Koszulity of cohomology = K(π,1)-ness + quasi-formality

Abstract

Abstract. This paper is a greatly expanded version of [37, Section 9.11]. 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(π, 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.

Generalized Bockstein maps and Massey products

“…The differential graded ring of continuous -valued -cochains is said to be formal if it is quasi-isomorphic to . The question of whether or not is always formal was raised by Hopkins and Wickelgren in their aforementioned work and answered in the negative by Positselski [Po]. If formality holds for some F and p , then a stronger version of Massey vanishing, that moreover the vanishing of the consecutive cup products yields definedness, holds in that instance.…”

Differential graded Koszul duality: An introductory survey

“…our Theorems 4.2 and 6.6, where the cohomological gradings are completely unbounded, but the notion of a filtered quasi‐isomorphism is used). See, for example, [67, section 3] for noncommutative versions of Quillen's theory.…”

Right-angled Artin groups and enhanced Koszul properties

“…This has great relevance in the context of Galois theory. Let K be a field containing a root of 1 of order p, and let G K denote the maximal pro-p Galois group of K -i.e., G K is the Galois group of the maximal pro-p-extension of K. In the last two decades, Koszulity has gained importance in Galois cohomology, thanks to the work of L. Positselski, especially in connection with the celebrated Bloch-Kato conjecture (see, e.g., [22,24,25]). In particular, Positselski conjectured that the F p -cohomology algebra of a maximal pro-p Galois group G K is Koszul, and this was shown to be true in some relevant cases (cf.…”