Right-angled Artin groups and enhanced Koszul properties

Abstract

Abstract Let 𝔽 {\mathbb{F}} be a finite field. We prove that the cohomology algebra H ∙ ⁢ ( G Γ , 𝔽 ) {H^{\bullet}(G_{\Gamma},\mathbb{F})} with coefficients in 𝔽 {\mathbb{F}} of a right-angled Artin group G Γ {G_{\Gamma}} is a strongly Koszul algebra for every finite graph Γ. Moreover, H ∙ ⁢ ( G Γ , 𝔽 ) {H^{\bullet}(G_{\Gamma},\mathbb{F})} is a universally Koszul algebra if, and only if, the graph Γ associated to the group G Γ {G_{\Gamma}} 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 pro-p Galois groups of fields formulated by J. Mináč et al.