Bass / Cassidy / Kovacic Contributions to Algebra

A Collection of Papers Dedicated to Ellis Kolchin
1. Auflage 2014
ISBN: 978-1-4832-6806-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 446 Seiten, Web PDF

Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin provides information pertinent to commutative algebra, linear algebraic group theory, and differential algebra. This book covers a variety of topics, including complex analysis, logic, K-theory, stochastic matrices, and differential geometry. Organized into 29 chapters, this book begins with an overview of the influence that Ellis Kolchin's work on the Galois theory of differential fields has had on the development of differential equations. This text then discusses the background model theoretic work in differential algebra and discusses the notion of model completions. Other chapters consider some properties of differential closures and some immediate consequences and include extensive notes with proofs. This book discusses as well the problems in finite group theory in finding the complex finite projective groups of a given degree. The final chapter deals with the finite forms of quasi-simple algebraic groups. This book is a valuable resource for students.

Autoren/Hrsg.

Bass, Hyman

Cassidy, Phyllis J.

Kovacic, Jerald

Inhaltsverzeichnis

1;Front Cover;1
2;Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;List of contributors;16
7;Preface;18
8;Chapter 1. Quadratic modules over polynomial rings;24
8.1;Introduction;24
8.2;1. Definitions, and background of the problem;25
8.3;2. Reduction by the Cancellation Theorem to low ranks;28
8.4;3. Quadratic spaces of low rank;30
8.5;4. Proof of the Cancellation Theorem;35
8.6;5. Proof of Karoubi's theorem;44
8.7;References;45
9;Chapter 2. The action of the universal modular group on certain boundary points;48
9.1;Text;48
9.2;References;58
10;Chapter 3. Differentially closed fields: a model-theoretic tour;60
10.1;Introduction;60
10.2;1. Background;61
10.3;2. Model completions;62
10.4;3. Main results;63
10.5;4. Some properties of differential closures and some immediate consequences;64
10.6;5. Notes;66
10.7;Appendix: Details of uniqueness proof filled in Definitions;78
10.8;References;83
11;Chapter 4. On finite projective groups;86
11.1;1. Introduction;86
11.2;2. Preliminaries;87
11.3;3. Finite strongly irreducible linear groups;89
11.4;4. Case I;92
11.5;5. Case II;97
11.6;6. Quasi-primitive linear groups;99
11.7;References;105
12;Chapter 5. Unipotent differential algebraic groups;106
12.1;Introduction of subject matter and notation;106
12.2;1. The structure of unipotent differential algebraic groups;109
12.3;2. Commutative linear unipotent differential algebraic groups;114
12.4;3. Differential rational cohomology;120
12.5;4. Extensions of differential algebraic groups;123
12.6;5. The groupsH.2(Ga, Ga) and Cent Ext.(Ga, Ga);131
12.7;References;138
13;Chapter 6. Solutions in the general solution;140
13.1;1. Introduction;140
13.2;2. Differentially closed places;141
13.3;3. Preliminary criteria;144
13.4;4. Algebraic extensions;145
13.5;5. Final criteria;150
13.6;6. Several indeterminates;150
13.7;References;151
14;Chapter 7. Folk theorems on elliptic equations;152
14.1;Text;152
14.2;References;157
15;Chapter 8. Limit properties of stochastic matrices;158
15.1;1. Preliminaries;159
15.2;2. Shrinking matrices;160
15.3;3. Analysis of supports;163
15.4;4. The case d = 1;165
15.5;5. The general case;169
16;Chapter 9. A fixed-point characterization of linearly reductive groups;174
16.1;1. Introduction;174
16.2;2. Proof of the theorem;175
16.3;References;178
17;Chapter 10. Orthogonal and unitary invariants of families of subspaces;180
17.1;1. Pairs of subspaces;181
17.2;2. Systems of lines;182
17.3;3. Representations of H*-algebras;184
17.4;4. Systems of subspaces;186
17.5;References;187
18;Chapter 11. The Macdonald–Kac formulas as a consequence of the Euler–Poincaré principle;188
18.1;Introduction;188
18.2;1. The Lie algebras of Kac and Moody;189
18.3;2. Quasisimple modules;192
18.4;3. Homology associated with F-parabolic subalgebras;192
18.5;4. Combinatorial identities;194
18.6;References;196
19;Chapter 12. The characters of reductive p-adic groups;198
19.1;Text;198
19.2;References;205
20;Chapter 13. Basic constructions in group extension theory;206
20.1;1. Introduction;206
20.2;2. Preliminaries;207
20.3;3. Group extensions;209
20.4;4. Crossed homomorphisms;212
20.5;5. Transgression;213
20.6;6. Inflation;215
20.7;7. Abelian kernels;217
20.8;8. Cohomology;222
21;Chapter 14. On the hyperalgebra of a semisimple algebraic group;226
21.1;1. The hyperalgebra;226
21.2;2. The algebra ur and its representations;227
21.3;3. Tensor products;231
21.4;4. Completely reducible G-modules;232
21.5;5. Mumford's conjecture;232
21.6;References;233
22;Chapter 15. A notion of regularity for differential local algebras;234
22.1;Introduction;234
22.2;1. Some preliminary facts about Kähler differentials;235
22.3;2. Regular differential algebras;239
22.4;3. A sufficient condition for regularity;242
22.5;4. Regular graded modules;244
22.6;5. The openness of the set of regular points;249
22.7;6. Regularity for .-fields;253
22.8;References;255
23;Chapter 16. The Engel–Kolchin theorem revisited;256
23.1;1. Introduction;256
23.2;2. A second unification of Kolchin's theorem and Levitzki's theorem;256
23.3;3. Two more theorems;257
23.4;4. Lie and Jordan analogues;259
23.5;5. The infinite case;259
23.6;References;260
24;Chapter 17. Prime differential ideals in differential rings;262
24.1;1. Special differential rings;262
24.2;2. The prime differential spectrum of a differential ring;268
24.3;References;272
25;Chapter 18. Constrained cohomology;274
25.1;Introduction;274
25.2;1. Constrained cohomology groups;275
25.3;2. Galois cohomology;279
25.4;3. Change of group;279
25.5;4. F-Cohomology;281
25.6;5. Change of field;282
25.7;6. The Hochschild–Serre sequence;285
25.8;References;289
26;Chapter 19. The integrability condition of deformations of CR structures;290
26.1;Introduction;290
26.2;1. The integrability condition of CR structures;291
26.3;2. The integrability condition of deformations;295
26.4;References;301
27;Chapter 20. Noetherian rings with many derivations;302
27.1;Text;302
27.2;References;316
28;Chapter 21. Hopf maps and quadratic forms over Z;318
28.1;1. Hurwitz triple and the map f;319
28.2;2. The map f and the map h;320
28.3;3. The family £;321
28.4;4. Hopf fibration S15 . S8 over Z;325
28.5;References;327
29;Chapter 22. Families of subgroup schemes of formal groups;328
29.1;1. Strict families of finite subgroup schemes of formal groups;329
29.2;2. Liftability of abelian varieties;339
29.3;References;341
30;Chapter 23. An effective lower bound on the "diophantine" approximation of algebraic functions by rational functions (II);344
30.1;Text;344
30.2;References;349
31;Chapter 24. On elementary, generalized elementary, and liouvillian extension fields;352
31.1;Text;352
31.2;References;365
32;Chapter 25. Derivations and valuation rings;366
32.1;Text;366
32.2;References;370
33;Chapter 26. On theorems of Lie–Kolchin, Borel, and Lang;372
33.1;Text;372
33.2;References;376
34;Chapter 27. A differential-algebraic study of the intrusion of logarithms into asymptotic expansions;378
34.1;A. Introduction;378
34.2;B. Some basic notations, procedures, and lemmas for the adjunction rank-rise problem;380
34.3;C. Instability ladders for differential polynomials of Class(V, r);384
34.4;D. Instability ladders for r-normal differential polynomials;386
34.5;E. Instability ladders for asymptotically nonsingular differential polynomials;389
34.6;F. Rank-rise results for the general first-order equation;390
34.7;G. Other rank-rise results [Note 23];393
34.8;Appendix;394
34.9;References;397
35;Chapter 28. A "theorem of Lie–Kolchin" for trees;400
35.1;Introduction;400
35.2;1. Trees;401
35.3;2. Various fixed-point properties;403
35.4;3. Solvable groups;404
35.5;4. Algebraic simple groups of relative rank = 2;406
35.6;5. The rank 1 case;408
35.7;References;411
36;Chapter 29. Regular elements in anisotropic tori;412
36.1;Introduction;412
36.2;1.;414
36.3;2.;415
36.4;3. Classical groups;416
36.5;4. Anisotropic tori in classical groups;418
36.6;5. Results for the classical groups;419
36.7;6. Type 1Al;419
36.8;7. Type1Bl;422
36.9;8. Type 1Cl;425
36.10;9. Type 1Dl;426
36.11;10. Type 2Al;427
36.12;11. Type 2Dl;428
36.13;12. The Suzuki groups 2C2(q2);428
36.14;13. Groups of type 3D4;429
36.15;14. Conjugacy classes in the Weyl group of F4;432
36.16;15. Groups of type 1F4;434
36.17;16. The Ree groups 2F4;436
36.18;17. Groups of type 1G2;438
36.19;18. Groups of type 2G2;440
36.20;19. Endomorphisms s with cyclic centralizer in W;441
36.21;20. Groups of type 1E6;443
36.22;21. Groups of type 2E6;444
36.23;22. Groups of type 1E7;445
36.24;23. Groups of type 1E8;446
36.25;References;447