Integral closure : rees algebras, multiplicities, algorithms整闭包
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作者： Wolmer Vasconcelos 著
出 版 社： 北京燕山出版社
出版时间： 2005-7-1字数：版次： 1页数： 519印刷时间： 2005/07/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540255406包装： 精装内容简介
Integral Closure gives an account of theoretical and algorithmic developments on the integral closure of algebraic structures. These are shared concerns in commutative algebra, algebraic geometry, number theory and the computational aspects of these fields. The overall goal is to determine and analyze the equations of the assemblages of the set of solutions that arise under various processes and algorithms. It gives a comprehensive treatment of Rees algebras and multiplicity theory - while pointing to applications in many other problem areas. Its main goal is to provide complexity estimates by tracking numerically invariants of the structures that may occur.
This book is intended for graduate students and researchers in the fields mentioned above. It contains, besides exercises aimed at giving insights, numerous research problems motivated by the developments reported.
目录
Preface
Introduction
1 Numerical Invariants of a Rees Algebra
1.1 Equations of a Rees Algebra
1.1.1 The Rees Algebra of an Ideal
1.1.2 Dimension of Symmetric and Rees Algebras of Modules
1.2 Rees Algebras and Reductions
1.2.1 Basic Properties of Reductions
1.2.2 Integrally Closed Ideals and Normal Ideals
1.3 Special Fiber and Analytic Spread
1.3.1 Special Fiber and Noether Normalization
1.3.2 Explicit Reduction Numbers
1.3.3 Analytic Spread and Codimension
1.4 Reduction Numbers of Ideals
1.5 Determinants and Ranks of Modules
1.5.1 Cayley-Hamilton Theorem
1.5.2 The Big Rank of a Module
1.6 Boundedness of Reduction Numbers
1.6.1 Arithmetic Degree
1.6.2 Global Bounds of Reduction Numbers
1.7 Intertwining Algebras
1.8 Briancon-Skoda Bounds
1.9 Exercises
2 Hilbert Functions and Multiplicities
2.1 Reduction Modules and Algebras
2.1.1 Structures Associated to Rees Algebras
2.1.2 Bounding Hilbert Functions
2.2 Maximal Hilbert Functions
2.2.1 The Eakin-Sathaye Theorem
2.2.2 Hilbert Functions of Primary Ideals
2.3 Degree Functions
2.3.1 ClassicalDegrees
2.3.2 Generalized Multiplicities of Graded Modules
2.4 Cohomological Degrees
2.4.1 Homological Degree
2.4.2 General Properties of Degs
2.5 Finiteness of Hilbert Functions
2.6 Numbers of Generators of Cohen-Macaulay Ideals
2.6.1 Estimating Number of Generators with Multiplicities
2.6.2 Number of Generators and the Socle
2.7 Multiplicities and Reduction Numbers
2.7.1 The Modulo Dimension One Technique
2.7.2 Special Fibers
2.7.3 Ideals of Dimension One and Two
2.8 Exercises
3 Depth and Cohomology of Rees Algebras
3.1 Settings of Cohen-Macaulayness
3.1.1 Systems of Parameters and Hypersurface Sections
3.1.2 Passing Cohen-Macaulayness Around
3.2 Cohen-Macaulayness of Proj (R) and Cohomology
3.2.1 Castelnuovo-Mumford Regularity and a-invariants
3.2.2 Vanishing of Cohomology
3.3 Reduction Number and Cohen-Macaulayness
3.4 Sk-Conditions on Rees Algebras
3.4.1 Detecting (Sk)
3.4.2 Rk-Conditions on Rees Algebras
3.5 Exercises
4 Divisors of a Rees Algebra
4.1 Divisors of an Algebra
4.2 Divisor Class Group
4.3 The Expected Canonical Module
4.4 The Fundamental Divisor
4.5 Cohen-Macaulay Divisors and Reduction Numbers
4.6 Exercises
5 Koszul Homology
5.1 Koszul Complexes of Ideals and Modules
5.2 Module Structure of Koszul Homology
5.3 Linkage and Residual Intersections
5.4 Approximation Complexes
5.5 Ideals with Good Reductions
……
6 Integral Closure of Algebras
7 Integral Closure and Normalization of Ideals
8 Integral Closure of Modules
9 HowTo
References
Notation and Terminology
Index
