Differential Tensor Algebras and their Module Categories (London Mathematical Society Lecture Note Series, Series Number 362)

This Volume Provides 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. T-algebras And Differentials -- Dutalgebras And Modules -- Bocses, Ditalgebras And Modules -- Layered Ditalgebras -- Exact Structures In A-mod -- Quotient Ditalgebras -- Frames And Roiter Ditalgebras -- Product Of Ditalgebras -- Hom-tensor Relations And Dual Basis -- Admissible Modules -- Complete Admissible Modules -- Bimodule Filtrations And Triangular Admissible Modules -- Free Bimodule Filtrations And Free Datalgebras -- Examples And Applications -- Passage From Ditalgebras To Finite-dimensional Algebras -- Scalar Extension And Ditalgebras -- Bimodules -- Parametrizing Bimodules And Wildness -- Nested And Seminested Ditalgebras -- Critical Diatalgebras -- Critical Ditalgebras -- Reduction Functors -- Reduction Functions -- Modules Over Non-wild Ditalgebras -- Tameness And Wildness -- Modules Over Non-wild Ditalgebras Revisited -- Modules Over Non-wild Algebras -- Absolute Wildness -- Generic Modules And Tameness -- Varieties Of Modules Over Ditalgebras -- Ditalgebras Of Partially Ordered Sets -- Further Examples Of Wild Ditalgebras -- Answers To Selected Exercises. R. Bautista, L. Salmerón And R. Zuazua. Includes Bibliographical References (p. 446-448) And Index. Differential Tensor Algebras and their Module Categories......Page 1 Contents......Page 6 Preface......Page 8 1. t-algebras and differentials......Page 11 2. Ditalgebras and modules......Page 14 3. Bocses, ditalgebras and modules......Page 23 4. Layered ditalgebras......Page 28 5. Triangular ditalgebras......Page 33 6. Exact structures in A-Mod......Page 42 7. Almost split conflations in A-Mod......Page 57 8. Quotient ditalgebras......Page 66 9. Frames and Roiter ditalgebras......Page 76 10. Product of ditalgebras......Page 81 11. Hom-tensor relations and dual basis......Page 84 12. Admissible modules......Page 92 13. Complete admissible modules......Page 108 14. Bimodule filtrations and triangular admissible modules......Page 118 15. Free bimodule filtrations and free ditalgebras......Page 127 16. AX is a Roiter ditalgebra, for suitable X......Page 138 17. Examples and applications......Page 143 18. The exact categories P(Λ), P1(Λ) and Λ-Mod......Page 154 19. Passage from ditalgebras to finite-dimensional algebras......Page 164 20. Scalar extension and ditalgebras......Page 179 21. Bimodules......Page 194 22. Parametrizing bimodules and wildness......Page 207 23. Nested and seminested ditalgebras......Page 225 24. Critical ditalgebras......Page 256 25. Reduction functors......Page 270 26. Modules over non-wild ditalgebras......Page 282 27. Tameness and wildness......Page 294 28. Modules over non-wild ditalgebras revisited......Page 302 29. Modules over non-wild algebras......Page 324 30. Absolute wildness......Page 337 31. Generic modules and tameness......Page 345 32. Almost split sequences and tameness......Page 355 33. Varieties of modules over ditalgebras......Page 370 34. Ditalgebras of partially ordered sets......Page 386 35. Further examples of wild ditalgebras......Page 394 36. Answers to selected exercises......Page 407 References......Page 456 Index......Page 459