Progress in commutative algebra. 2, Closures, finiteness and factorization /

This is the second of two volumes of a state-of-the-art survey article collection which emanates from three commutative algebra sessions atthe 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and...

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Bibliographic Details
Other Authors: Francisco, Christopher.
Format: eBook
Published: Berlin ; Boston : De Gruyter, 2012.
Series:Proceedings in mathematics.
Online Access:CONNECT
Table of Contents:
  • Preface; A Guide to Closure Operations in Commutative Algebra; 1 Introduction; 2 What Is a Closure Operation?; 2.1 The Basics; 2.2 Not-quite-closure Operations; 3 Constructing Closure Operations; 3.1 Standard Constructions; 3.2 Common Closures as Iterations of Standard Constructions; 4 Properties of Closures; 4.1 Star-, Semi-prime, and Prime Operations; 4.2 Closures Defined by Properties of (Generic) Forcing Algebras; 4.3 Persistence; 4.4 Axioms Related to the Homological Conjectures; 4.5 Tight Closure and Its Imitators; 4.6 (Homogeneous) Equational Closures and Localization.
  • 5 Reductions, Special Parts of Closures, Spreads, and Cores5.1 Nakayama Closures and Reductions; 5.2 Special Parts of Closures; 6 Classes of Rings Defined by Closed Ideals; 6.1 When Is the Zero Ideal Closed?; 6.2 When Are 0 and Principal Ideals Generated by Non-zerodivisors Closed?; 6.3 When Are Parameter Ideals Closed (Where R Is Local)?; 6.4 When Is Every Ideal Closed?; 7 Closure Operations on (Sub)modules; 7.1 Torsion Theories; A Survey of Test Ideals; 1 Introduction; 2 Characteristic p Preliminaries; 2.1 The Frobenius Endomorphism; 2.2 F-purity; 3 The Test Ideal.
  • 3.1 Test Ideals of Map-pairs3.2 Test Ideals of Rings; 3.3 Test Ideals in Gorenstein Local Rings; 4 Connections with Algebraic Geometry; 4.1 Characteristic 0 Preliminaries; 4.2 Reduction to Characteristic p> 0 and Multiplier Ideals; 4.3 Multiplier Ideals of Pairs; 4.4 Multiplier Ideals vs. Test Ideals of Divisor Pairs; 5 Tight Closure and Applications of Test Ideals; 5.1 The Briançon-Skoda Theorem; 5.2 Tight Closure for Modules and Test Elements; 6 Test Ideals for Pairs (R, at) and Applications; 6.1 Initial Definitions of at -test Ideals; 6.2 at -tight Closure; 6.3 Applications.
  • 7 Generalizations of Pairs: Algebras of Maps8 Other Measures of Singularities in Characteristic p; 8.1 F-rationality; 8.2 F-injectivity; 8.3 F-signature and F-splitting Ratio; 8.4 Hilbert-Kunz( -Monsky) Multiplicity; 8.5 F-ideals, F-stable Submodules, and F-pure Centers; A Canonical Modules and Duality; A.1 Canonical Modules, Cohen-Macaulay and Gorenstein Rings; A.2 Duality; B Divisors; C Glossary and Diagrams on Types of Singularities; C.1 Glossary of Terms; Finite-dimensional Vector Spaces with Frobenius Action; 1 Introduction; 2 A Noncommutative Principal Ideal Domain.
  • 3 Ideal Theory and Divisibility in Noncommutative PIDs3.1 Examples in K{F}; 4 Matrix Transformations over Noncommutative PIDs; 5 Module Theory over Noncommutative PIDs; 6 Computing the Invariant Factors; 6.1 Injective Frobenius Actions on Finite Dimensional Vector Spaces over a Perfect Field; 7 The Antinilpotent Case; Finiteness and Homological Conditions in Commutative Group Rings; 1 Introduction; 2 Finiteness Conditions; 3 Homological Dimensions and Regularity; 4 Zero Divisor Controlling Conditions; Regular Pullbacks; 1 Introduction; 2 Some Background; 3 Pullbacks of Noetherian Rings. 4 Pullbacks of Prüfer Rings.