Introduction to Affine Group Schemes

ISBN-13: 9780387904214 / Angielski / Twarda / 1979 / 164 str.

William C. Waterhouse; W. C. Waterhouse

Opis

Ah Love Could you and I with Him consl?ire To grasp this sorry Scheme of things entIre' KHAYYAM People investigating algebraic groups have studied the same objects in many different guises. My first goal thus has been to take three different viewpoints and demonstrate how they offer complementary intuitive insight into the subject. In Part I we begin with a functorial idea, discussing some familiar processes for constructing groups. These turn out to be equivalent to the ring-theoretic objects called Hopf algebras, with which we can then con- struct new examples. Study of their representations shows that they are closely related to groups of matrices, and closed sets in matrix space give us a geometric picture of some of the objects involved. This interplay of methods continues as we turn to specific results. In Part II, a geometric idea (connectedness) and one from classical matrix theory (Jordan decomposition) blend with the study of separable algebras. In Part III, a notion of differential prompted by the theory of Lie groups is used to prove the absence of nilpotents in certain Hopf algebras. The ring-theoretic work on faithful flatness in Part IV turns out to give the true explanation for the behavior of quotient group functors. Finally, the material is connected with other parts of algebra in Part V, which shows how twisted forms of any algebraic structure are governed by its automorphism group scheme.

Spis treści

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.