Relative nonhomogeneous Koszul duality

TL;DR

This work develops a detailed theory of nonhomogeneous Koszul duality over complex base rings, proving key theorems and establishing derived equivalences, with applications to differential operators and de Rham algebras.

Contribution

It extends Koszul duality theory to a nonhomogeneous, relative setting over complex base rings, including proofs of the Poincare-Birkhoff-Witt theorem and derived duality constructions.

Findings

Proved the Poincare-Birkhoff-Witt theorem in this context

Constructed triangulated equivalences of derived Koszul duality

Applied duality to differential operators and de Rham DG-algebras

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.