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

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