[Paper Review] Mutations of splitting maximal modifying modules: The case of reflexive polygons

Yusuke Nakajima|arXiv (Cornell University)|Jan 20, 2016

Algebraic structures and combinatorial models47 references3 citations

TL;DR

This paper establishes that all splitting maximal modifying modules over three-dimensional Gorenstein toric singularities associated with reflexive polygons are connected via iterated mutations, generalizing the notion of flops in crepant resolutions to non-commutative crepant resolutions (NCCRs). Using dimer models and quiver mutations, the authors prove the exchange graph of splitting MM modules is connected, providing a categorical analogue to the geometric connectedness of crepant resolutions via flops.

ABSTRACT

It is known that every three dimensional Gorenstein toric singularity has a crepant resolution. Although it is not unique, all crepant resolutions are connected by repeating the operation "flop". On the other hand, this singularity also has a non-commutative crepant resolution (= NCCR) which is constructed from a consistent dimer model. Such an NCCR is given as the endomorphism ring of a certain module which we call splitting maximal modifying module. In this paper, we show that all splitting maximal modifying modules are connected by repeating the operation "mutation" of splitting maximal modifying modules for the case of toric singularities associated with reflexive polygons.

Motivation & Objective

• To investigate the relationship between non-commutative crepant resolutions (NCCRs) of 3D Gorenstein toric singularities arising from dimer models.
• To determine whether all splitting maximal modifying (MM) modules over such singularities are connected via a categorical operation analogous to geometric flops.
• To establish that mutations of dimer models induce mutations of splitting MM modules, and that the resulting exchange graph is connected.
• To extend the known geometric connectedness of crepant resolutions (via flops) to the non-commutative setting using mutation operations.
• To verify the connectedness of the exchange graph of splitting MM modules for all reflexive polygon cases, confirming a non-commutative analogue of the Bondal-Orlov conjecture in this context.

Proposed method

• Constructs NCCRs from consistent dimer models dual to quivers with potential (QP), using the endomorphism ring of a splitting maximal modifying module.
• Applies the mutation operation on dimer models, defined as the dual of mutation of quivers with potential (DWZ mutation), to generate new dimer models.
• Translates dimer model mutations into mutations of splitting MM modules via the correspondence between dimer models and module generators.
• Uses the derived equivalence of NCCRs to show that mutations preserve the NCCR structure and connect different MM modules.
• Employs the cluster category framework and mutation of cluster tilting objects to analyze connectivity in the exchange graph.
• Applies a twist by rank-one reflexive modules to extend connectivity from a generating set to the full module category, using the class group Cl(R).

Experimental results

Research questions

• RQ1 Are all non-commutative crepant resolutions (NCCRs) of a 3D Gorenstein toric singularity associated with a reflexive polygon connected via a sequence of mutations?
• RQ2 Can the mutation operation on dimer models be lifted to a mutation of splitting maximal modifying modules, preserving the NCCR structure?
• RQ3 Is the exchange graph of splitting maximal modifying modules connected for all 3D Gorenstein toric singularities arising from reflexive polygons?
• RQ4 Does the existence of a connected exchange graph for a generating set of the class group imply full connectivity of the exchange graph of splitting MM modules?
• RQ5 Can the categorical mutation operation serve as a non-commutative analogue to the geometric flop operation in crepant resolutions?

Key findings

• All splitting maximal modifying modules over 3D Gorenstein toric singularities associated with reflexive polygons are connected via iterated mutations.
• The exchange graph EG(MM₁(R)) of splitting MM modules is connected for all such singularities, as proven via the mutation framework.
• For each reflexive polygon type (e.g., 4a, 8b), the exchange graph is explicitly described and shown to be connected through mutation sequences.
• The mutation operation on dimer models induces a well-defined mutation on splitting MM modules, preserving the NCCR structure.
• The connectedness of the exchange graph is established by showing that all modules are connected to a generating set via twist by reflexive ideals and mutation.
• The result confirms that the non-commutative version of the Bondal-Orlov conjecture holds in this setting: all NCCRs are connected via mutations.