[Paper Review] A New Triangulated Category for Rational Surface Singularities

Osamu Iyama, Michael Wemyss|arXiv (Cornell University)|May 25, 2009

Algebraic structures and combinatorial models16 references3 citations

TL;DR

This paper introduces a new triangulated category, the stable quotient of the Frobenius category of special Cohen-Macaulay (CM) modules over rational surface singularities, which serves as a geometrically meaningful substitute for the stable category of matrix factorizations in the non-Gorenstein case. The key contribution is the classification of relatively projective-injective objects in this category as corresponding precisely to the non-(-2)-curves in the minimal resolution’s dual graph, thereby isolating crepant divisors and enabling a homological understanding of reconstruction algebras.

ABSTRACT

In this short paper we introduce a new triangulated category for rational surface singularities which in the non-Gorenstein case acts as a substitute for the stable category of matrix factorizations. The category is formed as a Frobenius quotient of the category of special CM modules, and we classify the relatively projective-injective objects and thus describe the AR quiver of the quotient. Connections to the corresponding reconstruction algebras are also discussed.

Motivation & Objective

• To construct a manageable triangulated category for rational surface singularities that replaces the unstable stable category of matrix factorizations in the non-Gorenstein case.
• To classify the relatively projective-injective objects in the stable quotient of the Frobenius category of special CM modules.
• To clarify the geometric and homological structure of reconstruction algebras via this new category.
• To establish connections between the AR quiver of the quotient category and the dual graph of the minimal resolution.
• To characterize when the category of CM modules admits cluster tilting objects, particularly for n=1,2 and n>2.

Proposed method

• The authors define a Frobenius category structure on the category of special CM modules, SCM(R), using the duality and relative projectivity.
• They form the stable quotient category, denoted as \underline{\underline{\mathop{\rm SCM}\nolimits}}(R), by factoring out the relatively projective-injective objects.
• Using Esnault’s criterion for full sheaves, they relate geometric data (locally free sheaves on the resolution) to CM modules.
• They identify that the relatively projective-injective objects in SCM(R) are R and the special CM modules corresponding to non-(-2)-curves in the dual graph.
• They apply relative AR theory to describe the AR quiver of the quotient category, showing it only 'sees' crepant divisors.
• They use the structure of the quotient to derive relations in reconstruction algebras, showing that at vertices corresponding to (-2)-curves, the local relation is a preprojective relation.

Experimental results

Research questions

• RQ1 What is the structure of the stable quotient category of the Frobenius category of special CM modules for rational surface singularities?
• RQ2 Which special CM modules are relatively projective-injective in this category, and how do they relate to the geometry of the minimal resolution?
• RQ3 How does this new triangulated category relate to the classical triangulated category of singularities and to reconstruction algebras?
• RQ4 What is the AR quiver of the quotient category, and how does it reflect the dual graph of the resolution?
• RQ5 Under what conditions does the category of CM modules admit an n-cluster tilting object?

Key findings

• The relatively projective-injective objects in SCM(R) are precisely R and the special CM modules corresponding to non-(-2)-curves in the dual graph of the minimal resolution.
• The stable quotient category \underline{\underline{\mathop{\rm SCM}\nolimits}}(R) is always Krull-Schmidt, unlike the category of singularities in the non-Gorenstein case.
• The category \underline{\underline{\mathop{\rm SCM}\nolimits}}(R) is zero if and only if all exceptional curves in the resolution are (-2)-curves, i.e., no crepant divisors.
• At each vertex corresponding to a (-2)-curve in the minimal resolution, the reconstruction algebra has a single local relation that is a cycle, and this relation is locally preprojective.
• The category \underline{\underline{\mathop{\rm SCM}\nolimits}}(R) is equivalent to \underline{\mathop{\rm CM}\nolimits}(R') for some Gorenstein ring R' in many cases.
• The category CM(R) admits a 1-cluster tilting object if and only if R is a quotient singularity; it admits a 2-cluster tilting object if and only if R ≅ k[[x,y]]^{1/3(1,1)}; and it admits an n-cluster tilting object for n>2 only if R is regular.