Introduction to Affine Group Schemes

W. C. Waterhouse

ISBN 10: 1461262194 / ISBN 13: 9781461262190

Verlag: Springer New York, Springer New York Okt 2011, 2011

Beschreibung

Neuware -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.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 180 pp. Englisch.

Bestandsnummer des Verkäufers 9781461262190

Bibliografische Details

• Titel: Introduction to Affine Group Schemes
• Verlag: Springer New York, Springer New York Okt 2011
• Erscheinungsjahr: 2011
• Sprache: Englisch
• ISBN-10: 1461262194
• ISBN-13: 9781461262190
• Einband: Taschenbuch
• Zustand: Neu
• Abmessungen: 235B x 155 H x 11 T Millimeter Breite: 235 mm, Höhe: 155 mm, Tiefe: 11 mm
• Gewicht: 283 Gramm

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Inhaltsangabe

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.