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Progress in Commutative Algebra 2 Closures, Finiteness and Factorization

:	Christopher Francisco, Lee C. Klingler, Sean M. Sather-Wagstaff, Janet C. Vassilev
:	Progress in Commutative Algebra 2 Closures, Finiteness and Factorization
:	Walter de Gruyter GmbH& Co.KG
:	9783110278606
:	De Gruyter Proceedings in Mathematics
:	1
:	CHF 0.50
:

:	Allgemeines, Lexika
:	English

:	325
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

< >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 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,Nort Dakota State University, Fargo, North Dakota, USA;Janet Vassilev, University of New Mexico, Albuquerque, New Mexico, USA.

	Preface	6
	A Guide to Closure Operations in Commutative Algebra	12
	1 Introduction	12
	2 What Is a Closure Operation?	13
	2.1 The Basics	13
	2.2 Not-quite-closure Operations	17
	3 Constructing Closure Operations	18
	3.1 Standard Constructions	18
	3.2 Common Closures as Iterations of Standard Constructions	20
	4 Properties of Closures	21
	4.1 Star-, Semi-prime, and Prime Operations	21
	4.2 Closures Defined by Properties of (Generic) Forcing Algebras	27
	4.3 Persistence	28
	4.4 Axioms Related to the Homological Conjectures	29
	4.5 Tight Closure and Its Imitators	31
	4.6 (Homogeneous) Equational Closures and Localization	32
	5 Reductions, Special Parts of Closures, Spreads, and Cores	33
	5.1 Nakayama Closures and Reductions	33
	5.2 Special Parts of Closures	34
	6 Classes of Rings Defined by Closed Ideals	36
	6.1 When Is the Zero Ideal Closed?	37
	6.2 When Are 0 and Principal Ideals Generated by Non-zerodivisors Closed?	37
	6.3 When Are Parameter Ideals Closed (Where R Is Local)?	38
	6.4 When Is Every Ideal Closed?	39
	7 Closure Operations on (Sub)modules	40
	7.1 Torsion Theories	42
	A Survey of Test Ideals	50
	1 Introduction	50
	2 Characteristic p Preliminaries	52
	2.1 The Frobenius Endomorphism	52
	2.2 F-purity	53
	3 The Test Ideal	55
	3.1 Test Ideals of Map-pairs	55
	3.2 Test Ideals of Rings	58
	3.3 Test Ideals in Gorenstein Local Rings	59
	4 Connections with Algebraic Geometry	61
	4.1 Characteristic 0 Preliminaries	61
	4.2 Reduction to Characteristic p	61
	63	61
	4.3 Multiplier Ideals of Pairs	65
	4.4 Multiplier Ideals vs. Test Ideals of Divisor Pairs	67
	5 Tight Closure and Applications of Test Ideals	68
	5.1 The Briançon-Skoda Theorem	72
	5.2 Tight Closure for Modules and Test Elements	72
	6 Test Ideals for Pairs (R, at) and Applications	74
	6.1 Initial Definitions of at -test Ideals	74
	6.2 at -tight Closure	76
	6.3 Applications	77
	7 Generalizations of Pairs: Algebras of Maps	79
	8 Other Measures of Singularities in Characteristic p	82
	8.1 F-rationality	82
	8.2 F-injectivity	83
	8.3 F-signature and F-splitting Ratio	84
	8.4 Hilbert-Kunz(-Monsky) Multiplicity	86
	8.5 F-ideals, F-stable Submodules, and F-pure Centers	89
	A Canonical Modules and Duality	91
	A.1 Canonical Modules, Cohen-