Global-to-local implication for maximal modifying modules

Determine whether the existence of a maximal modification algebra for a three-dimensional normal affine variety with canonical singularities implies that the completion at every maximal ideal admits a maximal modification algebra.

Background

The paper notes that completion does not generally preserve maximal modifying modules: even when a module is maximal modifying globally, its completed localization need not be maximal modifying locally. Consequently, the authors leave unresolved whether global existence implies existence for all completed local rings.

References

Even if $M$ is an MM $R$-module, the module $\widehat{M_m}$ is not MM $\widehat{R_m}$-module in general Example 2.13, and we do not know whether the implication (1) $\Rightarrow$ (2) holds.

— 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