Kernel of φ_* generated by shifted line modules

Prove that the kernel of the right adjoint functor φ_*: Qgr T_q → Mod B_q is generated by the shifted line modules L_1(−1), …, L_n(−1), where L_i = T_q/ℓ_i are the line modules associated to the prime ideals ℓ_i and (−1) denotes the projective grading shift.

Background

The line modules L_i = T/ℓi form the exceptional locus in the geometric resolution and exhibit an A_n intersection pattern. The authors show that each L_i(−1) lies in ker(φ), where φ_ = H0_{Qgr T}(−) is the pushforward to Mod B.

They conjecture that these n shifted line modules generate the entire kernel of φ_*; proving this would clarify the structure of objects in Qgr T annihilated by the pushforward to B.

References

We conjecture that the modules $L_1(-1), \dots, L_n(-1)$ generate the kernel of $\phi_*$. This is the subject of ongoing research.

— Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities  (2510.07137 - Crawford et al., 8 Oct 2025) in Remark \ref{rem:kernel}, Section “Intersection theory”