Relative nonhomogeneous Koszul duality

Abstract

This book contains a detailed exposition of the nonhomogeneous Koszul duality theory in the relative situation over a noncentral, noncommutative, nonsemisimple base ring, as announced in Section 0.4 of arXiv:0708.3398. We prove the Poincare-Birkhoff-Witt theorem in this context and construct the triangulated equivalences of derived Koszul duality. The duality between the ring of differential operators and the de Rham DG-algebra, with the ring of functions as the base ring, is the thematic example. The moderate generality level makes the exposition in this book more accessible than the very heavily technical Chapter 11 of arXiv:0708.3398.

arXiv commentLaTeX 2e with mathtools.sty and tikz-cd; 220 pages, 7 commutative diagrams. v.7: Subsection 0.11 and new Section 9 inserted; v.8: Lemma 3.7 inserted, Subsections 10.8-10.10 added -- this is intended as a complete version; v.9: seven references added; v.10: small things corrected, references updated -- this is intended as the final arXiv version (the publisher's version is much more complete)

References in corpus (11)

• Localization of modules for a semisimple Lie algebra in prime characteristic
• Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence
• Nonhomogeneous quadratic duality and curvature
• Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories
• Weak Proregularity, Weak Stability, and the Noncommutative MGM Equivalence
• Dedualizing complexes and MGM duality
• Koszul property and Bogomolov's conjecture
• Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality
• Semiinfinite cohomology of associative algebras and bar duality
• Cocommutative coalgebras: homotopy theory and Koszul duality
• Koszul duality and homotopy theory of curved Lie algebras

Cited by in corpus (6)

• Contramodules
• Differential graded Koszul duality: an introductory survey
• Coderived and contraderived categories of locally presentable abelian DG-categories
• An explicit self-dual construction of complete cotorsion pairs in the relative context
• Contraderived categories of CDG-modules
• A bounded below, noncontractible, acyclic complex of projective modules