Strong crepancy of the algebraic categorical resolution via Λ_q

Determine whether the weakly crepant categorical resolution ψ: D(Λ_q) → D(B_q), where Λ_q = A_q ⋊ C_{n+1} and B_q = A_q^{C_{n+1}}, is strongly crepant; equivalently, prove that the relative dualising complex for this categorical resolution is trivial in the sense of Van den Bergh, so that D(Λ_q) is a strongly crepant categorical resolution of D(B_q).

Background

The paper considers the noncommutative surface singularity B_q = A_q{C_{n+1}} and the algebraic resolution Λq = A_q ⋊ C{n+1}. Using the adjoint pair (ψ*, ψ_*), the authors show that D(Λ_q) gives a weakly crepant categorical resolution of D(B_q).

Van den Bergh’s notion of a strongly crepant categorical resolution corresponds to the triviality of the relative dualising complex. While the weak crepancy is established, whether this categorical resolution is strongly crepant remains unresolved.

References

There is also a notion of a strongly crepant categorical resolution , corresponding to triviality of the relative dualising complex; we believe this resolution to be strongly crepant, but have not been able to prove it.

— Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities  (2510.07137 - Crawford et al., 8 Oct 2025) in Subsection “The algebraic resolution is categorical” (label CATRES), around Proposition \ref{prop:PsiCrepantCategoricalRes}