Leavitt Path Algebras and Classical K-Theory (Indian Statistical Institute Series)

Preface
Acknowledgements
Contents
Editors and Contributors
Part I Leavitt Path Algebras
1 A Survey of Some of the Recent Developments in Leavitt Path Algebras
	1.1 Introduction
	1.2 Preliminaries
	1.3 Leavitt Path Algebras Satisfying a Polynomial Identity
	1.4 Four Important Graphical Conditions
	1.5 Simple Modules over Leavitt Path Algebras
	1.6 Leavitt Path Algebras with Simple Modules Having Special Properties
	1.7 One-Sided Ideals in a Leavitt Path Algebra
	References
2 The Groupoid Approach to Leavitt Path Algebras
	2.1 Introduction
		2.1.1 Historical Overview: Groupoids, Graphs, and Their Algebras
		2.1.2 Background: Leavitt Path Algebras
		2.1.3 Background: Steinberg Algebras
		2.1.4 Background: Graph Groupoids
	2.2 The Steinberg Algebra of a Groupoid
		2.2.1 Groupoids
		2.2.2 Topological Groupoids
		2.2.3 Introducing Steinberg Algebras
		2.2.4 Properties of Steinberg Algebras
		2.2.5 First Examples
		2.2.6 Graded Groupoids and Graded Steinberg Algebras
	2.3 The Path Space and Boundary Path Groupoid  of a Graph
		2.3.1 Graphs
		2.3.2 The Path Space of a Graph
		2.3.3 The Boundary Path Groupoid
	2.4 The Leavitt Path Algebra of a Graph
		2.4.1 Introducing Leavitt Path Algebras
		2.4.2 Uniqueness Theorems for Leavitt Path Algebras
		2.4.3 The Steinberg Algebra Model
		2.4.4 Uniqueness Theorems for Steinberg Algebras
	References
3 Étale Groupoids and Steinberg Algebras a Concise Introduction
	3.1 Introduction
	3.2 Inverse Semigroups
		3.2.1 Inverse Semigroups
		3.2.2 Examples of Inverse Semigroups
	3.3 Groupoids
		3.3.1 Groupoids
		3.3.2 Topological Groupoids
		3.3.3 Examples of Groupoids
		3.3.4 Inverse Semigroup of Bisections of a Groupoid
		3.3.5 Graph Groupoids
	3.4 Steinberg Algebras
		3.4.1 Steinberg Algebras
		3.4.2 Convolution Algebra of Continuous Functions  From mathcalG to R
		3.4.3 Centre of a Steinberg Algebra
		3.4.4 Uniqueness Theorems
		3.4.5 Ideal Structures of Steinberg Algebras
	3.5 Combinatorial and Dynamical Invariants of étale Groupoids
		3.5.1 Full Groups
		3.5.2 Homology and K-Theory
	3.6 Partial Crossed Product Rings
	3.7 Non-Hausdorff Ample Groupoids
		3.7.1 Non-Hausdorff Simplicity
	References
4 The Injective and Projective Leavitt Complexes
	4.1 Introduction
	4.2 Preliminaries
		4.2.1 Triangulated Categories
		4.2.2 Compactly Generated Triangulated Categories
		4.2.3 Differential Graded Algebras
	4.3 The Injective Leavitt Complex of a Finite Graph Without Sinks
		4.3.1 The Injective Leavitt Complex
		4.3.2 The Independence of the Injective Leavitt Complex
	4.4 The Projective Leavitt Complex of a Finite Graph Without Sources
		4.4.1 The Projective Leavitt Complex
		4.4.2 The Independence of the Projective Leavitt Complex
	References
5 A Survey on the Ideal Structure  of Leavitt Path Algebras
	5.1 Introduction
	5.2 Preliminaries
		5.2.1 Graph Theory
		5.2.2 Leavitt Path Algebra
	5.3 Ideals in Leavitt Path Algebras
		5.3.1 Graded Ideals
		5.3.2 The Structure Theorem of Graded Ideals
		5.3.3 Structure of Two-Sided Ideals
		5.3.4 Prime and Primitive Ideals
		5.3.5 Maximal Ideals
	References
6 Gröbner Bases and Dimension Formulas for Ternary Partially Associative Operads
	6.1 Introduction
	6.2 Preliminaries
	6.3 Gröbner Bases and Dimension Formulas
	References
7 A Survey on Koszul Algebras and Koszul Duality
	7.1 Introduction
	7.2 Preliminaries
	7.3 Koszul Algebras
		7.3.1 Poincaré Series and Hilbert Series Formula
		7.3.2 Fröberg Formula
	7.4 Koszul Duality
	7.5 Koszul Phenomena in Literature
	7.6 Combinatorics
	7.7 Geometry
	7.8 Koszulness of Affine Semigroup Rings
	7.9 Koszulness and Pólya Frequency Sequences
	References
Part II Classical K-Theory
8 Symplectic Linearization  of an Alternating Polynomial Matrix
	8.1 Introduction
	8.2 Preliminaries
	8.3 Elementary Symplectic Linearization
	References
9 Actions on Alternating Matrices and Compound Matrices
	9.1 Introduction
	9.2 Preliminaries
	9.3 Associated Linear Transformations
	9.4 The 4 times4 Case
	9.5 Injectivity
	References
10 A Survey on the Non-injectivity  of the Vaserstein Symbol  in Dimension Three
	10.1 Introduction
	10.2 The Witt Group WG(A)
	10.3 The Vaserstein Symbol V: Um3(A)/E3(A) -3murightarrowWE(A)
	10.4 The Vaserstein Symbol in Dimension Three and Four
	10.5 An Uncountable Family of Singular Counterexamples
	10.6 Smooth Counterexamples
	10.7 Jean Fasel\'s Conjecture
	References
11 Two Approaches to the Bass–Suslin Conjecture
	11.1 Introduction
	11.2 The Suslin–Vaserstein Symbol
	11.3 The Suslin–Vaserstein Symbol
		11.3.1 The Elementary Unimodular Vector Witt Groups
		11.3.2 An Analogue of Karoubi\'s Linearization Process
		11.3.3 Analogue of Vaserstein\'s Theorem
	11.4 A Descent Approach
	References
12 The Pillars of Relative  Quillen–Suslin Theory
	12.1 Introduction
	12.2 Definitions and Notations
	12.3 Equivalence: Relative L-G Principle and Normality
	12.4 Relative L-G Principle for Transvection Subgroups
	References
13 The Quotient Unimodular Vector Group is Nilpotent
	13.1 Introduction
	13.2 Recap About the Suslin Matrix Sr(v, w)
	13.3 Computation of the Matrix of the Linear Transformation
	13.4 SUmr(R)/EUmr(R) is Nilpotent
	13.5 Abelian Quotients over Polynomial Extensions of a Local Ring
	References
14 On a Theorem of Suslin
	14.1 Introduction
	14.2 Some Preliminaries
	14.3 On Some Results on Cocycles
	14.4 On a Proof of Suslin\'s Theorem
	References
15 On an Algebraic Analogue of the Mayer–Vietoris Sequence
	15.1 Introduction
	15.2 Some Preliminaries
	15.3 The Group Γ(A)
	15.4 On the Group π1 (SL2(A))
	15.5 On Cocycles Associated to Alternating Matrices
	15.6 On Some Consequences of the Above Results
	References
16 On the Completability of Unimodular Rows of Length Three
	16.1 Introduction
	16.2 Some Preliminaries
	16.3 On the Euler Class Group
	16.4 On a Lemma of Suslin
	16.5 On Various Proofs of Seshadri\'s Theorem
	16.6 On a Result of Bhatwadekar–Keshari
	16.7 Seshadri\'s Theorem and Euler Class Groups
	References
17 On a Group Structure on Unimodular Rows of Length Three over a Two-Dimensional Ring
	17.1 Introduction
	17.2 Some Preliminaries
	17.3 On the Euler Class Group
	17.4 On Some Results of Bhatwadekar–Raja Sridharan
	17.5 The Definition of the Group Structure on Um3(A)/SL3(A)
	17.6 The Conclusion of the Proof of the Existence of a Group Structure
	References
18 Relating the Principles  of Quillen–Suslin Theory
	18.1 Introduction
	18.2 The Local–Global Principle and Normality
	18.3 The Monic Inversion Principle
	References