Modules over Non-Noetherian Domains

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1 Mathematical Surveys and Monographs Volume 84 Modules over Non-Noetherian Domains Laszlo Fuchs Luigi Sake American Mathematical Society

2 Table of Contents Preface List of Symbols xi xv Chapter I. Commutative Domains and Their Modules 1. Generalities on domains 1 2. Fractional ideals 9 3. Integral dependence Module categories Lemmas on Horn and Ext Lemmas on tensor and torsion products Divisibility and relative divisibility Pure submodules The exchange property Semilocal endomorphism rings 52 Notes 55 Chapter II. Valuation Domains 1. Fundamental properties of valuation domains Totally ordered abelian groups Valuations Ideals of valuation domains The class semigroup Maximal and almost maximal valuation domains Henselian valuation rings Strongly discrete valuation domains 86 Notes 89 Chapter III. Priifer Domains 1. Fundamental properties and characterizations Priifer domains of finite character The class semigroup Lattice-ordered abelian groups Bezout domains Elementary divisor domains Strongly discrete Priifer domains 119 Notes 121 vn

3 viii TABLE OF CONTENTS Chapter IV. More Non-Noetherian Domains 1. Krull domains Coherent domains /&-Local domains Matlis domains Reflexive domains 142 Notes 147 Chapter V. Finitely Generated Modules 1. Cyclic modules Finitely generated modules Finitely presented modules Finite presentations Finitely generated modules over valuation domains Indecomposable finitely generated modules Finitely generated modules with local endomorphism rings Decompositions of finitely generated modules Finitely generated modules without the Krull-Schmidt property Domains whose finitely generated modules are direct sums of cyclics 189 Notes 191 Chapter VI. Projectivity and Projective Dimension 1. Projective modules Projective dimension Projective dimension over valuation domains Global projective dimension of Priifer domains Tight submodules Modules of projective dimension one Equivalent presentations Stacked bases over /i-local Priifer domains Flat modules Weak dimension Quasi-projective modules Pure- and fld-projectivity 240 Notes 245 Chapter VII. Divisible Modules 1. Divisible modules /^Divisible modules, Matlis domains Divisible modules over valuation domains Categories of divisible modules Indecomposable divisible modules Superdecomposable divisible modules 269 Notes 272 Chapter VIII. Topology and Filtration 1. The fl-topology Complete torsion-free modules. The Matlis category equivalence.. 278

4 TABLE OF CONTENTS IX 3. Completions of ideals R-Completions over Matlis domains Cokernels of i?-completions Weakly cotorsion modules Linear compactness Filtration and ultracompleteness 300 Notes 302 Chapter IX. Injective Modules 1. Injectivity Indecomposable injectives Absolute purity Injectives over valuation and Priifer domains S-Injectives Injectives over Krull domains Injective dimension Quasi-injective modules 333 Notes 335 Chapter X. Uniserial Modules 1. Generalities on uniserial modules Endomorphism rings of uniserial modules Uniserial modules over valuation domains Existence of non-standard uniserial modules More on the existence of non-standard uniserial modules Kaplansky's problem The threshold submodules Life-span of uniserial modules Uniserial modules of the same level The monoid Unis R 373 Notes 379 Chapter XI. Heights, Invariants and Basic Submodules 1. Heights Equiheight, nice and balanced submodules Indicators Invariants Basic submodules Modules with trivial invariants 396 Notes 402 Chapter XII. Polyserial Modules 1. Polyserial and weakly polyserial modules Direct sums of uniserial modules Monoserial modules Episerial modules and their submodules Direct decompositions of weakly polyserial modules 419 Notes 421

5 X TABLE OF CONTENTS Chapter XIII. RD- and Pure-Injectivity 1. izd-injective modules Pure-injective modules Algebraic compactness Pure-injective modules over Priifer domains Pure-injective modules over valuation domains Pure-injectivity over coherent domains Hi-Compact modules Cotorsion modules 457 Notes 461 Chapter XIV. Torsion Modules 1. Decompositions of torsion modules Torsion modules of projective dimension one Simple presentation Balanced submodules Simply presented torsion modules Torsion-complete modules 483 Notes 487 Chapter XV. Torsion-Free Modules of Finite Rank 1. Preliminaries Direct sums of ideals Torsion-free modules over valuation domains D-Domains Indecomposable modules Indecomposability over valuation domains Direct decompositions of torsion-free modules Warfield domains Intrinsic characterization of Warfield domains 526 Notes 529 Chapter XVI. Infinite Rank Torsion-Free Modules 1. Chains of projective modules Almost projective modules Balancedness Balanced-projective dimension Separable modules Slender modules Large indecomposable modules Baer modules Butler modules over valuation domains Whitehead modules 576 Nctes 582 Appendix on Set Theory 585 Bibliography 591 Author Index 603 Subject Index 607

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