Strong crepancy of the geometric categorical resolution via Qgr T

Determine whether the categorical resolution φ: D^b(Qgr T_q) → D(B_q) constructed from the graded ring T_q is strongly crepant; equivalently, prove that D(Qgr T_q) is a strongly crepant categorical resolution of D(B_q) in the sense of Van den Bergh/Kuznetsov.

Background

The authors construct a geometric resolution category X_q ≃ Qgr T_q and show that φ = (φ*, φ_*) gives a weakly crepant categorical resolution of D(B_q). They further establish a derived equivalence Db(Qgr T_q) ≃ D(Λ_q).

Despite this, the strong crepancy of D(Qgr T_q) → D(B_q)—that is, triviality of the relative dualising complex for this resolution—remains unproven, mirroring the unresolved status for the algebraic resolution via Λ_q.

References

We believe that $D(\rQgr T)$ is also a strongly crepant categorical resolution of $ D(B)$, as with $ D(\Lambda)$, but again we have not been able to prove this.

— Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities  (2510.07137 - Crawford et al., 8 Oct 2025) in Section “A derived equivalence of resolutions” (after Theorem \ref{thm:iv})