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El. knyga: Progress in Commutative Algebra 2: Closures, Finiteness and Factorization

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This is the second of two volumes of a state-of-the-art survey article collection which emanates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. The current trends in two of the most active areas of commutative algebra are presented: non-noetherian rings (factorization, ideal theory, integrality), advances from the homological study of noetherian rings (the local theory, graded situation and its interactions with combinatorics and geometry). This second volume discusses closures, decompositions, and factorization.

"Christopher Francisco, Oklahoma State University, Stillwater, Oklahoma, USA; Lee C. Klingler, Florida Atlantic University, Boca Raton, Florida, USA; Sean M. Sather-Wagstaff, North Dakota State University, Fargo, North Dakota, USA; Janet Vassilev, University of New Mexico, Albuquerque, New Mexico, USA. "
Preface v
A Guide to Closure Operations in Commutative Algebra
Neil Epstein
1 Introduction
1(1)
2 What Is a Closure Operation?
2(5)
2.1 The Basics
2(4)
2.2 Not-quite-closure Operations
6(1)
3 Constructing Closure Operations
7(3)
3.1 Standard Constructions
7(2)
3.2 Common Closures as Iterations of Standard Constructions
9(1)
4 Properties of Closures
10(12)
4.1 Star-, Semi-prime, and Prime Operations
10(6)
4.2 Closures Defined by Properties of (Generic) Forcing Algebras
16(1)
4.3 Persistence
17(1)
4.4 Axioms Related to the Homological Conjectures
18(2)
4.5 Tight Closure and Its Imitators
20(1)
4.6 (Homogeneous) Equational Closures and Localization
21(1)
5 Reductions, Special Parts of Closures, Spreads, and Cores
22(3)
5.1 Nakayama Closures and Reductions
22(1)
5.2 Special Parts of Closures
23(2)
6 Classes of Rings Defined by Closed Ideals
25(4)
6.1 When Is the Zero Ideal Closed?
26(1)
6.2 When Are 0 and Principal Ideals Generated by Non-zerodivisors Closed?
26(1)
6.3 When Are Parameter Ideals Closed (Where R Is Local)?
27(1)
6.4 When Is Every Ideal Closed?
28(1)
7 Closure Operations on (Sub)modules
29(10)
7.1 Torsion Theories
31(8)
A Survey of Test Ideals
Karl Schwede
Kevin Tucker
1 Introduction
39(2)
2 Characteristic p Preliminaries
41(3)
2.1 The Frobenius Endomorphism
41(1)
2.2 F-purity
42(2)
3 The Test Ideal
44(6)
3.1 Test Ideals of Map-pairs
44(3)
3.2 Test Ideals of Rings
47(1)
3.3 Test Ideals in Gorenstein Local Rings
48(2)
4 Connections with Algebraic Geometry
50(7)
4.1 Characteristic 0 Preliminaries
50(2)
4.2 Reduction to Characteristic p >0 and Multiplier Ideals
52(2)
4.3 Multiplier Ideals of Pairs
54(2)
4.4 Multiplier Ideals vs. Test Ideals of Divisor Pairs
56(1)
5 Tight Closure and Applications of Test Ideals
57(6)
5.1 The Briancon--Skoda Theorem
61(1)
5.2 Tight Closure for Modules and Test Elements
61(2)
6 Test Ideals for Pairs (R, at) and Applications
63(5)
6.1 Initial Definitions of at-test Ideals
63(2)
6.2 at-tight Closure
65(1)
6.3 Applications
66(2)
7 Generalizations of Pairs: Algebras of Maps
68(3)
8 Other Measures of Singularities in Characteristic p
71(30)
8.1 F-rationality
71(1)
8.2 F-injectivity
72(1)
8.3 F-signature and F-splitting Ratio
73(2)
8.4 Hilbert--Kunz(--Monsky) Multiplicity
75(3)
8.5 F-ideals, F-stable Submodules, and F-pure Centers
78(2)
A Canonical Modules and Duality
80(1)
A.1 Canonical Modules, Cohen--Macaulay and Gorenstein Rings
80(1)
A.2 Duality
81(2)
B Divisors
83(2)
C Glossary and Diagrams on Types of Singularities
85(1)
C.1 Glossary of Terms
86(15)
Finite-dimensional Vector Spaces with Frobenius Action
Florian Enescu
1 Introduction
101(1)
2 A Noncommutative Principal Ideal Domain
102(2)
3 Ideal Theory and Divisibility in Noncommutative PIDs
104(5)
3.1 Examples in K{F}
107(2)
4 Matrix Transformations over Noncommutative PIDs
109(2)
5 Module Theory over Noncommutative PIDs
111(3)
6 Computing the Invariant Factors
114(7)
6.1 Injective Frobenius Actions on Finite Dimensional Vector Spaces over a Perfect Field
118(3)
7 The Antinilpotent Case
121(8)
Finiteness and Homological Conditions in Commutative Group Rings
Sarah Glaz
Ryan Schwarz
1 Introduction
129(1)
2 Finiteness Conditions
130(3)
3 Homological Dimensions and Regularity
133(3)
4 Zero Divisor Controlling Conditions
136(9)
Regular Pullbacks
Jason G. Boynton
Sean Sather-Wagstaff
1 Introduction
145(2)
2 Some Background
147(4)
3 Pullbacks of Noetherian Rings
151(2)
4 Pullbacks of Prufer Rings
153(3)
5 Pullbacks of Coherent Rings
156(3)
6 The n-generator Property in Pullbacks
159(6)
7 Factorization in Pullbacks
165(6)
Noetherian Rings without Finite Normalization
Bruce Olberding
1 Introduction
171(3)
2 Normalization and Completion
174(2)
3 Examples between DVRs
176(9)
4 Examples Birationally Dominating a Local Ring
185(4)
5 A Geometric Example
189(1)
6 Strongly Twisted Subrings of Local Noetherian Domains
190(15)
Krull Dimension of Polynomial and Power Series Rings
John J. Watkins
1 Introduction
205(2)
2 A Key Property of R[ x]
207(1)
3 The Main Theorem
208(4)
4 Additional Applications
212(5)
5 The Dimension of Power Series Rings
217(4)
The Projective Line over the Integers
Ela Celikbas
Christina Eubanks-Turner
1 Introduction
221(1)
2 Definitions and Background
222(4)
3 The Coefficient Subset and Radical Elements of Proj(Z[ h, k])
226(3)
4 The Conjecture for Proj(Z[ h, k]) and Previous Partial Results
229(3)
5 New Results Supporting the Conjecture
232(6)
6 Summary and Questions
238(3)
On Zero Divisor Graphs
Jim Coykendall
Sean Sather-Wagstaff
Laura Sheppardson
Sandy Spiroff
1 Introduction
241(2)
2 Survey of Past Research on Zero Divisor Graphs
243(7)
2.1 Beck's Zero Divisor Graph
243(1)
2.2 Anderson and Livingston's Zero Divisor Graph
244(2)
2.3 Mulay's Zero Divisor Graph
246(1)
2.4 Other Zero Divisor Graphs
247(3)
3 Star Graphs
250(20)
4 Graph Homomorphisms and Graphs Associated to Modules
270(3)
5 Cliques
273(9)
6 Girth and Cut Vertices
282(6)
6.1 Girth
282(4)
6.2 Cut Vertices
286(2)
7 Chromatic Numbers and Clique Numbers
288(13)
7.1 Chromatic/Clique Number 1
289(1)
7.2 Chromatic/Clique Number 2
290(1)
7.3 Chromatic/Clique Number 3
290(3)
A Tables for Example 3.14
293(2)
B Graph Theory
295(6)
A Closer Look at Non-unique Factorization via Atomic Decay and Strong Atoms
Scott T. Chapman
Ulrich Krause
1 Introduction
301(2)
2 Strong Atoms and Prime Ideals
303(2)
3 Atomic Decay in the Ring of Integers of an Algebraic Number Field
305(4)
4 The Fundamental Example of the Failure of Unique Factorization: Z[ √-5]
309(2)
5 A More Striking Example
311(3)
6 Concluding Remarks and Questions
314
Christopher Francisco, Oklahoma State University, Stillwater, Oklahoma, USA; Lee C. Klingler, Florida Atlantic University, Boca Raton, Florida, USA; Sean M. Sather-Wagstaff, North Dakota State University, Fargo, North Dakota, USA; Janet Vassilev, University of New Mexico, Albuquerque, New Mexico, USA.
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