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E-raamat: Slenderness: Volume 1, Abelian Categories

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Slenderness is a concept relevant to the fields of algebra, set theory, and topology. This first book on the subject is systematically presented and largely self-contained, making it ideal for researchers and graduate students. It provides over 350 exercises as well as many open problems to inspire further research.

Slenderness is a concept relevant to the fields of algebra, set theory, and topology. This first book on the subject is systematically presented and largely self-contained, making it ideal for researchers and graduate students. The appendix gives an introduction to the necessary set theory, in particular to the (non-)measurable cardinals, to help the reader make smooth progress through the text. A detailed index shows the numerous connections among the topics treated. Every chapter has a historical section to show the original sources for results and the subsequent development of ideas, and is rounded off with numerous exercises. More than 100 open problems and projects are presented, ready to inspire the keen graduate student or researcher. Many of the results are appearing in print for the first time, and many of the older results are presented in a new light.

Arvustused

'At the end of every chapter some exercises, problems and notes are gathered which could be very useful for understanding the theory and for future research on this subject.' George Ciprian Modoi, MathSciNet

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A leading expert presents a unified concept of slenderness in Abelian categories, with numerous open problems and exercises.
Preface ix
Introduction 1(40)
0.1 Categories and Functors
1(9)
0.2 Abelian Categories and some Categorical Constructions
10(6)
0.3 Products and Coproducts
16(7)
0.4 Limits and Colimits
23(12)
0.5 Exercises, Problems, and Notes
35(6)
1 Topological Rings and Modules and their Completions
41(50)
1.1 Topology and Algebra in Agreement
42(10)
1.2 Completions and MetrizabiUty
52(6)
1.3 The Product Topologies
58(3)
1.4 Existence of Non-Discrete Hausdorff Topologies
61(5)
1.5 Completions, Constructions, and Stable Properties*
66(6)
1.6 Rings of Continuous Functions and Manifolds*
72(3)
1.7 Exercises, Problems, and Notes
75(16)
2 Inverse Limits
91(28)
2.1 The Mittag-Leffler Condition
91(7)
2.2 Surjective Inverse Systems
98(6)
2.3 Sheaves and the Flabby Conditions
104(7)
2.4 Exercises, Problems, and Notes
111(8)
3 The Idea of Slenderness
119(31)
3.1 A Path to Slenderness
119(7)
3.2 Equivalent Definitions of Slenderness
126(5)
3.3 L-Slender Objects
131(2)
3.4 Some Properties and Examples
133(3)
3.5 Fundamental Characterizations
136(4)
3.6 Some Constructions and More Specific Examples
140(2)
3.7 Exercises, Problems, and Notes
142(8)
4 Objects of Type Π / Π
150(39)
4.1 Purity, Algebraic Compactness, and Cotorsion Modules
151(10)
4.2 Filter Quotients and Products
161(11)
4.3 Slenderness of Modules over Domains
172(5)
4.4 Exercises, Problems, and Notes
177(12)
5 Concrete Examples. Slender Rings
189(25)
5.1 Examples of Slender and Non-Slender Rings and Modules
189(4)
5.2 Relationships Between Slenderness in Different Categories
193(5)
5.3 Slenderness for Noetherian and Dedekind Rings
198(11)
5.4 Exercises, Problems, and Notes
209(5)
6 More Examples of Slender Objects
214(23)
6.1 Slender Boolean Rings
214(5)
6.2 (Pseudo-)Grading and Semigroup Algebras
219(6)
6.3 Slenderness of Rings of Functions
225(8)
6.4 Exercises, Problems, and Notes
233(4)
Appendix Ordered Sets and Measurable Cardinals
237(14)
A.1 Posets, Ordinals, and Cardinals
237(3)
A.2 Directed Sets
240(4)
A.3 Measurable Cardinals
244(7)
References 251(23)
Notation Index 274(6)
Name Index 280(8)
Subject Index 288
Radoslav Dimitric was the first to give a complete characterization of slender objects in the general setting of Abelian categories. His research has mostly been concerned with algebra and how it relates to topology and set theory, but his research also includes the history of mathematics, mathematics education and financial engineering.
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