Auslander-Buchweitz Approximations of Equivariant Modules (London Mathematical Society Lecture Note Series, Series Number 282)

This book, first published in 2000, focuses on homological aspects of equivariant modules. It presents a homological approximation theory in the category of equivariant modules, unifying the Cohen-Macaulay approximations in commutative ring theory and Ringel's theory of delta-good approximations for quasi-hereditary algebras and reductive groups. The book provides a detailed introduction to homological algebra, commutative ring theory and homological theory of comodules of co-algebras over an arbitrary base. It aims to overcome the difficulty of generalising known homological results in representation theory. This book will be of interest to researchers and graduate students in algebra, specialising in commutative ring theory and representation theory. Cover......Page 1 London Mathematical Society Lecture Note Series 282......Page 2 Auslander-Buchweitz Approximations of Equivariant Modules......Page 4 0521796962......Page 5 Contents......Page 8 Introduction......Page 12 1.1 Yoneda's lemma......Page 18 1.2 Adjoint functors and limits......Page 20 1.3 Exact categories......Page 22 1.4 Derived categories and derived functors......Page 26 1.5 Extensions and Ext groups......Page 29 1.6 The cobar resolution......Page 32 1.7 Grothendieck categories......Page 35 1.8 Grothendieck topology and sheaf theory......Page 37 1.9 Noetherian categories and locally noetherian categories......Page 41 1.10 Semisimple objects in a Grothendieck category......Page 42 1.11 Full subcategories of an abelian category......Page 45 1.12 χ-approximations and the Auslander-Buchweitz theory......Page 46 2.1 Flat modules and pure maps......Page 54 2.2 Mittag-Leffler modules......Page 58 2.3 Faithfully flat morphisms and descent theory......Page 61 2.4 The I-depth......Page 63 2.5 Cohen-Macaulay, Gorenstein, and regular rings......Page 65 2.6 Local cohomology......Page 67 2.7 Ring-theoretic properties of morphisms......Page 68 2.8 Betti numbers, Bass numbers and complete intersections......Page 73 2.9 Resolutions of perfect modules......Page 74 2.10 Dualizing complexes and canonical modules......Page 76 2.11 The duality of proper morphisms and rational singularities......Page 78 2.12 Summary of open loci results......Page 82 2.13 Normal flatness......Page 84 3 Hopf algebras over an arbitrary base......Page 88 3.1 Coalgebras and bialgebras......Page 89 3.2 Hopf algebras......Page 90 3.3 Comodules......Page 92 3.5 Bicomodules, Horn and \bigotimes......Page 93 3.6 The restriction and the induction......Page 98 3.7 Locally noetherian property......Page 103 3.8 The dual algebra of a coalgebra......Page 104 3.9 The dual coalgebra of an algebra......Page 107 3.10 Rational modules......Page 108 3.11 FPCP coalgebras and IFP coalgebras......Page 110 3.12 \bigotimes and Horn of modules and comodules over a Hopf algebra......Page 112 3.13 The dual Hopf algebra......Page 115 3.14 Module algebras and comodule algebras......Page 116 3.15 Coalgebras and comodules over a scheme......Page 117 4.1 Group schemes as faisceaux......Page 118 4.2 Rational representations of an algebraic group......Page 119 4.3 Algebraic tori......Page 122 4.4 Maximal tori, Borel subgroups, and reductive groups......Page 123 4.5 Split reductive groups......Page 124 4.6 General linear groups......Page 127 4.7 Representations of reductive groups over an algebraically closed field......Page 128 4.8 Universal module functors......Page 130 4.9 Tilting modules......Page 132 4.10 Cotilting modules......Page 133 5.1 Cocommutative Hopf algebra actions......Page 139 5.2 Tor^A and Ext_A as A#U-modules......Page 141 5.3 (G, A)-modules......Page 144 1.1 Construction of Ext_A......Page 148 1.2 Equivariant modules of a split torus......Page 153 1.3 FPCP groups and IFP groups......Page 154 2.1 Stability of various loci......Page 156 2.2 Universal density of hyperalgebras......Page 161 2.3 A generalization to equivariant sheaves......Page 163 2.4 Matijevic-Roberts type theorem......Page 170 1.1 Weak split highest weight coalgebras......Page 174 1.2 Weak highest weight theory......Page 179 1.3 Highest weight coalgebras and good comodules......Page 186 1.4 Weak highest weight coalgebras and good filtrations......Page 191 2.1 U-acyclicity of flat complexes......Page 195 2.2 The definition and the existence of a Donkin system......Page 201 2.3 Basic properties of the Donkin system......Page 208 3.1 Ringel's approximation over a field......Page 215 3.2 Tilting modules over a field......Page 219 4.1 Tilting modules over a commutative ring......Page 222 4.2 Minimal Ringel's approximations over local rings......Page 230 4.3 Cohen-Macaulay analogue of u-good module......Page 232 4.4 Cohen-Macaulay Ringel's approximation......Page 236 4.5 Applications to split reductive groups......Page 239 4.6 Good modules of a general linear group......Page 241 1.1 Graded G-algebras......Page 246 1.2 Reductive group actions on graded algebras......Page 252 1.3 Relative Ringel's approximation......Page 256 1.4 Relative Cohen-Macaulay Ringel's approximation......Page 262 2.1 Resolutions of determinantal rings......Page 267 2.2 Buchsbaum-Rim type resolutions......Page 271 2.3 Kempf 's construction......Page 273 Glossary......Page 278 Bibliography......Page 284 Index......Page 294 Cover; Title; Copyright; Dedication; Contents; Introduction; Conventions and terminology; I Background Materials; 1 From homological algebra; 1.1 Yoneda's lemma; 1.2 Adjoint functors and limits; 1.3 Exact categories; 1.4 Derived categories and derived functors; 1.5 Extensions and Ext groups; 1.6 The cobar resolution; 1.7 Grothendieck categories; 1.8 Grothendieck topology and sheaf theory; 1.9 Noetherian categories and locally noetherian categories; 1.10 Semisimple objects in a Grothendieck category ... ; 1.11 Full subcategories of an abelian category This book presents a new homological approximation theory in the category of equivariant modules, unifying the Cohen-Macaulay approximations in commutative ring theory and Ringel's theory of Delta-good approximations for quasi-hereditary algebras and reductive groups. The book provides a detailed introduction to homological algebra, commutative ring theory and homological theory of comodules of coalgebras over an arbitrary base. It aims to overcome the difficulty of generalizing known homological results in representation theory. This book, first published in 2000, focuses on homological aspects of equivariant modules and discusses interactions between commutative ring theory and representation theory. The book aims to unify two important examples of Auslander-Buchweitz approximations in these areas of algebra. It is primarily aimed at researchers but will also be suitable for graduate students. (1.1.1) We denote the category of sets by Set, the category of groups by Grp, and the category of abelian groups by Ab or by zM.