Differential Tensor Algebras and their Module Categories (London Mathematical Society Lecture Note Series)

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R. Bautista, L. Salmerón, R. ZuazuaCambridge University Press, 9/17/2009EAN 9780521757683, ISBN10: 0521757681Paperback, 462 pages, 22.7 x 15.2 x 2 cmLanguage: EnglishThis volume pr ovides a systematic presentation of the theory of differential tensor algebras and their categories of modules. It involves reduction techniques which have proved to be very useful in the development of representation theory of finite dimensional algebras. The main results obtained with these methods are presented in an elementary and self contained way. The authors provide a fresh point of view of well known facts on tame and wild differential tensor algebras, on tame and wild algebras, and on their modules. But there are also some new results and some new proofs. Their approach presents a formal alternative to the use of bocses (bimodules over categories with coalgebra structure) with underlying additive categories and pull-back reduction constructions. Professional mathematicians working in representation theory and related fields, and graduate students interested in homological algebra will find much of interest in this book.Preface1. t-algebras and differentials2. Ditalgebras and modules3. Bocses, ditalgebras and modules4. Layered ditalgebras5. Triangular ditalgebras6. Exact structures in A-Mod7. Almost split conflations in A-Mod8. Quotient ditalgebras9. Frames and Roiter ditalgebras10. Product of ditalgebras11. Hom-tensor relations and dual basis12. Admissible modules13. Complete admissible modules14. Bimodule ltrations and triangular admissible modules15. Free bimodule ltrations and free ditalgebras16. AX is a Roiter ditalgebra, for suitable X17. Examples and applications18. The exact categories P(Λ), P1(Λ) and Λ-Mod19. Passage from ditalgebras to finite dimensional algebras20. Scalar extension and ditalgebras21. Bimodules22. Parametrizing bimodules and wildness23. Nested and seminested ditalgebras24. Critical ditalgebras25. Reduction functors26. Modules over non-wild ditalgebras27. Tameness and wildness28. Modules over non-wild ditalgebras revisited29. Modules over non-wild algebras30. Absolute wildness31. Generic modules and tameness32. Almost split sequences and tameness33. Varieties of modules over ditalgebras34. Ditalgebras of partially ordered sets35. Further examples of wild ditalgebras36. Answers to selected exercisesReferencesIndex. Show more