[Paper Review] Maximal Cohen-Macaulay modules over surface singularities

Igor Burban, Yuriy Drozd|ArXiv.org|Mar 2, 2008

Algebraic structures and combinatorial models67 references17 citations

TL;DR

This paper provides a comprehensive survey on maximal Cohen-Macaulay (MCM) modules over surface singularities, focusing on their homological properties, representation types, and connections to the McKay correspondence. It establishes that A∞ and D∞ singularities have countable (discrete) MCM representation type via matrix factorization techniques and proves that Ext1(M,N) has finite length precisely when the singularity is isolated, linking homological finiteness to geometric isolation. The work unifies algebraic and geometric approaches to the McKay correspondence for quotient, simple, and minimally elliptic singularities, and confirms the Serre duality and Calabi-Yau structure in the stable category of MCM modules over Gorenstein isolated singularities.

ABSTRACT

This is a survey article about properties of Cohen-Macaulay modules over surface singularities. We discuss various results on the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay Correspondence. Finally, we describe matrix factorizations corresponding to the indecomposable Cohen-Macaulay modules over the non-isolated singularities $A_\infty$ and $D_\infty$.

Motivation & Objective

• To survey the structure and representation theory of maximal Cohen-Macaulay (MCM) modules over surface singularities, particularly in relation to the McKay correspondence.
• To clarify the role of the Macaulayfication functor and reflexive modules in the context of simple, quotient, and minimally elliptic singularities.
• To establish a new proof of the countable (discrete) MCM representation type for non-isolated singularities A∞ and D∞ using matrix factorizations.
• To unify algebraic and geometric approaches to the McKay correspondence for quotient, simply elliptic, and cusp singularities.
• To demonstrate that the stable category of MCM modules over an isolated Gorenstein surface singularity is (d−1)-Calabi-Yau, with a Serre duality structure.

Proposed method

• Utilizes the Macaulayfication functor to relate arbitrary modules to MCM modules, especially in two-dimensional Cohen-Macaulay singularities.
• Applies the theory of reflexive modules and their duality to analyze MCM modules over normal surface singularities.
• Employs matrix factorizations to describe indecomposable MCM modules over A∞ and D∞ singularities.
• Applies Buchweitz's framework to interpret the stable category of MCM modules as a triangulated category.
• Uses the isomorphism Ext1(M,N) has finite length if and only if the singularity is isolated to characterize isolated singularities homologically.
• Applies computer algebra (Singular) to compute Ext and Hom dimensions for explicit modules over hypersurface singularities.

Experimental results

Research questions

• RQ1 What is the structure of the category of maximal Cohen-Macaulay modules over surface singularities, and how does it relate to the geometry of the singularity?
• RQ2 How do the algebraic and geometric McKay correspondences relate to each other for quotient surface singularities?
• RQ3 What is the representation type of MCM modules over non-isolated singularities such as A∞ and D∞, and how can it be proven via matrix factorizations?
• RQ4 Under what conditions does Ext1(M,N) have finite length for MCM modules M and N, and what does this imply about the singularity?
• RQ5 What is the Calabi-Yau structure of the stable category of MCM modules over an isolated Gorenstein surface singularity?

Key findings

• The stable category of maximal Cohen-Macaulay modules over an isolated Gorenstein surface singularity of Krull dimension d is a (d−1)-Calabi-Yau triangulated category.
• The canonical functor from the category of MCM modules over a local ring to its completion is fully faithful, preserving the structure of MCM modules.
• For any isolated Gorenstein singularity, there exists a bifunctorial isomorphism Ext1(M, syzd(Tr(N))*) ≅ D(Hom(M,N)), establishing Serre duality in the stable category.
• The surface singularities A∞ and D∞ have countable (discrete) MCM representation type, as confirmed by a new proof using matrix factorizations.
• Ext1(M,N) has finite length for all MCM modules M and N if and only if the singularity is isolated, providing a homological characterization of isolated singularities.
• For a Gorenstein k-algebra A of Krull dimension d that is an isolated singularity, the functor S = syz1−d is a Serre functor on the stable category of MCM modules, yielding a bifunctorial isomorphism Hom(M,N) ≅ Hom(N,S(M))∗.