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.
