内容

目次

Preface; 1. Results on topological spaces; 1.1 Irreducible sets andspaces; 1.2 Dimension; 1.3 Noetherian spaces; 1.4 Constructible sets; 1.5Gluing topological spaces; 2. Rings and modules; 2.1 Ideals; 2.2 Prime andmaximal ideals; 2.3 Rings of fractions and localization; 2.4 Localization ofmodules; 2.5 Radical of an ideal; 2.6 Local rings; 2.7 Noetherian rings andmodules; 2.8 Derivations; 2.9 Module of differentials; 3. Integralextensions; 3.1 Integral dependence; 3.2 Integrally closed rings; 3.3Extensions of prime ideals; 4. Factorial rings; 4.1 Generalities; 4.2 Uniquefactorization; 4.3 Principal ideal domains and Euclidean domains; 4.4Polynomial and factorial rings; 4.5 Symmetric polynomials; 4.6 Resultant anddiscriminant; 5. Field extensions; 5.1 Extensions; 5.2 Algebraic andtranscendental elements; 5.3 Algebraic extensions; 5.4 Transcendence basis;5.5 Norm and trace; 5.6 Theorem of the primitive element; 5.7 Going DownTheorem; 5.8 Fields and derivations; 5.9 Conductor; 6. Finitely generatedalgebras; 6.1 Dimension; 6.2 Noether's Normalization Theorem; 6.3 Krull'sPrincipal Ideal Theorem; 6.4 Maximal ideals; 6.5 Zariski topology; 7.Gradings and filtrations; 7.1 Graded rings and graded modules; 7.2 Gradedsubmodules; 7.3 Applications; 7.4 Filtrations; 7.5 Grading associated to afiltration; 8. Inductive limits; 8.1 Generalities; 8.2 Inductive systems ofmaps; 8.3 Inductive systems of magmas, groups and rings; 8.4 An example; 8.5Inductive systems of algebras; 9. Sheaves of functions; 9.1 Sheaves; 9.2Morphisms; 9.3 Sheaf associated to a presheaf; 9.4 Gluing; 9.5 Ringed space;10. Jordan decomposition and some basic results on groups; 10.1 Jordandecomposition; 10.2 Generalities on groups; 10.3 Commutators; 10.4 Solvablegroups; 10.5 Nilpotent groups; 10.6 Group actions; 10.7 Generalities onrepresentations; 10.8 Examples; 11. Algebraic sets; 11.1 Affine algebraicsets; 11.2 Zariski topology; 11.3 Regular functions; 11.4 Morphisms; 11.5Examples of morphisms; 11.6 Abstract algebraic sets; 11.7 Principal opensubsets; 11.8 Products of algebraic sets; 12. Prevarieties and varieties;12.1 Structure sheaf; 12.2 Algebraic prevarieties; 12.3 Morphisms ofprevarieties; 12.4 Products of prevarieties; 12.5 Algebraic varieties; 12.6Gluing; 12.7 Rational functions; 12.8 Local rings of a variety; 13.Projective varieties; 13.1 Projective spaces; 13.2 Projective spaces andvarieties; 13.3 Cones and projective varieties; 13.4 Complete varieties; 13.5Products; 13.6 Grassmannian variety; 14. Dimension; 14.1 Dimension ofvarieties; 14.2 Dimension and the number of equations; 14.3 System ofparameters; 14.4 Counterexamples; 15. Morphisms and dimension; 15.1 Criterionof affineness; 15.2 Affine morphisms; 15.3 Finite morphisms; 15.4Factorization and applications; 15.5 Dimension of fibres of a morphism; 15.6An example; 16. Tangent spaces; 16.1 A first approach; 16.2 Zariski tangentspace; 16.3 Differential of a morphism; 16.4 Some lemmas; 16.5 Smooth points;17. Normal varieties; 17.1 Normal varieties; 17.2 Normalization; 17.3Products of normal varieties; 17.4 Properties of normal varieties; 18. Rootsystems; 18.1 Reflections; 18.2 Root systems; 18.3 Root systems and bilinearforms; 18.4 Passage to the field of real numbers; 18.5 Relation between tworoots; 18.6 Base of a root system; 18.7 Weyl chambers; 18.8 Highest root;18.9 Closed subsets of roots; 18.10 Weights; 18.11 Graphs; 18.12 Dynkindiagrams; 18.13 Classification of root systems; 19. Lie algebras; 19.1Generalities on Lie algebras; 19.2 Representations; 19.3 Nilpotent Liealgebras; 19.4 Solvable Lie algebras; 19.5 Radical and the largest nilpotentideal; 19.6 Nilpotent radical; 19.7 Regular linear forms; 19.8 Cartansubalgebras; 20. Semisimple and reductive Lie algebras; 20.1 Semisimple Liealgebras; 20.2 Examples; 20.3 Semisimplicity of representations; 20.4Semisimple and nilpotent elements; 20.5 Reductive Lie algebras; 20.6 Resultson the structure of semisimple Lie algebras; 20.7 Subalgebras of semisimpleLie algebras; 20.8 Parabolic subalgebras; 21. Algebraic groups; 21.1Generalities; 21.2 Subgroups and morphisms; 21.3 Connectedness; 21.4 Actionsof an algebraic group; 21.5 Modules; 21.6 Group closure; 22. Affine algebraicgroups; 22.1 Translations of functions; 22.2 Jordan decomposition; 22.3Unipotent groups; 22.4 Characters and weights; 22.5 Tori and diagonalizablegroups; 22.6 Groups of dimension one; 23. Lie algebra of an algebraic group;23.1 An associative algebra; 23.2 Lie algebras; 23.3 Examples; 23.4 Computingdifferentials; 23.5 Adjoint representation; 23.6 Jordan decomposition; 24.Correspondence between groups and Lie algebras; 24.1 Notations; 24.2 Analgebraic subgroup; 24.3 Invariants; 24.4 Functorial properties; 24.5Algebraic Lie subalgebras; 24.6 A particular case; 24.7 Examples; 24.8Algebraic adjoint group; 25. Homogeneous spaces and quotients; 25.1Homogeneous spaces; 25.2 Some remarks; 25.3 Geometric quotients; 25.4Quotient by a subgroup; 25.5 The case of finite groups; 26. Solvable groups;26.1 Conjugacy classes; 26.2 Actions of diagonalizable groups; 26.3 Fixedpoints; 26.4 Properties of solvable groups; 26.5 Structure of solvablegroups; 27. Reductive groups; 27.1 Radical and unipotent radical; 27.2Semisimple and reductive groups; 27.3 Representations; 27.4 Finitenessproperties; 27.5 Algebraic quotients; 27.6 Characters; 28. Borel subgroups,parabolic subgroups and Cartan subgroups; 28.1 Borel subgroups; 28.2 Theoremsof density; 28.3 Centralizers and tori; 28.4 Properties of parabolicsubgroups; 28.5 Cartan subgroups; 29. Cartan subalgebras, Borel subalgebrasand parabolic subalgebras; 29.1 Generalities; 29.2 Cartan subalgebras; 29.3Application to semisimple Lie algebras; 29.4 Borel subalgebras; 29.5Properties of parabolic subalgebras; 29.6 More on reductive Lie algebras;29.7 Other applications; 29.8 Maximal subalgebras; 30. Representations ofsemisimple Lie algebras; 30.1 Enveloping algebra; 30.2 Weights and primitiveelements; 30.3 Finite-dimensional modules; 30.4 Verma modules; 30.5 Resultson existence and uniqueness; 30.6 A property of the Weyl group; 31. Symmetricinvariants; 31.1 Invariants of finite groups; 31.2 Invariant polynomialfunctions; 31.3 A free module; 32. S-triples; 32.1 Theorem ofJacobson-Morosov; 32.2 Some lemmas; 32.3 Conjugation of S-triples; 32.4Characteristic; 32.5 Regular and principal elements; 33. Polarizations; 33.1Definition of polarizations; 33.2 Polarizations in the semisimple case; 33.3A non-polarizable element; 33.4 Polarizable elements; 33.5 Theorem ofRichardson; 34. Results on orbits; 34.1 Notations; 34.2 Some lemmas; 34.3Generalities on orbits; 34.4 Minimal nilpotent orbit; 34.5 Subregularnilpotent orbit; 34.6 Dimension of nilpotent orbits; 34.7 Prehomogeneousspaces of parabolic type; 35. Centralizers; 35.1 Distinguished elements; 35.2Distinguished parabolic subalgebras; 35.3 Double centralizers; 35.4Normalizers; 35.5 A semisimple Lie subalgebra; 35.6 Centralizers and regularelements; 36. s -root systems; 36.1 Definition; 36.2 Restricted root systems;36.3 Restriction of a root; 37. Symmetric Lie algebras; 37.1 Primarysubalgebras; 37.2 Definition of symmetric Lie algebras; 37.3 Naturalsubalgebras; 37.4 Cartan subspaces; 37.5 The case of reductive Lie algebras;37.6 Linear forms; 38. Semisimple symmetric Lie algebras; 38.1 Notations;38.2 Iwasawa decomposition; 38.3 Coroots; 38.4 Centralizers; 38.5 S-triples;38.6 Orbits; 38.7 Symmetric invariants; 38.8 Double centralizers; 38.9Normalizers; 38.10 Distinguished elements; 39. Sheets of Lie algebras; 39.1Jordan classes; 39.2 Topology of Jordan classes; 39.3 Sheets; 39.4 Dixmiersheets; 39.5 Jordan classes in the symmetric case; 39.6 Sheets in thesymmetric case; 40. Index and linear forms; 40.1 Stable linear forms; 40.2Index of a representation; 40.3 Some useful inequalities; 40.4 Index andsemi-direct products; 40.5 Heisenberg algebras in semisimple Lie algebras;40.6 Index of Lie subalgebras of Borel subalgebras; 40.7 Seaweed Liealgebras; 40.8 An upper bound for the index; 40.9 Cases where the bound isexact; 40.10 On the index of parabolic subalgebras; References; Index ofnotations; Index of terminologies