Overview
- 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
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (13 chapters)
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Front Matter
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Back Matter
Keywords
About this book
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.
Authors and Affiliations
About the authors
Peter Schenzel retired from Martin Luther University Halle-Wittenberg in 2015. There, he gained his PhD in 1975, under the supervision of Wolfgang Vogel and his Habilitation in 1979. The latter was published in the Springer Lecture Notes in Mathematics. He published further more than 90 research papers. He was invited and speaker of several international conferences. For three years he was a member of the Max Planck Institute for Mathematics.
Anne Marie Simon is professor at the Free University of Brussels. She received her PhD, under the guidance of Jacques Tits at the same university in 1969. She was a visitor at Brown University for one year. She published more than 20 research papers. At least two of them dealt with the study of completion and local homology and have been quoted in several research papers.
Bibliographic Information
Book Title: Completion, Čech and Local Homology and Cohomology
Book Subtitle: Interactions Between Them
Authors: Peter Schenzel, Anne-Marie Simon
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-319-96517-8
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2018
Hardcover ISBN: 978-3-319-96516-1Published: 25 September 2018
Softcover ISBN: 978-3-030-07207-0Published: 28 December 2018
eBook ISBN: 978-3-319-96517-8Published: 15 September 2018
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XIX, 346
Number of Illustrations: 21 b/w illustrations
Topics: Commutative Rings and Algebras