This research monograph develops the theory of relative nonhomogeneous
Koszul duality. Koszul duality is a fundamental phenomenon in
homological algebra and related areas of mathematics, such as algebraic
topology, algebraic geometry, and representation theory. Koszul duality
is a popular subject of contemporary research.
This book, written by one of the world's leading experts in the area,
includes the homogeneous and nonhomogeneous quadratic duality theory
over a nonsemisimple, noncommutative base ring, the
Poincare-Birkhoff-Witt theorem generalized to this context, and
triangulated equivalences between suitable exotic derived categories of
modules, curved DG comodules, and curved DG contramodules. The thematic
example, meaning the classical duality between the ring of differential
operators and the de Rham DG algebra of differential forms, involves
some of the most important objects of study in the contemporary
algebraic and differential geometry. For the first time in the history
of Koszul duality the derived D-\Omega duality is included into a
general framework. Examples highly relevant for algebraic and
differential geometry are discussed in detail.