# Right-angled Artin groups and enhanced Koszul properties

## TL;DR

This paper investigates the Koszul properties of cohomology algebras of right-angled Artin groups over finite fields, revealing conditions for strong and universal Koszulity linked to the underlying graph's properties.

## Contribution

It establishes that these cohomology algebras are strongly Koszul for all graphs and universally Koszul if the graph has the diagonal property, providing new examples related to Galois cohomology.

## Key findings

- Cohomology algebra is strongly Koszul for all graphs.
- Universal Koszulity occurs iff the graph has the diagonal property.
- Provides new examples supporting a Galois cohomology conjecture.

## Abstract

Let F be a finite field. We prove that the cohomology algebra with coefficients in F of a right-angled Artin group is a strongly Koszul algebra for every finite graph ${\Gamma}$. Moreover, the same algebra is a universally Koszul algebra if, and only if, the graph ${\Gamma}$ associated to the right-angled Artin group has the diagonal property. From this we obtain several new examples of pro-p groups, for a prime number p, whose continuous cochain cohomology algebra with coefficients in the field of p elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal prop Galois groups of fields formulated by J. Min\'a\v{c} et al.