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 Cech 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.
General
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