Derived Equivalence of Maximal Orthogonal Subcategories

Definitions and mathematical statement

Let $\Lambda$ be the isolated-singularity $R$-order in the source, and let $\operatorname{CM}\Lambda$ denote its category of finitely generated modules that are maximal Cohen--Macaulay over $R$. In a standard local convention, $R$ is a commutative Noetherian Cohen--Macaulay local ring, an $R$-order is a finite $R$-algebra maximal Cohen--Macaulay as an $R$-module, and isolated singularity requires nonsingularity away from the maximal ideal. A customary nonsingular-order condition there is $\operatorname{gl.dim}\Lambda_{\mathfrak p}=\dim R_{\mathfrak p}$. The report's exact base-ring, completeness, dimension, and order conventions are not supplied and must be checked; these conventions do not assert the conjecture for every ring fitting this broad template.

For $M\in\operatorname{CM}\Lambda$, let $\operatorname{add}M$ be the full subcategory of direct summands of finite direct sums of $M$. Under the usual maximal orthogonality convention, $\mathcal C=\operatorname{add}M$ is functorially finite and \[ \mathcal C=\{X\in\operatorname{CM}\Lambda:\operatorname{Ext}^i_\Lambda(X,C)=0 \text{ for every }C\in\mathcal C,\ 1\leq i\leq n-1\} \] and \[ \mathcal C=\{X\in\operatorname{CM}\Lambda:\operatorname{Ext}^i_\Lambda(C,X)=0 \text{ for every }C\in\mathcal C,\ 1\leq i\leq n-1\}. \] Functorial finiteness means that every object has both a right and a left $\mathcal C$-approximation. Maximal does not merely mean inclusion-maximal among pairwise Ext-orthogonal subcategories.

The source conjecture, attributed to Iyama by the supplied literature field, asks whether for any two such generators $M,N$ for the same $\Lambda$, \[ D^b(\operatorname{mod}\operatorname{End}_\Lambda(M)) \simeq D^b(\operatorname{mod}\operatorname{End}_\Lambda(N)) \] as triangulated categories. Here $\operatorname{mod}$ denotes finitely generated modules with a consistent side convention. These are the ordinary endomorphism rings, not the stable endomorphism quotients. The permitted $n$ and its relation to $\dim R$, and any intended $R$-linearity of the equivalence, are not specified in the extracted record.

The supplied assessment checked 2026-08-21 reports results for maximal $1$-orthogonal subcategories and for several low-dimensional or additional-structure cases, but says the general statement is not established. The exact hypotheses of those cases are absent; the secondary reference is labeled as concerning stable endomorphism rings and must not automatically be applied to the ordinary rings above.

Short English statement

For one isolated-singularity order, are the endomorphism rings of any two additive generators of maximal $(n-1)$-orthogonal Cohen--Macaulay subcategories derived equivalent? The base-ring, dimension, and orthogonality conventions need source verification.

Sources

• Local UnsolvedMath registry, permanent ID 30000182, catalogue entry, Derived Equivalence of Maximal Orthogonal Subcategories.
• Iyama, contribution labeled Higher dimensional Auslander correspondence, in Representation Theory of Finite-Dimensional Algebras (2005), Oberwolfach Reports, pp. 319--406, DOI https://doi.org/10.4171/owr/2005/06.
• Supplied reference labeled "recent work on derived equivalence classes of stable endomorphism rings": https://doi.org/10.1007/s00209-020-02475-y.
• Supplied literature triage checked 2026-08-21 and supplied statement-verification claim of comparison with the official report.