Unipotent

Abstract: 2. Sufficient condi t ions for hypoe l l ip t i c i ty . . . . . . . . . . . . . . . . . . . . . 251 8. Graded a n d free Lie a lgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 255 4. H a r m o n i c analysis on iV a n d the p roof of T h eo rem 2 . . . . . . . . . . . . . . . 257 5. Di la t ions a n d h o m o g e n e i t y on groups . . . . . . . . . . . . . . . . . . . . . . 261 6. Smoo th ly va ry ing families of f u n d a m e n t a l solut ions . . . . . . . . . . . . . . . . 265

Abstract: I The Basic Subject Matter.- 1 Affine Group Schemes.- 1.1 What We Are Talking About.- 1.2 Representable Functors.- 1.3 Natural Maps and Yoneda's Lemma.- 1.4 Hopf Algebras.- 1.5 Translating from Groups to Algebras.- 1.6 Base Change.- 2 Affine Group Schemes: Examples.- 2.1 Closed Subgroups and Homomorphisms.- 2.2 Diagonalizable Group Schemes.- 2.3 Finite Constant Groups.- 2.4 Cartier Duals.- 3 Representations.- 3.1 Actions and Linear Representations.- 3.2 Comodules.- 3.3 Finiteness Theorems.- 3.4 Realization as Matrix Groups.- 3.5 Construction of All Representations.- 4 Algebraic Matrix Groups.- 4.1 Closed Sets in kn.- 4.2 Algebraic Matrix Groups.- 4.3 Matrix Groups and Their Closures.- 4.4 From Closed Sets to Functors.- 4.5 Rings of Functions.- 4.6 Diagonalizability.- II Decomposition Theorems.- 5 Irreducible and Connected Components.- 5.1 Irreducible Components in kn.- 5.2 Connected Components of Algcbraic Matrix Groups.- 5.3 Components That Coalesce.- 5.4 Spec A.- 5.5 The Algebraic Meaning of Connectedness.- 5 6 Vista: Schemes.- 6 Connected Components and Separable Algebras.- 6.1 Components That Decompose.- 6.2 Separable Algebras.- 6.3 Classification of Separable Algebras.- 6.4 Etale Group Schemes 49 6 5 Separable Subalgcbras.- 6.5 Separable Subalgcbras.- 6.6 Connected Group Schemes.- 6.7 Connected Components of Group Schemes.- 6.8 Finite Groups over Perfect Fields.- 7 Groups of Multiplicative Type.- 7.1 Separable Matrices.- 7.2 Groups of Multiplicative Type.- 7.3 Character Groups.- 7.4 Anisotropic and Split Tori.- 7.5 Examples of Tori.- 7.6 Some Automorphism Group Schcmes.- 7.7 A Rigidity Theorem.- 8 Unipotent Groups.- 8.1 Unipotent Matrices.- 8 2 The Kolchin Fixed Point Theorem.- 8.3 Unipotent Group Schemes.- 8.4 Endomorphisms of Ga..- 8.5 Finite Unipotent Groups.- 9 Jordan Decomposition.- 9.1 Jordan Decomposition of a Matrix.- 9.2 Decomposition in Algebraic Matrix Groups.- 9.3 Decomposition of Abelian Algebraic Matrix Groups.- 9.4 Irreducible Representations of Abelian Group Schemes.- 9.5 Decomposition of Abelian Group Schemes.- 10 Nilpotent and Solvable Groups.- 10.1 Derived Subgroups.- 10.2 The Lie-Kolchin Triangularization Theorem.- 10.3 The Unipotent Subgroup.- 10.4 Decomposition of Nilpotent Groups.- 10.5 Vista: Borel Subgroups.- 10.6 Vista: Differential Algebra.- III The Infinitesimal Theory.- 11 Differentials.- 11.1 Derivations and Differentials.- 11.2 Simple Properties of Differentials.- 11.3 Differentials of Hopf Algebras.- 11.4 No Nilpotents in Characteristic Zero.- 11.5 Differentials of Field Extensions.- 11.6 Smooth Group Schemes.- 11.7 Vista: The Algebro-Geomctric Meaning of Smoothness.- 11.8 Vista: Formal Groups.- 12 Lie Algebras.- 12.1 Invariant Operators and Lie Algebras.- 12.2 Computation or Lie Algebras.- 12.3 Examples.- 12.4 Subgroups and Invariant Subspaces.- 12.5 Vista: Reductive and Semisimple Groups.- IV Faithful Flatness and Quotients.- 13 Faithful Flatness.- 13.1 Definition of Faithful Flatness.- 13.2 Localization Properties.- 13.3 Transition Properties.- 13.4 Generic Faithful Flatness.- 13.5 Proof of the Smoothness Theorem.- 14 Faithful Flatness of Hopf Algebras.- 14.1 Proof in the Smooth Case.- 14.2 Proof with Nilpotents Present.- 14.3 Simple Applications.- 14.4 Structure of Finite Connected Groups.- 15 Quotient Maps.- 15.1 Quotient Maps.- 15.2 Matrix Groups over$$ bar k $$/k.- 15.3 Injections and Closed Kmbeddings.- 15.4 Universal Property of Quotients.- 15.5 Sheaf Property of Quotients.- 15.6 Coverings and Sheaves.- 15.7 Vista: The Etale Topology.- 16 Construction of Quotients.- 16.1 Subgroups as Stabilizers.- 16.2 Difficulties with Coset Spaces.- 16.3 Construction of Quotients.- 16.4 Vista: Invariant Theory.- V Descent Theory.- 17 Descent Theory Formalism.- 17.1 Descent Data.- 17.2 The Descent Theorem.- 17.3 Descent of Algebraic Structure.- 17.4 Example: Zariski Coverings.- 17.5 Construction of Twisted Forms.- 17.6 Twisted Forms and Cohomology.- 17.7 Finite Galois Extensions.- 17.8 Infinite Galois Extensions.- 18 Descent Theory Computations.- 18.1 A Cohomology Exact Sequence.- 18.2 Sample Computations.- 18.3 Principal Homogeneous Spaces.- 18.4 Principal Homogeneous Spaces and Cohomology.- 18.5 Existence of Separable Splitting Fields.- 18.6 Example: Central Simple Algebras.- 18.7 Example: Quadratic Forms and the Arf Invariant.- 18.8 Vanishing Cohomology over Finite Fields.- 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.

Abstract: Introduction I. The Jacobian Conjecture 1. Statement of the Jacobian Problem; first observations 2. Some history of the Jacobian Conjecture 3. Faulty proofs 4. The use of stabilization and of formal methods II. The Reduction Theorem 1. Notation 2. Statement of the Reduction Theorem 3. Reduction to degree 3 4. Proof of the Reduction Theorem 5. r-linearization and unipotent reduction 6. Nilpotent rank 1 Jacobians III. The Formal Inverse 1. Notation 2. Abhyankar's Inversion Formula 3. The terms Gj 4. The tree expansion G\ = lT(\/a(T))lfPTf 5. Calculations References