Introduction to Affine Group Schemes

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.