Overview
- Provides an in-depth discussion of Koszul duality in the relative context over a base ring
- Presents generalization of the Poincare-Birkhoff-Witt theorem to the relative context
- Adds a whole new "relative" dimension to the Koszul duality studies existing in the literature
Part of the book series: Frontiers in Mathematics (FM)
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Table of contents (10 chapters)
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Front Matter
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Back Matter
Keywords
- quadratic and Koszul graded rings
- relative quadratic duality
- Koszul duality over a base ring
- Poincare-Birkhoff-Witt theorem over a base ring
- curved DG rings and curved DG modules
- graded comodules and contramodules
- rings of differential operators
- de Rham DG algebra of differential forms
- D-Omega duality
About this book
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 timein 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.
Reviews
Authors and Affiliations
About the author
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).
Bibliographic Information
Book Title: Relative Nonhomogeneous Koszul Duality
Authors: Leonid Positselski
Series Title: Frontiers in Mathematics
DOI: https://doi.org/10.1007/978-3-030-89540-2
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Softcover ISBN: 978-3-030-89539-6Published: 11 February 2022
eBook ISBN: 978-3-030-89540-2Published: 10 February 2022
Series ISSN: 1660-8046
Series E-ISSN: 1660-8054
Edition Number: 1
Number of Pages: XXIX, 278
Number of Illustrations: 1 b/w illustrations