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Differential Tensor Algebras and their Module Categories

$75.99 (C)

Part of London Mathematical Society Lecture Note Series

  • Date Published: October 2009
  • availability: Available
  • format: Paperback
  • isbn: 9780521757683

$ 75.99 (C)

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About the Authors
  • 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.

    • Includes central results not covered in existing books
    • Suitable for professional mathematicians and graduate students with only a basic knowledge of module theory
    • Contains over 90 exercises for the reader to test their understanding
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    Reviews & endorsements

    "The authors provide all minute details of every proof. The work is a remarkable example of what the reviewer would call "open source" mathematics. The reviewer feels that the publication of this important book will serve as a catalyst for further study of bocses and related structures."
    Alex Martsinkovsky, Mathematical Reviews

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    Product details

    • Date Published: October 2009
    • format: Paperback
    • isbn: 9780521757683
    • length: 462 pages
    • dimensions: 227 x 152 x 20 mm
    • weight: 0.65kg
    • contains: 90 exercises
    • availability: Available
  • Table of Contents

    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(Λ), P1(Λ) and Λ-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

  • Authors

    R. Bautista, National University of Mexico
    R. Bautista is a Professor in the Institute of Mathematics at the National University of Mexico, Morelia.

    L. Salmerón, National University of Mexico
    L. Salmerón is a Professor in the Institute of Mathematics at the National University of Mexico, Morelia.

    R. Zuazua, National University of Mexico
    R. Zuazua is a Professor in the Mathematics Department of the Faculty of Sciences at the National University of Mexico, Mexico City.

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