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Introduction to Affine Group Schemes

  • Textbook
  • © 1979

Overview

Authors:
  1. William C. Waterhouse

    1. Department of Mathematics, The Pennsylvania State University, University Park, USA

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Part of the book series: Graduate Texts in Mathematics (GTM, volume 66)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xi
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  2. The Basic Subject Matter

    1. Front Matter

      Pages 1-1
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    2. Affine Group Schemes

      • William C. Waterhouse
      Pages 3-12

    3. Affine Group Schemes: Examples

      • William C. Waterhouse
      Pages 13-20

    4. Representations

      • William C. Waterhouse
      Pages 21-27

    5. Algebraic Matrix Groups

      • William C. Waterhouse
      Pages 28-35

  3. Decomposition Theorems

    1. Front Matter

      Pages 37-37
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    2. Irreducible and Connected Components

      • William C. Waterhouse
      Pages 39-45

    3. Connected Components and Separable Algebras

      • William C. Waterhouse
      Pages 46-53

    4. Groups of Multiplicative Type

      • William C. Waterhouse
      Pages 54-61

    5. Unipotent Groups

      • William C. Waterhouse
      Pages 62-67

    6. Jordan Decomposition

      • William C. Waterhouse
      Pages 68-72

    7. Nilpotent and Solvable Groups

      • William C. Waterhouse
      Pages 73-79

  4. The Infinitesimal Theory

    1. Front Matter

      Pages 81-81
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    2. Differentials

      • William C. Waterhouse
      Pages 83-91

    3. Lie Algebras

      • William C. Waterhouse
      Pages 92-100

  5. Faithful Flatness and Quotients

    1. Front Matter

      Pages 101-101
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    2. Faithful Flatness

      • William C. Waterhouse
      Pages 103-108

    3. Faithful Flatness of Hopf Algebras

      • William C. Waterhouse
      Pages 109-113

    4. Quotient Maps

      • William C. Waterhouse
      Pages 114-120
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Keywords

About this book

Ah Love! Could you and I with Him consl?ire To grasp this sorry Scheme of things entIre' KHAYYAM People investigating algebraic groups have studied the same objects in many different guises. My first goal thus has been to take three different viewpoints and demonstrate how they offer complementary intuitive insight into the subject. In Part I we begin with a functorial idea, discussing some familiar processes for constructing groups. These turn out to be equivalent to the ring-theoretic objects called Hopf algebras, with which we can then con­ struct new examples. Study of their representations shows that they are closely related to groups of matrices, and closed sets in matrix space give us a geometric picture of some of the objects involved. This interplay of methods continues as we turn to specific results. In Part II, a geometric idea (connectedness) and one from classical matrix theory (Jordan decomposition) blend with the study of separable algebras. In Part III, a notion of differential prompted by the theory of Lie groups is used to prove the absence of nilpotents in certain Hopf algebras. The ring-theoretic work on faithful flatness in Part IV turns out to give the true explanation for the behavior of quotient group functors. Finally, the material is connected with other parts of algebra in Part V, which shows how twisted forms of any algebraic structure are governed by its automorphism group scheme.

Authors and Affiliations

  • Department of Mathematics, The Pennsylvania State University, University Park, USA

    William C. Waterhouse

Bibliographic Information

  • Book Title: Introduction to Affine Group Schemes

  • Authors: William C. Waterhouse

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-6217-6

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York Inc. 1979

  • Hardcover ISBN: 978-0-387-90421-4Published: 13 November 1979

  • Softcover ISBN: 978-1-4612-6219-0Published: 12 October 2011

  • eBook ISBN: 978-1-4612-6217-6Published: 06 December 2012

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: XII, 164

  • Topics: Group Theory and Generalizations, Algebra

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