Differential Tensor Algebras and their Module Categories

A detailed account of main results in the theory of differential tensor algebras.

R. Bautista (Author), L. Salmerón (Author), R. Zuazua (Author)

9780521757683, Cambridge University Press

Paperback, published 17 September 2009

462 pages, 90 exercises
22.7 x 15.2 x 2 cm, 0.65 kg

'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 In addition, they include numerous historical references … Many sections of the book exercises, and solutions to some of those can be found in the last section.'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 Reviews

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.

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(?), P1(?) and ?-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.

Subject Areas: Algebra [PBF]