
Approximations and Endomorphism Algebras of Modules
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This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2).
It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C , is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C . This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications.
The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S -complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C .
The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.
Reviews / Votes
"I strongly recommend the monograph to anyone who is interested in the modern theory of modules."
(pruz), EMS Newsletter 9/2007
"All in all, I highly recommend the book to everyone interested in cotorsion pairs, approximation theory, realization of algebras or application of set theory to algebra."
Gábor Braun, Zentralblatt MATH 1121/2007
"The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory."
L'Enseignement Mathematique 3-4/2006
"As was true for the first edition this book provides a good introduction into the subject for self-study at a graduate level and it also provides a very comprehensive survey on the subjects presenting the state-of-the-art. Both volumes have been written in a very clear and self-explaining way and the contents shows the expertise of the two authors in the field. [.] The book by Göbel and Trlifaj is certainly one of the most comprehensive elaborations on module theory and its interaction with set-theory and more generally logic. It shows once more that the two authors are strong experts in their fields. New and recent topics are covered in the same brilliant way of writing as before and bring the reader up to date. [.] Approximations and Endomorphism Algebras by Göbel and Trlifaj is a marvelous work that can be used either for self-study introducing the reader to a very interesting field of research or as the main reference book covering a wide scope of results and techniques on topics in module theory and set-theoretic applications to it. I can only recommend it to anyone interested in these fields." Zentralblatt für Mathematik
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Content
- Intro
- CONTENTS
- Volume 1
- Introduction
- List of Symbols
- PART I: SOME USEFUL CLASSES OF MODULES
- 1 S-completions
- 1.1 Support of elements in B - a first step
- 1.2 Uncountable S incompletions
- 1.3 Modules of cardinality ? 2 ?0
- 2 Pure-injective modules
- 2.1 Direct limits, finitely presented modules and pure submodules
- 2.2 Characterizations of pure-injective modules
- 3 Mittag-Leffler modules
- 3.1 Characterizations of Mittag-Leffler modules
- 3.2 Flat Mittag-Leffler modules
- 3.3 Unions of countable pure chains of pure-projective modules
- 3.4 Locally projective modules
- 3.5 Open problems
- 4 Slender modules
- 4.1 Factors of products and slender modules
- 4.2 Slender modules over Dedekind domains
- 4.3 Open problems
- PART II: APPROXIMATIONS AND COTORSION PAIRS
- 5 Approximations of modules
- 5.1 Preenvelopes and precovers
- 5.2 Cotorsion pairs and Tor-pairs
- 5.3 Minimal approximations
- 5.4 Open problems
- 6 Complete cotorsion pairs
- 6.1 Ext and direct limits
- 6.2 The abundance of complete cotorsion pairs
- 6.3 Ext and inverse limits
- 6.4 Open problems
- 7 Hill lemma and its applications
- 7.1 The general version of the Hill Lemma
- 7.2 Kaplansky theorem for cotorsion pairs
- 7.3 C-socle sequences and Filt.(C)-precovers
- 7.4 Singular compactness for C-filtered modules
- 7.5 Ascending and descending properties of modules
- 7.6 The rank version of the Hill Lemma
- 7.7 Matlis cotorsion and strongly flat modules
- 7.8 Open problems
- 8 Deconstruction of the roots of Ext
- 8.1 Approximations by modules of finite homological dimensions
- 8.2 Closure properties providing for deconstruction
- 8.3 The closure of a cotorsion pair
- 9 Modules of projective dimension one
- 9.1 Structure of P 1 and WI for semiprime Goldie rings
- 9.2 The class lim P 1
- 9.3 Open problems
- 10 Kaplansky classes and abstract elementary classes
- 10.1 Kaplansky classes and deconstructibility
- 10.2 Flat Mittag-Leffler modules revisited
- 10.3 Abstract elementary classes of the roots of Ext
- 10.4 Open problems
- 11 Independence results for cotorsion pairs
- 11.1 Completeness of cotorsion pairs under the Diamond Principle
- 11.2 Uniformisation and cotorsion pairs not generated by a set
- 11.3 Open problems
- 12 The lattice of cotorsion pairs
- 12.1 Ultra-cotorsion-free modules and the Strong Black Box
- 12.2 Rational cotorsion pairs
- 12.3 Embedding posets into the lattice of cotorsion pairs
- PART III: TILTING AND COTILTING APPROXIMATIONS
- 13 Tilting approximations
- 13.1 Tilting modules
- 13.2 Classes of finite type
- 13.3 Localisation of tilting modules
- 13.4 Product-completeness of tilting modules
- 13.5 Open problems
- 14 1-tilting modules and their applications
- 14.1 Tilting torsion classes
- 14.2 The structure of 1-tilting modules and classes over particular rings
- 14.3 Baer modules
- 14.4 Matlis localisations
- 14.5 Open problems
- 15 Cotilting classes
- 15.1 Cotilting classes and the classes of cofinite type
- 15.2 1-cotilting modules and cotilting torsion-free classes
- 15.3 Cotilting over Prüfer domains
- 15.4 Ext-rigid systems
- 15.5 Open problems
- 16 Tilting and cotilting classes over commutative noetherian rings
- 16.1 Cotilting classes and characteristic sequences
- 16.2 Tilting classes over commutative noetherian rings
- 16.3 Tilting and cotilting modules over 1-Gorenstein rings
- 16.4 Tor-pairs over hereditary rings
- 16.5 Open problems
- 17 Tilting approximations and the finitistic dimension conjectures
- 17.1 Finitistic dimension conjectures and the tilting module T f
- 17.2 A formula for the little finitistic dimension of right artinian rings
- 17.3 Artinian rings with P <? contravariantly finite
- 17.4 Open problems
- Bibliography
- Index
- Volume II
- PART IV: PREDICTION PRINCIPLES
- 18 Survey of prediction principles using ZFC and more
- 18.1 The closed unbounded filter
- 18.2 The Diamond Principle
- 18.3 TheWeak Diamond Principle
- 18.4 Surprisingly short
- 18.5 A first application of {} ? E
- 19 Prediction principles in ZFC: the Black Boxes and others
- 19.1 The Easy Black Box
- 19.2 The Strong Black Box
- 19.3 The Strong Black Box for endomorphism rings
- 19.4 The Strong Black Box for ultra-cotorsion-free modules
- 19.5 The General Black Box
- 19.6 The General Box for E(R)-algebras
- 19.7 The Shelah Elevator
- 19.8 Shelah's absolutely rigid family of trees
- 19.9 Absolutely rigid trees
- PART V. ENDOMORPHISM ALGEBRAS AND AUTOMORPHISM GROUPS
- 20 Realising algebras - by algebraically independent elements and by prediction principles
- 20.1 Realising algebras of size = 2?0
- 20.2 Realising all cotorsion-free algebras
- 20.3 Algebras of row-and-column-finite matrices
- 21 Automorphism groups of torsion-free abelian groups
- 21.1 The Hirsch-Zassenhaus-Theorem
- 21.2 An extension to the torsion case
- 21.3 Involutions of the group of units
- 21.4 Open problems
- 22 Modules with distinguished submodules
- 22.1 The five-submodule theorem, an easy application of the Elevator
- 22.2 The four-submodule theorem, a harder case
- 22.3 Absolutely rigid abelian groups and E-rings of size ?.?/ do not exist
- 22.4 Absolutely indecomposable R 4 -modules
- 22.5 A discussion of representations of posets
- 22.6 Open problems
- 23 R-modules and fields from modules with distinguished submodules
- 23.1 Modules: the unrestricted case and the absolute case
- 23.2 A topological realisation from Theorem 22.21
- 23.3 Appendix
- 23.4 Prescribing endomorphism monoids and automorphism groups of fields
- 23.5 Absolutely rigid fields
- 23.6 Open problems
- 24 Endomorphism algebras of @ n -free modules
- 24.1 ? 1 -free modules of cardinality ? 1
- 24.2 The next step: ? n -free modules for any natural number n ? N
- 24.3 The basics for the new Combinatorial Black Box
- 24.4 ? n -free A-modules
- 24.5 The triple-homomorphism ? and freeness
- 24.6 Chains of triples
- 24.7 The Step Lemma
- 24.8 Application of the Strong Black Box
- 24.9 Fully rigid systems of Nk - free R-modules with a prescribed R-algebra
- 24.10 Open problems
- PART VI: MODULES AND RINGS RELATED TO ALGEBRAIC TOPOLOGY
- 25 Localisations and cellular covers, the general theory for R-modules
- 25.1 A sketch of the categorical settings
- 25.2 Localisations not represented by morphisms
- 25.3 Constructing localisations of R-modules
- 25.4 Cellular covers of R-modules
- 25.5 Some additions and cosmetics on cellular covers and localisations
- 25.6 Excursion on localisations and cellular covers of non-commutative groups
- 25.7 Open problems
- 26 Tame and wild localisations of size = 2 ?0
- 26.1 Tame localisations
- 26.2 The wild case: classical E(R)-algebras
- 26.3 Characterizations of E(R)-algebras
- 26.4 Construction of cotorsion-free E(R)-algebras of rank = 2 ?0
- 26.5 E(R)-algebras and uniquely transitive modules
- 27 Tame cellular covers
- 27.1 Tame cellular covers of abelian groups
- 27.2 Cellular covers of divisible groups
- 27.3 Cellular covers of torsion and mixed groups
- 27.4 When the only cellular covers are the trivial ones
- 27.5 Open problems
- 28 Wild cellular covers
- 28.1 Cellular covers of subgroups of Q
- 28.2 The kernels of rank <2 ?0 for cellular covering maps
- 28.3 Characterizing kernels of cellular covers of abelian groups
- 29 Absolute E-rings
- 29.1 A very simple example shows the idea of the proof
- 29.2 Constructing strongly rigid coloured trees
- 29.3 The construction of E-rings
- 29.4 Invariant principal ideals of R
- 29.5 The existence of absolute E-modules
- 29.6 Open problems
- PART VII: CELLULAR COVERS, LOCALISATIONS AND E (R) -ALGEBRAS
- 30 Large kernels of cellular covers and large localisations
- 30.1 Large kernels of cellular covers
- 30.2 Large cotorsion-free E(R)-algebras
- 30.3 Localisations of cotorsion-free modules
- 30.4 Open problems
- 31 Mixed E(R)-modules over Dedekind domains
- 31.1 The construction of mixed E(R)-modules
- 32 E(R)-modules with cotorsion
- 32.1 The constructions of E(R) -algebras with cotorsion
- 32.2 Open problems
- PART VIII: SOME USEFUL CLASSES OF ALGEBRAS
- 33 Generalised E(R)-algebras
- 33.1 Background, strategy, the basic setting and the main result
- 33.2 The theory of skeletons
- 33.3 Bodies
- 33.4 Types
- 33.5 Small cancellation of types
- 33.6 Arbitrarily large free skeletons
- 33.7 The algebraic structure of free bodies B Y
- 33.8 The Step Lemma
- 33.9 The main construction using the diamond principle
- 33.10 Proof of the Main Theorem with {} ? E
- 33.11 The main construction in ZFC
- 33.12 Proof of the Main Theorem with the General Black Box
- 33.13 Appendix: rigid systems of generalised E(R)-algebras
- 33.14 Open problems
- 34 Some more useful classes of algebras
- 34.1 Leavitt typerings: the discrete case
- 34.2 Algebras with a Hausdorff topology
- 34.3 Realising particular algebras as endomorphism algebras
- Bibliography
- Index
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