Maximal Modifying Algebras for Canonical Threefold Singularities

Definitions and mathematical statement

In the usual noetherian normal Cohen--Macaulay local setting, let $(R,\mathfrak m_R)$ have Krull dimension $3$. A finitely generated $R$-module $M$ is reflexive if the natural map \[ M\longrightarrow \operatorname{Hom}_R(\operatorname{Hom}_R(M,R),R) \] is an isomorphism. Write $\operatorname{add}_R(M)$ for the finitely generated modules isomorphic to direct summands of finite direct sums of copies of $M$. A nonzero reflexive module $M$ is modifying if $\operatorname{End}_R(M)$ is maximal Cohen--Macaulay as an $R$-module, meaning that its depth is $\dim R$. It is maximal modifying if it is modifying and, for every finitely generated reflexive $R$-module $N$, \[ \operatorname{End}_R(M\oplus N)\text{ is maximal Cohen--Macaulay over }R \quad\Longrightarrow\quad N\in\operatorname{add}_R(M). \] An algebra of the form $\operatorname{End}_R(M)$ for such a maximal modifying module is called a maximal modifying algebra.

Reconstructed conjectural statement. For every local ring $(R,\mathfrak m_R)$ of a three-dimensional Gorenstein canonical singularity, there exists a nonzero finitely generated reflexive maximal modifying $R$-module $M$. Equivalently, every such $R$ admits a maximal modifying algebra in the preceding sense. The reconstructed statement imposes no crepant-resolution hypothesis.

Here Gorenstein has its usual noetherian local meaning of finite injective dimension as an $R$-module. In the usual geometric setting where canonical discrepancies are defined, canonical means that the discrepancies $a_E$ in \[ K_Y=g^*K_{\operatorname{Spec}R}+\sum_E a_E E \] are nonnegative for exceptional prime divisors on resolutions $g$. The supplied record does not specify the base field, characteristic, algebraic or analytic category, or whether the local rings are complete or essentially of finite type. Consequently, these definitions explain the conventional terminology but do not resolve the intended range of the universal quantifier.

Different assertion in the original extraction. The supplied original text begins with an isolated "conjecture" label and then states "Corollary 2": if $(R,\mathfrak m_R)$ is a local ring of a three-dimensional Gorenstein canonical singularity admitting a crepant resolution $f\colon X\to\operatorname{Spec}R$, then \[ b_3\left(f^{-1}(\mathfrak m_R)\right)\ne 0 \quad\Longrightarrow\quad R\text{ admits no noncommutative crepant resolution}. \] A crepant resolution is a resolution satisfying $K_X=f^*K_{\operatorname{Spec}R}$. In the conventional normal Gorenstein setting, a noncommutative crepant resolution is an endomorphism algebra $\Lambda=\operatorname{End}_R(P)$ of a nonzero finitely generated reflexive module that is maximal Cohen--Macaulay over $R$ and is a nonsingular order: $\operatorname{gl.dim}\Lambda_{\mathfrak p}=\dim R_{\mathfrak p}$ for every prime $\mathfrak p$ of $R$. The cohomology theory and coefficients defining the displayed third Betti number are not supplied. This extracted corollary is a distinct assertion, not an equivalent formulation of the reconstructed existence conjecture; their connection requires source review.

Short English statement

The reconstructed conjecture asks whether every local three-dimensional Gorenstein canonical singularity has a reflexive module whose endomorphism algebra is Cohen--Macaulay and which cannot be enlarged while retaining that property, except by adjoining summands already generated by it. The source record labels this conjecture open. Its original extraction instead contains a conditional obstruction to a noncommutative crepant resolution. The intended geometric domain and the relationship between these passages require human review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006062$, source problem code catalogue entry, title "Maximal Modifying Algebras for Canonical Threefold Singularities." No individual proposer is supplied; the registry gives proposed year $2024$.
• Supplied report citation: MFO-RIMS Tandem Workshop: Algebraic Geometry and Noncommutative Projective Varieties (2025), Oberwolfach Reports, pp. 2551--2606, DOI 10.4171/owr/2024/44. https://doi.org/10.4171/owr/2024/44. The literature-source list calls this "Oberwolfach Report 44/2024." The discrepancy between the parenthetical year $2025$ and the $2024$ identifiers remains unresolved.
• Supplied literature reference: Iyama--Wemyss, Maximal Modifications and Auslander--Reiten Duality. https://arxiv.org/abs/1007.1296.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the problem open, describes the extracted record as an isolated conjecture label, identifies the surrounding topic as existence of maximal modifying algebras for all three-dimensional Gorenstein canonical singularities, and reports that no general proof was located. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels its reconstructed statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim, not an independent verification performed for this draft. The original extraction contains "Corollary 2" concerning the third Betti number of the exceptional fibre and nonexistence of a noncommutative crepant resolution.