Approximations and Endomorphism Algebras of Modules (De Gruyter Expositions in Mathematics 41)

This monograph provides a thorough treatment of two important parts of contemporary module theory: approximations of modules and their applications, notably to infinite dimensional tilting theory, and realizations of algebras as endomorphism algebras of groups and modules. Attention is also given to E-rings and free modules with distinguished submodules. The monograph starts from basic facts and gradually develops the theory to its present frontiers. It is suitable for graduate students interested in algebra as well as experts in module and representation theory. Chapter 1 Some useful classes of modules 26 1.1 S–completions 26 1.2 Pure–injective modules 45 1.3 Locally projective modules 68 1.4 Factors of products and slender modules 83 1.5 Slender modules over Dedekind domains 109 Chapter 2 Approximations of modules 119 2.1 Preenvelopes and precovers 119 2.2 Cotorsion pairs and Tor–pairs 124 2.3 Minimal approximations 131 Chapter 3 Complete cotorsion pairs 137 3.1 Ext and direct limits 137 3.2 The variety of complete cotorsion pairs 142 3.3 Ext and inverse limits 150 Chapter 4 Deconstruction of cotorsion pairs 159 4.1 Approximations by modules of finite homological dimensions 159 4.2 Hill Lemma and Kaplansky Theorem for cotorsion pairs 167 4.3 Closure properties providing for completeness 174 4.4 Matlis cotorsion and strongly flat modules 188 4.5 The closure of a cotorsion pair 203 Chapter 5 Tilting approximations 213 5.1 Tilting modules 213 5.2 Classes of finite type 226 5.3 Injectivity properties of tilting modules 244 Chapter 6 1–tilting modules and their applications 249 6.1 Tilting torsion classes 249 6.2 The structure of tilting modules and classes over particular rings 253 6.3 Matlis localizations 267 Chapter 7 Tilting approximations and the finitistic dimension conjectures 280 7.1 Finitistic dimension conjectures and the tilting module 280 7.2 A formula for the little finitistic dimension of right artinian rings 288 7/3 Artinian rings with P