Differential Tensor Algebras and Their Module Categories

Differential Tensor Algebras and Their Module Categories

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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.show more

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'The authors provide all minute details of every proof. the work is a remarkable example of what the reviewer would call 'open source' mathematics. In addition, they include numerous historical references ... Many sections of the book exercises, and solutions to some of those can be found in the last section.' Alex Martinskovsky, Mathematical Reviewsshow more

About R. Bautista

R. Bautista is a Professor in the Institute of Mathematics at the National University of Mexico, Morelia. L. Salmeron is a Professor in the Institute of Mathematics at the National University of Mexico, Morelia. R. Zuazua is a Professor in the Mathematics Department of the Faculty of Sciences at the National University of Mexico, Mexico City.show more

Table of contents

Preface; 1. t-algebras and differentials; 2. Ditalgebras and modules; 3. Bocses, ditalgebras and modules; 4. Layered ditalgebras; 5. Triangular ditalgebras; 6. Exact structures in A-Mod; 7. Almost split conflations in A-Mod; 8. Quotient ditalgebras; 9. Frames and Roiter ditalgebras; 10. Product of ditalgebras; 11. Hom-tensor relations and dual basis; 12. Admissible modules; 13. Complete admissible modules; 14. Bimodule ltrations and triangular admissible modules; 15. Free bimodule ltrations and free ditalgebras; 16. AX is a Roiter ditalgebra, for suitable X; 17. Examples and applications; 18. The exact categories P(LAMBDA), P1(LAMBDA) and LAMBDA-Mod; 19. Passage from ditalgebras to finite dimensional algebras; 20. Scalar extension and ditalgebras; 21. Bimodules; 22. Parametrizing bimodules and wildness; 23. Nested and seminested ditalgebras; 24. Critical ditalgebras; 25. Reduction functors; 26. Modules over non-wild ditalgebras; 27. Tameness and wildness; 28. Modules over non-wild ditalgebras revisited; 29. Modules over non-wild algebras; 30. Absolute wildness; 31. Generic modules and tameness; 32. Almost split sequences and tameness; 33. Varieties of modules over ditalgebras; 34. Ditalgebras of partially ordered sets; 35. Further examples of wild ditalgebras; 36. Answers to selected exercises; References; Index.show more