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.

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Leonid Positselski received his Ph.D. in Mathematics from Harvard University in 1998. He did his postdocs at the Institute for Advanced Study (Princeton), Institut des Hautes Etudes Scientifiques (Bures-sur-Yvette), Max-Planck-Institut fuer Mathematik (Bonn), the University of Stockholm, and the Independent University of Moscow in 1998-2003. He taught as an Associate Professor at the Mathematics Faculty of the National Research University Higher School of Economics in Moscow in 2011-2014. In Spring 2014 he moved from Russia to Israel, and since 2018 he work as a Researcher at the Institute of Mathematics of the Czech Academy of Sciences in Prague.

He is an algebraist specializing in homological algebra. His research papers span a wide area including algebraic geometry, representation theory, commutative algebra, algebraic K-theory, and algebraic number theory.

He is the author of four books and memoirs, including "Quadratic Algebras" (joint with A. Polishchuk, AMS University Lecture Series, 2005), "Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures" (Monografie Matematyczne IMPAN, Birkhauser Basel, 2010), "Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence" (AMS Memoir, 2011), and "Weakly curved A-infinity algebras over a topological local ring" (Memoir of the French Math. Society, 2018-19).

Inhalt
Preface.- Prologue.- Introduction.- Homogeneous Quadratic Duality over a Base Ring.- Flat and Finitely Projective Koszulity.- Relative Nonhomogeneous Quadratic Duality.- The Poincare-Birkhoff-Witt Theorem.- Comodules and Contramodules over Graded Rings.- Relative Nonhomogeneous Derived Koszul Duality: the Comodule Side.- Relative Nonhomogeneous Derived Koszul Duality: the Contramodule Side.- The Co-Contra Correspondence.- Koszul Duality and Conversion Functor.- Examples.- References.