Handbook of Tilting Theory (London Mathematical Society Lecture Note Series, Series Number 332)

Tilting Theory Originates In The Representation Theory Of Finite Dimensional Algebras. Today The Subject Is Of Much Interest In Various Areas Of Mathematics, Such As Finite And Algebraic Group Theory, Commutative And Non-commutative Algebraic Geometry, And Algebraic Topology. The Aim Of This Book Is To Present The Basic Concepts Of Tilting Theory As Well As The Variety Of Applications. It Contains A Collection Of Key Articles, Which Together Form A Handbook Of The Subject, And Provide Both An Introduction And Reference For Newcomers And Experts Alike. Edited By Lidia Angeleri Hügel, Dieter Happel, Henning Krause. Includes Bibliographical References. Cover......Page 1 Title Page......Page 4 Copyright......Page 5 Contents......Page 6 1 Introduction page......Page 10 2 Basic results of classical tilting theory......Page 18 References......Page 21 1 Introduction......Page 24 2 Notation......Page 25 3 Representation-finite algebras......Page 27 4 Critical algebras......Page 33 5 Tame algebras......Page 35 References......Page 37 1 Introduction......Page 40 2 Tilting modules......Page 41 3 Tilting functors, spectral sequences and filtrations......Page 44 4 Applications......Page 52 5 Edge effects, and the case t ..2......Page 55 References......Page 56 1 Introduction......Page 58 2 Derived categories......Page 60 3 Derived functors......Page 72 4 Tilting and derived equivalences......Page 75 5 Triangulated categories......Page 81 6 Morita theory for derived categories......Page 87 7 Comparison of t-structures, spectral sequences......Page 92 8 Algebraic triangulated categories and dg algebras......Page 99 References......Page 106 6 Hereditary categories......Page 114 1 Fundamental concepts......Page 115 2 Examples of hereditary categories......Page 117 3 Repetitive shape of the derived category......Page 121 4 Perpendicular categories......Page 123 5 Exceptional objects......Page 124 6 Piecewise hereditary algebras and Happel’s theorem......Page 126 8 Modules over hereditary algebras......Page 130 9 Spectral properties of hereditary categories......Page 133 10 Weighted projective lines......Page 134 11 Quasitilted algebras......Page 151 References......Page 152 1 Some background......Page 156 3 Basics on Fourier-Mukai transforms......Page 158 4 The reconstruction theorem......Page 164 5 Curves and surfaces......Page 168 6 Threefolds and higher dimensional varieties......Page 175 7 Non-commutative rings in algebraic geometry......Page 179 References......Page 182 8 Tilting theory and homologically finite subcategories with applications to quasihereditary algebras......Page 188 1 The Basic Ingredients......Page 190 2 The Correspondence Theorem......Page 200 3 Quasihereditary algebras and their generalizations......Page 209 4 Generalizations......Page 216 References......Page 220 9 Tilting modules for algebraic groups and finite dimensional algebras......Page 224 1 Quasi-hereditary algebras......Page 226 2 Coalgebras and Comodules......Page 229 3 Linear Algebraic Groups......Page 234 4 Reductive Groups......Page 237 5 Infinitesimal Methods......Page 242 6 Some support for tilting modules......Page 247 7 Invariant theory......Page 248 8 General Linear Groups......Page 250 9 Connections with symmetric groups and Hecke algebras......Page 253 10 Some recent applications to Hecke algebras......Page 256 References......Page 263 1 Introduction......Page 268 2 The partial order of tilting modules......Page 269 3 The quiver of tilting modules......Page 270 4 The simplicial complex of tilting modules......Page 279 References......Page 284 11 Infinite dimensional tilting modules and cotorsion pairs......Page 288 1 Cotorsion pairs and approximations of modules......Page 290 2 Tilting cotorsion pairs......Page 301 3 Cotilting cotorsion pairs......Page 307 4 Finite type, duality, and some examples......Page 313 5 Tilting modules and the finitistic dimension conjectures......Page 321 References......Page 325 12 Infinite dimensional tilting modules over finite dimensional algebras......Page 332 1 Definitions and preliminaries......Page 333 2 The subcategory correspondence......Page 336 3 The finitistic dimension conjectures......Page 341 4 Complements of tilting and cotilting modules......Page 345 5 Classification of all cotilting modules......Page 349 References......Page 350 13 Cotilting dualities......Page 354 1 Generalized Morita Duality and Finitistic Cotilting Modules......Page 357 2 Cotilting Modules and Bimodules......Page 359 3 Weak Morita Duality......Page 362 4 Pure Injectivity of Cotilting Modules and Reflexivity......Page 364 References......Page 365 1 A brief introduction to modular representation theory......Page 368 2 The abelian defect group conjecture......Page 369 3 Symmetric algebras......Page 370 4 Characters and derived equivalence......Page 375 5 Splendid equivalences......Page 379 6 Derived equivalence and stable equivalence......Page 382 7 Lifting stable equivalences......Page 384 8 Clifford theory......Page 385 9 Cases for which the Abelian Defect Group Conjecture has been verified......Page 387 10 Nonabelian defect groups......Page 392 References......Page 393 1 Introduction......Page 402 2 Spectral Algebra......Page 405 3 Quillen model categories......Page 408 4 Differential graded algebras......Page 412 5 Two topologically equivalent DGAs......Page 415 References......Page 418 Appendix Some remarks concerning tilting modules and tilted algebras. Origin. Relevance. Future......Page 422 1 Basic Setting......Page 423 2 Connections......Page 432 3 The new cluster tilting approach......Page 455 References......Page 479 Cover 1 Title Page 4 Copyright 5 Contents 6 1 Introduction page 10 2 Basic results of classical tilting theory 18 References 21 3 Classification of representation-finite algebras and their modules 24 1 Introduction 24 2 Notation 25 3 Representation-finite algebras 27 4 Critical algebras 33 5 Tame algebras 35 References 37 4 A spectral sequence analysis of classical tilting functors 40 1 Introduction 40 2 Tilting modules 41 3 Tilting functors, spectral sequences and filtrations 44 4 Applications 52 5 Edge effects, and the case t ..2 55 References 56 5 Derived categories and tilting 58 1 Introduction 58 2 Derived categories 60 3 Derived functors 72 4 Tilting and derived equivalences 75 5 Triangulated categories 81 6 Morita theory for derived categories 87 7 Comparison of t-structures, spectral sequences 92 8 Algebraic triangulated categories and dg algebras 99 References 106 6 Hereditary categories 114 1 Fundamental concepts 115 2 Examples of hereditary categories 117 3 Repetitive shape of the derived category 121 4 Perpendicular categories 123 5 Exceptional objects 124 6 Piecewise hereditary algebras and Happel’s theorem 126 7 Derived equivalence of hereditary categories 130 8 Modules over hereditary algebras 130 9 Spectral properties of hereditary categories 133 10 Weighted projective lines 134 11 Quasitilted algebras 151 References 152 7 Fourier-Mukai transforms 156 1 Some background 156 2 Notations and conventions 158 3 Basics on Fourier-Mukai transforms 158 4 The reconstruction theorem 164 5 Curves and surfaces 168 6 Threefolds and higher dimensional varieties 175 7 Non-commutative rings in algebraic geometry 179 References 182 8 Tilting theory and homologically finite subcategories with applications to quasihereditary algebras 188 1 The Basic Ingredients 190 2 The Correspondence Theorem 200 3 Quasihereditary algebras and their generalizations 209 4 Generalizations 216 References 220 9 Tilting modules for algebraic groups and finite dimensional algebras 224 1 Quasi-hereditary algebras 226 2 Coalgebras and Comodules 229 3 Linear Algebraic Groups 234 4 Reductive Groups 237 5 Infinitesimal Methods 242 6 Some support for tilting modules 247 7 Invariant theory 248 8 General Linear Groups 250 9 Connections with symmetric groups and Hecke algebras 253 10 Some recent applications to Hecke algebras 256 References 263 10 Combinatorial aspects of the set of tilting modules 268 1 Introduction 268 2 The partial order of tilting modules 269 3 The quiver of tilting modules 270 4 The simplicial complex of tilting modules 279 References 284 11 Infinite dimensional tilting modules and cotorsion pairs 288 1 Cotorsion pairs and approximations of modules 290 2 Tilting cotorsion pairs 301 3 Cotilting cotorsion pairs 307 4 Finite type, duality, and some examples 313 5 Tilting modules and the finitistic dimension conjectures 321 References 325 12 Infinite dimensional tilting modules over finite dimensional algebras 332 1 Definitions and preliminaries 333 2 The subcategory correspondence 336 3 The finitistic dimension conjectures 341 4 Complements of tilting and cotilting modules 345 5 Classification of all cotilting modules 349 References 350 13 Cotilting dualities 354 1 Generalized Morita Duality and Finitistic Cotilting Modules 357 2 Cotilting Modules and Bimodules 359 3 Weak Morita Duality 362 4 Pure Injectivity of Cotilting Modules and Reflexivity 364 References 365 14 Representations of finite groups and tilting 368 1 A brief introduction to modular representation theory 368 2 The abelian defect group conjecture 369 3 Symmetric algebras 370 4 Characters and derived equivalence 375 5 Splendid equivalences 379 6 Derived equivalence and stable equivalence 382 7 Lifting stable equivalences 384 8 Clifford theory 385 9 Cases for which the Abelian Defect Group Conjecture has been verified 387 10 Nonabelian defect groups 392 References 393 15 Morita theory in stable homotopy theory 402 1 Introduction 402 2 Spectral Algebra 405 3 Quillen model categories 408 4 Differential graded algebras 412 5 Two topologically equivalent DGAs 415 References 418 Appendix Some remarks concerning tilting modules and tilted algebras. Origin. Relevance. Future 422 1 Basic Setting 423 2 Connections 432 3 The new cluster tilting approach 455 References 479 052168045X,9780521680455,9780511735134,0511735138 Cambridge University Press A handbook of key articles providing both an introduction and reference for newcomers and experts alike