Introduction to Affine Group Schemes / by William C. Waterhouse
(Graduate Texts in Mathematics ; 66)
| データ種別 | 電子ブック |
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| 出版情報 | New York, NY : Springer New York , 1979 |
| 本文言語 | 英語 |
| 大きさ | XII, 164 p : online resource |
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| 内容注記 | I The Basic Subject Matter 1 Affine Group Schemes 2 Affine Group Schemes: Examples 3 Representations 4 Algebraic Matrix Groups II Decomposition Theorems 5 Irreducible and Connected Components 6 Connected Components and Separable Algebras 7 Groups of Multiplicative Type 8 Unipotent Groups 9 Jordan Decomposition 10 Nilpotent and Solvable Groups III The Infinitesimal Theory 11 Differentials 12 Lie Algebras IV Faithful Flatness and Quotients 13 Faithful Flatness 14 Faithful Flatness of Hopf Algebras 15 Quotient Maps 16 Construction of Quotients V Descent Theory 17 Descent Theory Formalism 18 Descent Theory Computations Appendix: Subsidiary Information A.1 Directed Sets and Limits A.2 Exterior Powers A.3 Localization. Primes, and Nilpotents A.4 Noetherian Rings A.5 The Hilbert Basis Theorem A.6 The Krull Intersection Theorem A.7 The Nocthcr Normalization Lemma A.8 The Hilbert Nullstellensatz A.9 Separably Generated Fields A.10 Rudimentary Topological Terminology Further Reading Index of Symbols |
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| 一般注記 | 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 |
| 著者標目 | *Waterhouse, William C. author SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:Algebra LCSH:Group theory FREE:Mathematics FREE:Group Theory and Generalizations FREE:Algebra |
| 分 類 | DC23:512.2 |
| 巻冊次 | ISBN:9781461262176 
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| ISBN | 9781461262176 |
| URL | http://dx.doi.org/10.1007/978-1-4612-6217-6 |
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