© 2018

Completion, Čech and Local Homology and Cohomology

Interactions Between Them

  • 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 Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Modules

    1. Front Matter
      Pages 1-1
    2. Peter Schenzel, Anne-Marie Simon
      Pages 3-21
    3. Peter Schenzel, Anne-Marie Simon
      Pages 23-65
    4. Peter Schenzel, Anne-Marie Simon
      Pages 67-83
  3. Complexes

    1. Front Matter
      Pages 85-85
    2. Peter Schenzel, Anne-Marie Simon
      Pages 87-116
    3. Peter Schenzel, Anne-Marie Simon
      Pages 117-133
    4. Peter Schenzel, Anne-Marie Simon
      Pages 135-163
    5. Peter Schenzel, Anne-Marie Simon
      Pages 165-199
    6. Peter Schenzel, Anne-Marie Simon
      Pages 201-213
    7. Peter Schenzel, Anne-Marie Simon
      Pages 215-247
  4. Duality

    1. Front Matter
      Pages 249-249
    2. Peter Schenzel, Anne-Marie Simon
      Pages 251-270
    3. Peter Schenzel, Anne-Marie Simon
      Pages 271-294
    4. Peter Schenzel, Anne-Marie Simon
      Pages 295-316
  5. Peter Schenzel, Anne-Marie Simon
    Pages C1-C1
  6. Back Matter
    Pages 317-346

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.


Completion Functor Derived Functor of Completion Cech (co-)Homology Duality Local Cohomology Local Homology

Authors and affiliations

  1. 1.Institut für InformatikMartin-Luther-Universität Halle-WittenbergHalleGermany
  2. 2.Service de Geometrie DifferentielleUniversité Libre de BruxellesBrusselsBelgium

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
  • Series Abbreviated Title Springer Monographs in Mathematics
  • DOI
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-96516-1
  • Softcover ISBN 978-3-030-07207-0
  • eBook ISBN 978-3-319-96517-8
  • Series ISSN 1439-7382
  • Series E-ISSN 2196-9922
  • Edition Number 1
  • Number of Pages XIX, 346
  • Number of Illustrations 21 b/w illustrations, 0 illustrations in colour
  • Topics Commutative Rings and Algebras
  • Buy this book on publisher's site