Koszulity of cohomology = $K(\pi ,1)$-ness + quasi-formality

Abstract

This paper is a greatly expanded version of Section 9.11 in arXiv:1006.4343. A series of definitions and results illustrating the thesis in the title (where quasi-formality means vanishing of a certain kind of Massey multiplications in the cohomology) is presented. In particular, we include a categorical interpretation of the "Koszulity implies $K(\pi ,1)$" claim, discuss the differences between two versions of Massey operations, and apply the derived nonhomogeneous Koszul duality theory in order to deduce the main theorem. In the end we demonstrate a counterexample providing a negative answer to a question of Hopkins and Wickelgren about formality of the cochain DG-algebras of absolute Galois groups, thus showing that quasi-formality cannot be strengthened to formality in the title assertion.

LaTeX 2e with pb-diagram and xy-pic, 39 pages, 1 commutative diagram; v.6: new section 5 inserted, explanations added here and there, several misprints corrected; v.7: several misprints corrected in sections 1 and 5; v.8: several misprints corrected, references updated -- this is intended as the final version

References in corpus (3)

• Nonhomogeneous quadratic duality and curvature
• Koszul property and Bogomolov's conjecture
• A-infinity structure on Ext-algebras

Cited by in corpus (9)

• The Massey vanishing conjecture for number fields
• Contramodules
• Differential graded Koszul duality: an introductory survey
• Smooth duality and co-contra correspondence
• Right-angled Artin groups and enhanced Koszul properties
• Mild pro-p groups and the Koszulity conjectures
• On the Bott-Chern and Aeppli cohomology
• Higher Yoneda product structures and Iwasawa algebras modulo $p$
• Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors