Local-to-global implication for maximal modifying modules

Determine whether the existence of maximal modification algebras for all completed local rings at maximal ideals implies the existence of a maximal modification algebra for the original global ring.

Background

The converse direction is also unresolved. Although a modifying global module whose completions are maximal modifying would itself be maximal modifying, the authors do not know whether local maximal modifying modules arise as completions of one common global modifying module.

References

However, this does not necessarially imply the direction (2) $\Rightarrow$ (1), since we do not know a priori that each MM module of $\widehat{R_m}$ is given as the completion of a common modifying $R$-module.

— Equivariant noncommutative crepant resolutions and $G$-maximal modifying modules  (2610.11708 - Hirano, 8 Oct 2026) in Remark immediately following Conjecture 1.1 in Section 1