Contributions to Algebra

Contributions to Algebra

A Collection of Papers Dedicated to Ellis Kolchin

1st Edition - January 1, 1977

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  • Editors: Hyman Bass, Phyllis J. Cassidy, Jerald Kovacic
  • eBook ISBN: 9781483268064

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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


    Quadratic Modules Over Polynomial Rings


    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


    The Action of the Universal Modular Group on Certain Boundary Points



    Differentially Closed Fields: A Model-Theoretic Tour


    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


    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


    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)


    Solutions in the General Solution

    1. Introduction

    2. Differentially Closed Places

    3. Preliminary Criteria

    4. Algebraic Extensions

    5. Final Criteria

    6. Several Indeterminates


    Folk Theorems on Elliptic Equations



    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


    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


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


    1. The Lie Algebras of Kac and Moody

    2. Quasisimple Modules

    3. Homology Associated with F-parabolic Subalgebras

    4. Combinatorial Identities


    The Characters of Reductive P-Adic Groups



    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


    A Notion of Regularity for Differential Local Algebras


    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


    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


    Prime Differential Ideals in Differential Rings

    1. Special Differential Rings

    2. The Prime Differential Spectrum of a Differential Ring


    Constrained Cohomology



    1. Constrained Cohomology Groups

    2. Galois Cohomology

    3. Change of Group

    4. F-Cohomology

    5. Change of Field

    6. The Hochschild-Serre Sequence


    The Integrability Condition of Deformations of CR Structures


    1. The Integrability Condition of CR Structures

    2. The Integrability Condition of Deformations


    Noetherian Rings with Many Derivations



    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


    Families of Subgroup Schemes of Formal Groups

    1. Strict Families of Finite Subgroup Schemes of Formal Groups

    2. Liftability of Abelian Varieties


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



    On Elementary, Generalized Elementary, and Liouvillian Extension Fields



    Derivations and Valuation Rings



    On Theorems of Lie-Kolchin, Borel, and Lang



    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



    A "Theorem of Lie-Kolchin" for Trees


    1. Trees

    2. Various Fixed-Point Properties

    3. Solvable Groups

    4. Algebraic Simple Groups of Relative Rank ≥ 2

    5. The Rank 1 Case


    Regular Elements in Anisotropic Tori




    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


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

Phyllis J. Cassidy

Jerald Kovacic

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