# Introduction to Affine Group Schemes

• William C. Waterhouse
Book

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

1. Front Matter
Pages i-xi
2. ### The Basic Subject Matter

1. Front Matter
Pages 1-1
2. William C. Waterhouse
Pages 3-12
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Pages 13-20
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Pages 21-27
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Pages 28-35
3. ### Decomposition Theorems

1. Front Matter
Pages 37-37
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Pages 39-45
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4. ### The Infinitesimal Theory

1. Front Matter
Pages 81-81
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5. ### Faithful Flatness and Quotients

1. Front Matter
Pages 101-101
2. William C. Waterhouse
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### Introduction

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.

### Keywords

Abelian group Algebra Algebraic structure Derivation Finite Gruppenschema Invariant Morphism Topology function proof theorem

#### Authors and affiliations

• William C. Waterhouse
• 1
1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-6217-6
• Copyright Information Springer-Verlag New York 1979
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4612-6219-0
• Online ISBN 978-1-4612-6217-6
• Series Print ISSN 0072-5285