An Introduction to Algebraic Geometry and Algebraic Groups

by Geck, Meinolf

• ISBN13: 9780198528319
• ISBN10: 0198528310
• Format: Hardcover
• Copyright: 2004-01-15
• Publisher: Clarendon Press

Summary

An accessible text introducing algebraic geometry 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 geometries 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 and Lie type. The text covers the conjugacy of borel subgroups and maximal tori, the theory of algebraic groups with a BN-pair, a thorough treatment of Frobenius maps on affine varieties and algebraic groups, zeta functions, and Lefschetz numbers for varieties over finite fields. Experts in the field will enjoy some of the new proofs. The text uses algebraic groups as the main examples, including worked out examples, instructuve exercises, as well as bibliographical and historical remarks.

Table of Contents

1 Algebraic sets and algebraic groups	1	(101)
1.1 The Zariski topology on affine space	1	(9)
1.2 Groebner bases and the Hilbert polynomial	10	(13)
1.3 Regular maps, direct products, and algebraic groups	23	(13)
1.4 The tangent space and non-singular points	36	(12)
1.5 The Lie algebra of a linear algebraic group	48	(12)
1.6 Groups with a split BN-pair	60	(14)
1.7 BN-pairs in symplectic and orthogonal groups	74	(13)
1.8 Bibliographic remarks and exercises	87	(15)
2 Affine varieties and finite morphisms	102	(63)
2.1 Hilbert's nullstellensatz and abstract affine varieties	103	(8)
2.2 Finite morphisms and Chevalley's theorem	111	(10)
2.3 Birational equivalences and normal varieties	121	(11)
2.4 Linearization and generation of algebraic groups	132	(8)
2.5 Group actions on affine varieties	140	(11)
2.6 The unipotent variety of the special linear groups	151	(7)
2.7 Bibliographic remarks and exercises	158	(7)
3 Algebraic representations and Borel subgroups	165	(59)
3.1 Algebraic representations, solvable groups, and theory	166	(8)
3.2 The main theorem of elimination theory	174	(10)
3.3 Grassmannian varieties and flag varieties	184	(11)
3.4 Parabolic subgroups and Borel subgroups	195	(10)
3.5 On the structure of Borel subgroups	205	(11)
3.6 Bibliographic remarks and exercises	216	(8)
4 Frobenius maps and finite groups of Lie type	224	(81)
4.1 Frobenius maps and rational structures	225	(10)
4.2 Frobenius maps and BN-pairs	235	(10)
4.3 Further applications of the Lang-Steinberg theorem	245	(12)
4.4 Counting points on varieties over finite fields	257	(10)
4.5 The virtual characters of Deligne and Lusztig	267	(11)
4.6 An example: the characters of the Suzuki groups	278	(11)
4.7 Bibliographic remarks and exercises	289	(16)
Index	305