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Completion, Čech and Local Homology and Cohomology

Interactions Between Them

  • Peter Schenzel
  • Anne-Marie Simon

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-xvii
  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. Back Matter
    Pages 317-346

About this book

Introduction

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.


Keywords

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

Authors and affiliations

  • Peter Schenzel
    • 1
  • Anne-Marie Simon
    • 2
  1. 1.Institut für InformatikMartin-Luther-Universität Halle-WittenbergHalleGermany
  2. 2.Service de Geometrie DifferentielleUniversité Libre de BruxellesBrusselsBelgium

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-96517-8
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-96516-1
  • Online ISBN 978-3-319-96517-8
  • Series Print ISSN 1439-7382
  • Series Online ISSN 2196-9922
  • Buy this book on publisher's site