Approximations and
Endomorphism Algebras
of Modules
by
Riidiger Gobel and Jan Trlifaj
W
DE
G
Walter de Gruyter • Berlin • New York
© 2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network.
Contents
Introduction xi List of Symbols xxi 1 Some useful classes of modules 1 1.1 S-completions 1 Support of elements in B — a first step 9 Uncountable S in completions 11
Modules of cardinality < 2N° 13
1.2 Pure-injective modules 20 Direct limits, finitely presented modules and pure submodules . 21 Characterizations of pure-injective modules 34 1.3 Locally projective modules 43 1.4 Factors of products and slender modules 58 1.5 Slender modules over Dedekind domains 84 2 Approximations of modules 94 2.1 Preenvelopes and precovers 94 2.2 Cotorsion pairs and Tor-pairs 99 2.3 Minimal approximations 106 3 Complete cotorsion pairs 112 3.1 Ext and direct limits 112 3.2 The variety of complete cotorsion pairs 117 3.3 Ext and inverse limits 125 4 Deconstruction of cotorsion pairs 134 4.1 Approximations by modules of finite homological dimensions . . 134 4.2 Hill Lemma and Kaplansky Theorem for cotorsion pairs 142 4.3 Closure properties providing for completeness 149 The tilting case 150 The cotilting case 157 4.4 Matlis cotorsion and strongly flat modules 163
viii Contents
Strongly flat modules over valuation domains 173 4.5 The closure of a cotorsion pair 178 Direct limits of modules of projective dimension < 1 184 5 Tilting approximations 188 5.1 Tilting modules 188 5.2 Classes of finite type 201 Deconstruction to countable type 202 Definability and the Mittag-Leffler condition 208 Finite type and resolving subcategories 213 5.3 Injectivity properties of tilting modules 219 6 1-tilting modules and their applications 224 6.1 Tilting torsion classes 224 6.2 The structure of tilting modules and classes over particular rings . 228 1-tilting classes over artin algebras 228 Tilting modules and classes over Priifer domains 231 The case of valuation and Dedekind domains 240 6.3 Matlis localizations 242 7 Tilting approximations and the finitistic dimension conjectures . . . 255 7.1 Finitistic dimension conjectures and the tilting module Tf . . . . 255 7.2 A formula for the little finitistic dimension of right artinian rings 263
7.3 Artinian rings with V<w contravariantly finite 267
8 Cotilting modules 274 8.1 Cotilting classes and the classes of cofinite type 274 8.2 1-cotilting modules and cotilting torsion-free classes 281 Cotilting modules and classes over Dedekind domains 284 Ext-rigid systems 289 9 The Black Box and its relatives 293 9.1 Survey of prediction principles using ZFC and more 293 Three equivalent versions of the Diamond Principle 294 The Weak Diamond Principle 301 Applications: the existence of almost free i?-modules with a prescribed endomorphism ring 302 9.2 The Black Boxes 312 The Strong Black Box 313 The more General Black Box 340 9.3 The Shelah Elevator 353
Contents ix The combinatorial part of the elevator 353 10 Independence results for cotorsion pairs 359 10.1 Completeness of cotorsion pairs under the Diamond Principle . . 360 10.2 Uniformization and cotorsion pairs not generated by a set . . . . 364 11 The lattice of cotorsion pairs 374 11.1 Ultra-cotorsion-free modules and the Strong Black Box 374 11.2 Rational cotorsion pairs 383 11.3 Embedding posets into the lattice of cotorsion pairs 392 12 Realizing algebras 402
12.1 Realizing algebras of size < 2H° 402
12.2 ^i-free modules of cardinality Ki 410 The construction of modules 412 The Pigeon-hole Lemma 417 Comparing branching points 420 Proof of the theorem 424 12.3 Realizing all cotorsion-free algebras 426 The main realization theorem and the Strong Black Box . . . . 428 The main realization theorem and the General Black Box . . . 436 Cotorsion-free modules 447 Almost cotorsion-free, separable, slender and Hi-free modules 448 Other classes of torsion-free modules 457 A discussion of realization theorems for torsion and mixed mod-ules 457 12.4 Algebras of row-and-column-finite matrices 459 13 E(R)-algebras 462 13.1 Classical £(i?)-algebras 462 Excursion: localizations and cellular complexes 462 Classical £"(i?)-algebras, the continuation 467
13.2 Constructing torsion-free, reduced .E(i?)-algebras of rank < 2K° 470
13.3 i?(i?)-algebras and uniquely transitive modules 472 UT-modules over principal ideal domains 474 Pure-invertible algebras 475 The inductive step for the construction of UT-modules 477 The construction of UT-modules 479 13.4 J5(J?)-algebras and the Strong Black Box 481
13.5 Discussing Hi-free £1(ii)-algebras of cardinality Hi 487
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The construction of mixed £'(i?)-modules 492 13.7 E(i?)-modules with cotorsion 496 The construction of £'(il)-algebras with cotorsion 503 13.8 Generalized £(.R)-algebras 506 13.9 Model theory for generalized E(R)-algebras 507 Discussion 507 The notion of terms 508 Model theory of skeletons via A-calculus 509 Model theory of bodies 516 From skeleton to the bodies 519 13.10 Constructing proper generalized E(i2)-algebras 521 Technical tools for the construction 521 Preparing the Step Lemmas for the construction 524 The three Step Lemmas for constructing generalized E(R)-algebras 526 The final stage: construction of generalized E(R)-algebras . . 530 14 Modules with distinguished submodules 535 14.1 The five-submodule theorem, an easy application of the elevator 535 14.2 The four-submodule theorem, a harder case 541 14.3 A discussion of representations of posets 555 14.4 Absolutely indecomposable modules 564 Rigid families of trees and the first w-Erdos cardinal 565 The main construction 567 Extension to fully rigid systems 570 Passing to absolutely fully rigid systems of i?5-modules . . . . 572 14.5 Passing to i?-modules 575 14.6 A topological realization from Theorem 14.2.12 581 15 Some useful classes of algebras 587 15.1 Leavitt type rings: the discrete case 587 15.2 Automorphism groups of torsion-free abelian groups 595 15.3 Algebras with a Hausdorff topology 597 15.4 Realizing particular algebras as endomorphism algebras 603 References 611 Index 634
