Extend distinguished local cohomology vanishing beyond P-algebras

Determine whether, for a finite-type connected graded coalgebra over a field whose dual graded algebra is not assumed to be a \(\mathcal{P}\)-algebra, every rational graded module has vanishing higher distinguished local cohomology \(H^n_{\mathrm{dist}}\) for all \(n>0\).

Background

The paper proves that if Γ\Gamma is a finite-type connected graded coalgebra over a field and Γ∗\Gamma^* is a P\mathcal{P}-algebra, then every rational graded Γ∗\Gamma^*-module has trivial higher distinguished local cohomology. This vanishing is a key technical input for comparing Ext-groups in graded comodules with Ext-groups in graded modules.

The authors explicitly leave unresolved whether the P\mathcal{P}-algebra hypothesis is necessary. Resolving this would enlarge the class of graded coalgebras for which rationality implies acyclicity with respect to distinguished local cohomology and could extend the paper’s Ext-comparison theorem.

References

The author does not know whether Proposition \ref{better version of main thm 1} remains true without the hypothesis that $\Gamma*$ is a $\mathcal{P}$-algebra.

— Ext-groups of graded comodules  (2609.26465 - Salch, 22 Sep 2026) in Section “Technical results on distinguished local cohomology,” immediately after Proposition \ref{better version of main thm 1}