Ring Theory And Algebraic Geometry

✍ Scribed by A. Granja; J.A. Hermida Alonso; A Verschoren

• Publisher: CRC Press
• Year: 2001
• Tongue: English
• Leaves: 361
• Category: Library

✦ Synopsis

Focuses on the interaction between algebra and algebraic geometry, including high-level research papers and surveys contributed by over 40 top specialists representing more than 15 countries worldwide. Describes abelian groups and lattices, algebras and binomial ideals, cones and fans, affine and projective algebraic varieties, simplicial and cellular complexes, polytopes, and arithmetics.

✦ Table of Contents

Ring Theory and Algebraic Geometry
Copyright Info
Preface
TOC
Contributors
Conference Participants
Chapter 1. Frobenius and Maschke Type Theorems for Doi- Hopf Modules and Entwined Modules Revisited: A Unified Approach
1 INTRODUCTION
2 SEPARABLE FUNCTORS AND FROBENIUS PAIRS OF FUNCTORS
3 ENTWINED MODULES AND DOI- HOPF MODULES
4 THE FUNCTOR FORGETTING THE COACTION
5 THE FUNCTOR FORGETTING THE A- ACTION
6 THE SMASH PRODUCT
REFERENCES
Chapter 2. Computing the Gelfand-Kirillov Dimension II
1 INTRODUCTION
2 ADMISSIBLE ORDERS IN MONOIDEALS AND STABLE SUBSETS
3 PBW ALGEBRAS, QUANTUM RELATIONS AND FILTRATIONS
4 CONSEQUENCES AND EXAMPLES
5 GROBNER BASES FOR MODULES
6 HOMOGENEOUS GROBNER BASES
7 THE GELFAND-KIRILLOV DIMENSION
References
Chapter 3. Some Problems About Nilpotent Lie Algebras
1 INTRODUCTION
2 FILIFORM LIE ALGEBRAS
2.1 Obtaining laws of families of filiform Lie algebras
2.2 Low-dimensional filiform Lie algebras
2.3 /c-abelian filiform Lie algebras
3 THE FAMILY OF p-FILIFORM LIE ALGEBRAS
3.1 p- filiform Lie algebras with p > n — 3
3.2 (n — 4)-filiform Lie algebras
3.3 p-filiform Lie algebras with n — 6 < p < n — 5
4 LIE ALGEBRAS WITH SMALL NILINDEX
4.1 Metabelian Lie algebras
4.2 Lie algebras with nilindex 3
5 NATURALLY GRADED NILPOTENT LIE ALGEBRAS
5.1 Naturally Graded filiform and Quasi- filiform Lie Algebras
5.2 Naturally Graded 3-filiform Lie Algebras
6 LENGTH OF NILPOTENT LIE ALGEBRAS
6.1 Connected gradations
6.2 Filiform Lie Algebra of maximum Length
6.3 Quasi- filiform Lie algebras of length greater than their nilindex
7 SYMBOLIC CALCULUS ON LIE ALGEBRAS
REFERENCES
Chapter 4. On L-Triples and Jordan H-Pairs
1 PREVIOUS RESULTS ON //-TRIPLES
2 PREVIOUS RESULTS ON JORDAN -PAIRS
3 MAIN RESULTS
REFERENCES
Chapter 5. Toric Mathematics from Semigroup Viewpoint
I INTRODUCTION
2 SEMIGROUP AND GENERATORS OF TORIC GEOMETRY
3 ABELIAN GROUPS AND LATTICES
4 SEMIGROUP IDEALS AND ALGEBRAS
5 CONES AND FANS
6 AFFINE AND PROJECTIVE TORIC VARIETIES
7 POLYTOPES, SIMPLICIAL AND CELLULAR COMPLEXES
8 MULTINUMERICAL SEMIGROUPS
9 APPLICATIONS
REFERENCES
Chapter 6. Canonical Forms for Linear Dynamical Systems over Commutative Rings: The Local Case
1 INTRODUCTION
2 LINEAR DYNAMICAL SYSTEMS OVER COMMUTATIVE RINGS: THE FEEDBACK GROUP
3 CANONICAL FORM FOR SYSTEMS OVER FIELDS
4 DEALING WITH THE LOCAL CASE
REFERENCES
Chapter 7. An introduction to Janet bases and Grobner bases
I INTRODUCTION
2 MONOMIALS
2.1 Janet modules
2.2 Multiplicative variables. Classes
3 COMPLETELY INTEGRABLE SYSTEMS. JANET BASES
4 JANET BASES AND GROBNER BASES
4.1 Homogeneous systems
4.2 Non-homogeneous systems
REFERENCES
Chapter 8. Invariants of Coalgebras
1 INTRODUCTION
2 PRELIMINARIES
3 THE PICARD GROUP
3.1 Definitions and properties
3.2 The Aut-Pic property
4 THE BRAUER GROUP OF A COCOMMUTATIVE COALGEBRA
4.1 Definitions and properties
4.2 Torsioness in the Brauer group
4.3 Subgroups of the Brauer group
Acknowledgments
REFERENCES
Chapter 9. Multiplication Objects
1 INTRODUCTION
2 MONOIDAL CATEGORIES
3 GENERAL PROPERTIES OF MULTIPLICATION OBJECTS
4 ENDOMORPHISMS OF MULTIPLICATION OBJECTS
REFERENCES
Chapter 10. Krull-Schmidt Theorem and Semilocal Endomorphism Rings
1 SEMILOCAL RINGS AND MODULES WHOSE ENDOMORPHISM RING IS SEMILOCAL
2 K0 OF A SEMILOCAL RING
3 UNISERIAL MODULES
4 HOMOGENEOUS SEMILOCAL RINGS AND MODULES WHOSE ENDOMORPHISM RING IS HOMOGENEOUS SEMILOCAL
REFERENCES
Chapter 11. On Suslin's Stability Theorem for R[XI, ... ,xm]
1 INTRODUCTION
2 CONSTRUCTIONS IN R[x]
2.1 Maximal ideals
2.2 Ideal of principal coefficients
2.3 Normalization of unimodular vectors
3 APPLICATIONS TO K-THEORY
REFERENCES
Chapter 12. Characterization of Rings Using Socle-Fine and Radical-Fine Notions
1 INTRODUCTION
2 PERFECT RINGS AND PSEUDO-FROBENIUS RINGS
3 RINGS WHOSE CLASS OF FINITE-DIMENSIONAL MODULES IS SOCLE-FINE
4 RADICAL-FINE CHARACTERIZATION OF RINGS
REFERENCES
Chapter 13. About Bernstein Algebras
1 PRELIMINARY RESULTS
2 IDEMPOTENTS
3 RELATIONS WITH OTHER CLASSES OF ALGEBRAS
4 BERNSTEIN PROBLEM
5 AUTOMORPHISMS AND DERIVATIONS
6 SOME OTHER ASPECTS
REFERENCES
Chapter 14. About an Algorithm of T. Oaku
1 INTRODUCTION
2 HOMOGENIZATION OF DIFFERENTIAL OPERATORS
3 COMPUTATION OF THE BERNSTEIN POLYNOMIAL
REFERENCES
Chapter 15. Minimal Injective Resolutions: Old and New
I INTRODUCTION
2 GRADE AND AUSLANDER-GORENSTEIN RINGS
3 COHEN-MACAULAY CONDITION
4 RESULTS
REFERENCES
Chapter 16. Special Divisors of Blowup Algebras
1 INTRODUCTION
2 DIVISORS
3 DIVISOR CLASS GROUP
4 THE EXPECTED CANONICAL MODULE
5 THE FUNDAMENTAL DIVISOR
6 COHEN-MACAULAY DIVISORS AND REDUCTION NUMBERS
7 VANISHING OF COHOMOLOGY
REFERENCES
Chapter 17. Existence of Euler Vector Fields for Curves with Binomial Ideal
1 INTRODUCTION
2 IRREDUCIBLE MONOMIAL CURVES
3 REDUCED MONOMIAL CURVES
4 MONOMIAL CURVES AND EULER VECTOR FIELDS
5 ALGORITHM
REFERENCES
Chapter 18. An Amitsur Cohomology Exact Sequence for Invo-lutive Brauer Groups of the Second Kind 1
1 INTRODUCTION
2 GENERALITIES
3 INVOLUTIVE INVARIANTS OF THE SECOND KIND
4 AMITSUR COHOMOLOGY
REFERENCES
Chapter 19. Computation of the Slopes of a D-Module of Type Dr/N
1 INTRODUCTION
2 DEFINITIONS
3 FINITENESS OF THE NUMBER OF SLOPES
4 A WAY OF COMPUTING AL M
5 THE ALGORITHM TO FIND SLOPES
6 ABOUT THE COMPUTATIONS IN V.
7 EXAMPLES
7.1 Slopes of O [ l / f ] / O .
7.2 Looking for slopes in a syzygy module
7.3 Slopes and direct sums of ideals
REFERENCES
Chapter 20. Symmetric Closed Categories and Involutive Brauer Groups1
1 INTRODUCTION
2 SOME BACKGROUND ON CLOSED CATEGORIESo
3 MONOIDS WITH INVOLUTION
4 THE INVOLUTIVE BRAUER GROUP
5 FUNCTORIAL BEHAVIOUR
REFERENCES