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Springer Monographs in Mathematics

Completion, Čech and Local Homology and Cohomology

Interactions Between Them

Authors: Schenzel, Peter, Simon, Anne-Marie

  • 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 Čech 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 Čech (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
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eBook $109.00
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  • ISBN 978-3-319-96517-8
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Hardcover $139.99
price for USA in USD
  • ISBN 978-3-319-96516-1
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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 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.


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.

Table of contents (12 chapters)

  • Preliminaries and Auxiliary Results

    Schenzel, Peter (et al.)

    Pages 3-21

  • Adic Topology and Completion

    Schenzel, Peter (et al.)

    Pages 23-65

  • Ext-Tor Vanishing and Completeness Criteria

    Schenzel, Peter (et al.)

    Pages 67-83

  • Homological Preliminaries

    Schenzel, Peter (et al.)

    Pages 87-116

  • Koszul Complexes, Depth and Codepth

    Schenzel, Peter (et al.)

    Pages 117-133

Buy this book

eBook $109.00
price for USA in USD (gross)
  • ISBN 978-3-319-96517-8
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.99
price for USA in USD
  • ISBN 978-3-319-96516-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Completion, Čech and Local Homology and Cohomology
Book Subtitle
Interactions Between Them
Authors
Series Title
Springer Monographs in Mathematics
Copyright
2018
Publisher
Springer International Publishing
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-319-96517-8
DOI
10.1007/978-3-319-96517-8
Hardcover ISBN
978-3-319-96516-1
Series ISSN
1439-7382
Edition Number
1
Number of Pages
XVII, 346
Topics