings of integers
		2.4.2 Group rings and integrality; character degrees
		2.4.3 Burnside’s pa qb -theorem
		2.4.4 The mathematical principle of the Dixon-Schneider algorithm
	2.5 Application to number theory; Dirichlet’s theorem on arithmetic progressions
	2.6 Exercises
3 Tensor products, Mackey formulas and Clifford theory
	3.1 The tensor product
		3.1.1 The definition of a tensor product
		3.1.2 Homomorphisms and tensor product
		3.1.3 The case of tensor products over fields
		3.1.4 Additional structure, algebras and modules
	3.2 Using tensor products, induced modules
	3.3 Mackey’s formula
	3.4 Clifford theory
		3.4.1 Inertia group
		3.4.2 Factor sets, second group cohomology
		3.4.3 Clifford’s main theorem
		3.4.4 Small and big inertia groups: Blichfeldt’s and Ito’s theorem
	3.5 Exercises
4 Bilinear forms on modules
	4.1 Invariant bilinear forms
	4.2 The Frobenius Schur indicator
	4.3 Quadratic modules in characteristic 2
		4.3.1 First group cohomology
		4.3.2 Properties of symmetric and exterior products
		4.3.3 Quadratic modules and group cohomology
	4.7 Exercises
5 Brauer induction, Brauer’s splitting field theorem
	5.1 Grothendieck and character ring
	5.2 Brauer induction formula
	5.3 Brauer’s splitting field theorem
	5.4 The semisimple algebra case
	5.5 Exercises
6 Some homological algebra methods in ring theory
	6.1 Some facts about Noetherian and artinian modules
		6.1.1 Elementary properties and definitions
		6.1.2 Composition series
	6.2 On projective modules
	6.3 Extension groups
		6.3.1 Degree 1 via syzygies
		6.3.2 Higher Ext-groups and applications
	6.4 Hereditary algebras
	6.5 Flatness
	6.6 Exercises
7 Some algebraic number theory
	7.1 Some supplements on algebraic integers
	7.2 Primary decomposition
	7.3 Discrete valuation rings
		7.3.1 Ideal structure of discrete valuation rings
		7.3.2 Valuations
		7.3.3 Completions
		7.3.4 Extension of valuations
	7.4 Fractional ideals
	7.5 Dedekind domains
	7.6 The strong approximation theorem
	7.7 Exercises
8 Some notions of integral representations
	8.1 Classical orders and their lattices
		8.1.1 Basic definitions and examples
		8.1.2 Orders as pullbacks
		8.1.3 Orders, lattices and localisation
	8.2 Reduced norms, traces, characteristic polynomials
	8.3 Maximal orders
		8.3.1 Definition, existence and examples
		8.3.2 Maximal orders are hereditary: the Auslander Buchsbaum theorem
	8.4 The Jordan Zassenhaus theorem
	8.5 Class groups for orders
		8.5.1 Definitions and elementary properties
		8.5.2 Idèle class groups
	8.6 Swan’s example
	8.7 Galois module structure
		8.7.1 Ramifications
		8.7.2 Taylor’s results on Galois module structure
	8.8 Exercises
9 Solution to selected exercises
Bibliography
Index
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