Handbook of Tilting Theory(傾斜理論)

內(nèi)容概要

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.

書籍目錄

1 Introduction   2 Basic results of classical tilting theory L. Angeleri Hiigel, D. Happel, and H. Krause  REFERENCES3 Classification of representation-finite algebras and their modules  T. Bruistle  1 Introduction  2 Notation  3 Representation-finite algebras  4 Critical algebras  5 Tame algebras  REFERENCES4 A spectral sequence analysis of classical tilting func-tors  S. Brenner and M. C. R. Butler  1 Introduction  2 Tilting modules  3 Tilting functors, spectral sequences and filtrations  4 Applications  5 Edge effects, and the case t=2  REFERENCES5 Derived categories and tilting  B. Keller  1 Introduction  2 Derived categories  3 Derived functors  4 Tilting and derived equivalences  5 Triangulated categories  6 Morita theory for derived categories  7 comparison of t-structures, spectral sequences  8 Algebraic triangulated categories and dg algebras  REFERENCES6 Hereditary categoriesH. Lenzing  1 Fundamental concepts  2 Examples of hereditary categories  3 Repetitive shape of the derived category  4 Perpendicular categories  5 Exceptional objects  6 Piecewise hereditary algebras and Happel's theorem  7 Derived equivalence of hereditary categories  8 Modules over hereditary algebras  9 Spectral properties of hereditary categories  10 Weighted projective lines  11 Quasitilted algebras  REFERENCES7 Fourier-Mukai transforms L. Hille and M. Van den Bergh  1 Some background  2 Notations and conventions  3 Basics on Fourier-Mukai transforms  4 The reconstruction theorem  5 Curves and surfaces  6 Threefolds and higher dimensional varieties  7 Non-commutative rings in algebraic geometry  REFERENCES8 Tilting theory and homologically finite subcategories with applications to quasihereditary algebras  I. Reiten  1 The Basic Ingredients  2 The Correspondence Theorem  3 Quasihereditary algebras and their generalizations  4 Generalizations  REFERENCES9 Tilting modules for algebraic groups and finite dimensional algebras  S.Donkin10 Combinatorial aspects of the set of titling modules   L.Unger11 Infinete dimensional tilting modules and cotorsion pairs  J.Trlifaj12 Infinite dimensional titing modules over finite dimensional algebras  ф.Solberg13 Cotiting dualities  R.Colpi and K.R.Fuller14 Representations of finite groups and tilting   J.Chuang and J.Rickard15 Morita theory in stable homotopy theory   B.ShipleyAppendix Some remarks concerning tilting modules and tilted algebras. Origin.Relevance.Future  C.M.Ringel

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