Weigel’s b2-conjecture for Koszul algebras

Determine whether every Koszul algebra A of cohomological dimension d over a field k satisfies the inequality \(\dim(\operatorname{Ext}^{2,2}_A(k,k))\leq \frac{d-1}{2d}\dim(\operatorname{Ext}^{1,1}_A(k,k))^2\).

Background

The paper identifies an unresolved question raised by Weigel concerning a quadratic upper bound on the degree-two Ext-dimension of a Koszul algebra in terms of its degree-one Ext-dimension and cohomological dimension. The question is motivated by the study of the invariant ω\omega for finitely presented graded Lie algebras and by known positive results for Koszul algebras whose eigenvalues are all real, including the case of cohomological dimension two.

The paper further explains that the inequality is known for right-angled Artin Lie algebras, but does not establish it for arbitrary Koszul algebras. A graph-theoretic reformulation is subsequently posed for Bestvina–Brady Lie algebras associated with finite graphs having acyclic flag complexes.

References

The following question was raised by Weigel in 1pt{weig}: Let $A$ be a Koszul algebra of cohomological dimension $d$ over a field $k$. Is it true that \dim(Ext{2,2}_A(k,k))\leq \frac{d-1}{2d}\dim(Ext{1,1}_A(k,k))2?

— Subgroups of Bestvina-Brady groups  (2502.03215 - Blumer, 5 Feb 2025) in Section 3, subsection “Bloch-Kato version of the b2-conjecture,” Question \ref{quest:b2}