# Contributions to Algebra

### A Collection of Papers Dedicated to Ellis Kolchin

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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.

## Table of Contents

List of contributors

Preface

Quadratic Modules Over Polynomial Rings

Introduction

1. Definitions, and Background of the Problem

2. Reduction by the Cancellation Theorem to Low Ranks

3. Quadratic Spaces of Low Rank

4. Proof of the Cancellation Theorem

References

The Action of the Universal Modular Group on Certain Boundary Points

Text

References

Differentially Closed Fields: A Model-Theoretic Tour

Introduction

1. Background

2. Model Completions

3. Main Results

4. Some Properties of Differential Closures and Some Immediate Consequences

5. Notes

Appendix: Details of Uniqueness Proof Filled in

References

On Finite Projective Groups

1. Introduction

2. Preliminaries

3. Finite Strongly Irreducible Linear Groups

4. Case I

5. Case II

6. Quasi-Primitive Linear Groups

References

Unipotent Differential Algebraic Groups

Introduction of Subject Matter and Notation

1. The Structure of Unipotent Differential Algebraic Groups

2. Commutative Linear Unipotent Differential Algebraic Groups

3. Differential Rational Cohomology

4. Extensions of Differential Algebraic Groups

5. The Groups HΔ2(Ga, Ga) and Cent ExtΔ(Ga, Ga)

References

Solutions in the General Solution

1. Introduction

2. Differentially Closed Places

3. Preliminary Criteria

4. Algebraic Extensions

5. Final Criteria

6. Several Indeterminates

References

Folk Theorems on Elliptic Equations

Text

References

Limit Properties of Stochastic Matrices

1. Preliminaries

2. Shrinking Matrices

3. Analysis of Supports

4. The Case d = 1

5. The General Case

A Fixed-Point Characterization of Linearly Reductive Groups

1. Introduction

2. Proof of the Theorem

References

Orthogonal and Unitary Invariants of Families of Subspaces

1. Pairs of Subspaces

2. Systems of Lines

3. Representations of H*-Algebras

4. Systems of Subspaces

References

The Macdonald-Kac Formulas as a Consequence of the Euler-Poincare Principle

Introduction

1. The Lie Algebras of Kac and Moody

2. Quasisimple Modules

3. Homology Associated with F-parabolic Subalgebras

4. Combinatorial Identities

References

The Characters of Reductive P-Adic Groups

Text

References

Basic Constructions in Group Extension Theory

1. Introduction

2. Preliminaries

3. Group Extensions

4. Crossed Homomorphisms

5. Transgression

6. Inflation

7. Abelian Kernels

8. Cohomology

On the Hyperalgebra of a Semisimple Algebraic Group

1. The Hyperalgebra

2. The Algebra u, and its Representations

3. Tensor Products

4. Completely Reducible G-Modules

5. Mumford's Conjecture

References

A Notion of Regularity for Differential Local Algebras

Introduction

1. Some Preliminary Facts About Kahler Differentials

2. Regular Differential Algebras

3. A Sufficient Condition for Regularity

4. Regular Graded Modules

5. The Openness of the Set of Regular Points

6. Regularity for Δ-Fields

References

The Engel-Kolchin Theorem Revisited

1. Introduction

2. A Second Unification of Kolchin's Theorem and Levitzki's Theorem

3. Two More Theorems

4. Lie and Jordan Analogues

5. The Infinite Case

References

Prime Differential Ideals in Differential Rings

1. Special Differential Rings

2. The Prime Differential Spectrum of a Differential Ring

References

Constrained Cohomology

Introduction

Notation

1. Constrained Cohomology Groups

2. Galois Cohomology

3. Change of Group

4. F-Cohomology

5. Change of Field

6. The Hochschild-Serre Sequence

References

The Integrability Condition of Deformations of CR Structures

Introduction

1. The Integrability Condition of CR Structures

2. The Integrability Condition of Deformations

References

Noetherian Rings with Many Derivations

Text

References

Hopf Maps and Quadratic Forms Over Z

1. Hurwitz Triple and the Map ƒ

2. The Map ƒ and the Map h

3. The Family se

4. Hopf Fibration S15 → S8 Over Z

References

Families of Subgroup Schemes of Formal Groups

1. Strict Families of Finite Subgroup Schemes of Formal Groups

2. Liftability of Abelian Varieties

References

An Effective Lower Bound on the "Diophantine" Approximation of Algebraic Functions by Rational Functions (II)

Text

References

On Elementary, Generalized Elementary, and Liouvillian Extension Fields

Text

References

Derivations and Valuation Rings

Text

References

On Theorems of Lie-Kolchin, Borel, and Lang

Text

References

A Differential-Algebraic Study of the Intrusion of Logarithms into Asymptotic Expansions

A. Introduction

B. Some Basic Notations, Procedures, and Lemmas for the Adjunction Rank-Rise Problem

C. Instability Ladders for Differential Polynomials of Class (V, r)

D. Instability Ladders for R-Normal Differential Polynomials

E. Instability Ladders for Asymptotically Nonsingular Differential Polynomials

F. Rank-Rise Results for the General First-Order Equation

G. Other Rank-Rise Results

Appendix

References

A "Theorem of Lie-Kolchin" for Trees

Introduction

1. Trees

2. Various Fixed-Point Properties

3. Solvable Groups

4. Algebraic Simple Groups of Relative Rank ≥ 2

5. The Rank 1 Case

References

Regular Elements in Anisotropic Tori

Introduction

1.

2.

3. Classical Groups

4. Anisotropic Tori in Classical Groups

5. Results for the classical Groups

6. Type 1Al

7. Type 1Bl

8. Type 1Cl

9. Type 1Dl

10. Type 2Al

11. Type 2Dl

12. The Suzuki Groups 2C2(q2)

13. Groups of Type 3D4

14. Conjugacy Classes in the Weyl Group of F4

15. Groups of Type 1F4

16. The Ree Groups 2F4

17. Groups of Type 1G2

18. Groups of Type 2G2

19. Endomorphisms σ with Cyclic Centralizer in W

20. Groups of Type 1E6

21. Groups of Type 2E6

22. Groups of Type 1Ε7

23. Groups of Type 1E8

References

## Product details

- No. of pages: 446
- Language: English
- Copyright: © Academic Press 1977
- Published: January 1, 1977
- Imprint: Academic Press
- eBook ISBN: 9781483268064

## About the Editors

### Hyman Bass

#### Affiliations and Expertise

Department of Mathematics, Columbia University, New York, New York