Completion, Čech and Local Homology and Cohomology
Interactions Between Them
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Beschreibung
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.
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 withduality, 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.
Provides a comprehensive study of the adic completion and its left-derived (the local homology functors) for modules and complexes over commutative rings Studies the relation between Cech and local homology, respectively cohomology, mainly in the setting of an ideal generated by a weakly pro-regular sequence and unbounded complexes Provides the duality between Cech (respectively local) homology and cohomology and further dualities in various new aspects including dualizing complexes Contains various criteria about completions and an analysis of the role of the assumptions illustrated by several examples
Autor*in
Peter Schenzel
Themen in »Completion, Čech and Local Homology and Cohomology«
Completion Functor Derived Functor of Completion Cech (co-)Homology Duality Local Cohomology Local Homology