Derived equivalence of all maximal modification algebras

Prove that all maximal modification algebras of a three-dimensional equi-codimensional normal Cohen–Macaulay domain with a canonical module are derived equivalent.

Background

The paper identifies the derived equivalence of maximal modification algebras as a noncommutative version of the Bondal–Orlov conjecture. The statement is known in several special cases, including when the ring admits an NCCR or is Gorenstein, while the paper proves it for Gorenstein MMAs over three-dimensional QQ-Gorenstein rings and derives the index-two case. The general assertion is explicitly presented as a conjecture.

References

The following is a version of noncommutative Bondal-Orlov conjecture.

Let $R$ be a three-dimensional Cohen--Macaulay equi-codimensional normal domain with a canonical module. Then all MMAs are derived equivalent.

— Equivariant noncommutative crepant resolutions and $G$-maximal modifying modules  (2610.11708 - Hirano, 8 Oct 2026) in Conjecture 1.2 in Section 1, subsection “$G$-maximal modifying modules”