Completion, Čech and Local Homology and Cohomology: Interactions Between Them (Springer Monographs in Mathematics)

Description

The aim of the present monograph is a thorough study of the adic-completion, its left derived functors and their relations to the local cohomology functors, as well as several completeness criteria, related questions and various dualities formulas. A basic construction is the Čech complex with respect to a system of elements and its free resolution. The study of its homology and cohomology will play a crucial role in order to understand left derived functors of completion and right derived functors of torsion. This is useful for the extension and refinement of results known for modules to unbounded complexes in the more general setting of not necessarily Noetherian rings. The book is divided into three parts. The first one is devoted to modules, where the adic-completion functor is presented in full details with generalizations of some previous completeness criteria for modules. Part II is devoted to the study of complexes. Part III is mainly concerned with duality, starting with those between completion and torsion and leading to new aspects of various dualizing complexes. The Appendix covers various additional and complementary aspects of the previous investigations and also provides examples showing the necessity of the assumptions. The book is directed to readers interested in recent progress in Homological and Commutative Algebra. Necessary prerequisites include some knowledge of Commutative Algebra and a familiarity with basic Homological Algebra. The book could be used as base for seminars with graduate students interested in Homological Algebra with a view towards recent research. Introduction Contents Part I Modules 1 Preliminaries and Auxiliary Results 1.1 Complexes 1.2 Inverse Limits 1.3 Direct Limits 1.4 Ext-Tor Duality and General Matlis Duality 1.5 Cones and Fibers 2 Adic Topology and Completion 2.1 Topological Preliminaries 2.2 The Case of Finitely Generated Ideals 2.3 Noetherian Rings and Matlis Duality 2.4 Completions of Flat Modules over a Noetherian Ring 2.5 The Left-Derived Functors of Completion 2.6 Relative Flatness and Completion in the General Case 2.7 Relatively Injective and Torsion Modules 2.8 Some Examples 3 Ext-Tor Vanishing and Completeness Criteria 3.1 Completeness and Pseudo-completeness Criteria 3.2 Modules of Infinite Co-depth 3.3 When is a Finitely Generated Module Complete? 3.4 Ext-Depth and Tor-Codepth with Local (Co-)Homology Part II Complexes 4 Homological Preliminaries 4.1 Double Complexes and Truncations 4.2 The Microscope 4.3 The Telescope 4.4 Special Resolutions and Their Uses 4.5 Minimal Injective Resolutions for Unbounded Complexes 4.6 Ext and Tor with Inverse and Direct Limits 5 Koszul Complexes, Depth and Codepth 5.1 Ext-Depth and Tor-Codepth 5.2 Basics About Koszul Complexes 5.3 The Ext-Depth Tor-Codepth Sensitivity of the Koszul Complex 5.4 Koszul Homology of Modules 6 Čech Complexes, Čech Homology and Cohomology 6.1 The Čech Complex 6.2 A Free Resolution of the Čech Complex 6.3 Čech Homology and Cohomology 6.4 Some Classes Related to the Čech Complex 6.5 Composites 6.6 Depth, Codepth and Čech Complexes 7 Local Cohomology and Local Homology 7.1 The General Case 7.2 First Vanishing Results with Applications to the Class mathcalTmathfraka 7.3 Weakly Pro-regular Sequences 7.4 Local and Čech Cohomology with Telescope 7.5 Local and Čech Homology with Microscope 7.6 Depth and Codepth with Local (Co-)Homology 8 The Formal Power Series Koszul Complex 8.1 Čech Homology and Koszul Complexes 8.2 Applications to Weakly Pro-regular Sequences 8.3 Applications to Koszul Homology 8.4 The Case of a Single Element 9 Complements and Applications 9.1 Composites 9.2 Adjointness and Duality 9.3 Some Endomorphisms 9.4 Mayer–Vietoris Sequences for Local and Čech (Co-)Homology 9.5 On the use of the classes mathcalCmathfraka and mathcalBmathfraka 9.6 Homologically Complete and Cohomologically Torsion Complexes 9.7 Homological Completeness and Cosupport 9.8 Change of Rings Part III Duality 10 Čech and Local Duality 10.1 Bounded Injective Complexes with Finitely Generated Cohomology 10.2 Čech Cohomology and Duality 10.3 Canonical Modules 10.4 Local Duality over Cohen–Macaulay Local Rings 10.5 On Gorenstein Local Rings and Duality 10.6 Local Cohomology over Finite Local Gorenstein Algebras 11 Dualizing Complexes 11.1 Evaluation Morphisms of Complexes 11.2 Definition of Dualizing Complexes for Noetherian Rings 11.3 First Change of Rings 11.4 Characterization and Uniqueness 11.5 Flat Change of Rings, Dualizing Complexes and Completion 11.6 Further Properties of Dualizing Complexes 12 Local Duality with Dualizing Complexes and Other Dualities 12.1 General Dualities and Hartshorne's Affine Duality 12.2 Local Duality with Dualizing Complexes 12.3 Extensions of Dualities, Complexes with Mini-Max Homology 12.4 Local Duality for Arbitrary Complexes 12.5 Greenlees' Warwick Duality Appendix A.1 Grothendieck's Non-vanishing Theorem A.2 Pro-regular or Weakly Pro-regular Sequences, Some Examples A.3 The Right-Derived Functors of the Completion Functor A.4 Specific Rings A.5 Some Examples of Modules References Subject Index Notation