An introduction to algebraic geometry and algebraic groups [[electronic resource] /] / Meinolf Geck

• Record Nr.: UNINA9910786146003321
• Autore: Geck Meinolf
• Pubbl/distr/stampa: Oxford ; ; New York, : Oxford University Press, 2003
• ISBN: 1-383-02466-9; 1-299-13293-6; 0-19-166372-7
• Descrizione fisica: 1 online resource (320 p.)
• Collana: Oxford Graduate Texts in Mathematics ; ; v.10; Oxford graduate texts in mathematics ; ; 10
• Disciplina: 516.35; 516.3/5
• Soggetti: Geometry, Algebraic; Linear algebraic groups
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: First published in paperback 2013.
• Nota di bibliografia: Includes bibliographical references (p. [298]-303) and index.
• Nota di contenuto: Cover; Contents; 1 Algebraic sets and algebraic groups; 1.1 The Zariski topology on affine space; 1.2 Groebner bases and the Hilbert polynomial; 1.3 Regular maps, direct products, and algebraic groups; 1.4 The tangent space and non-singular points; 1.5 The Lie algebra of a linear algebraic group; 1.6 Groups with a split BN-pair; 1.7 BN-pairs in symplectic and orthogonal groups; 1.8 Bibliographic remarks and exercises; 2 Affine varieties and finite morphisms; 2.1 Hilbert's nullstellensatz and abstract affine varieties; 2.2 Finite morphisms and Chevalley's theorem; 2.3 Birational equivalences and normal varieties2.4 Linearization and generation of algebraic groups; 2.5 Group actions on affine varieties; 2.6 The unipotent variety of the special linear groups; 2.7 Bibliographic remarks and exercises; 3 Algebraic representations and Borel subgroups; 3.1 Algebraic representations, solvable groups, and tori; 3.2 The main theorem of elimination theory; 3.3 Grassmannian varieties and flag varieties; 3.4 Parabolic subgroups and Borel subgroups; 3.5 On the structure of Borel subgroups; 3.6 Bibliographic remarks and exercises; 4 Frobenius maps and finite groups of Lie type4.1 Frobenius maps and rational structures; 4.2 Frobenius maps and BN-pairs; 4.3 Further applications of the Lang-Steinberg theorem; 4.4 Counting points on varieties over finite fields; 4.5 The virtual characters of Deligne and Lusztig; 4.6 An example: the characters of the Suzuki groups; 4.7 Bibliographic remarks and exercises; Index; A; B; C; D; E; F; G; H; I; J; K; L; M; N; O; P; R; S; T; U; V; W; Z
• Sommario/riassunto: An accessible text introducing algebraic geometries and algebraic groups at advanced undergraduate and early graduate level, this book develops the language of algebraic geometry from scratch and uses it to set up the theory of affine algebraic groups from first principles.Building on the background material from algebraic geometry and algebraic groups, the text provides an introduction to more advanced and specialised material. An example is the representation theory of finite groups of Lie type.The text covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups