Stability over cDV singularities and other complete local rings

TL;DR

This paper characterizes the stability of modules over noncommutative minimal models of cDV singularities, revealing a polyhedral fan structure derived from silting theory and providing explicit deformation descriptions in the isolated case.

Contribution

It extends silting theory and stability correspondence from finite-dimensional algebras to complete local Noetherian rings, including non-isolated cDV singularities.

Findings

Stability controlled by an infinite polyhedral fan from silting theory

Explicit deformation descriptions for stable modules in the isolated case

Generalization of silting-stability correspondence to broader algebraic settings

Abstract

We characterise subcategories of semistable modules for noncommutative minimal models of compound Du Val singularities, including the non-isolated case. We find that the stability is controlled by an infinite polyhedral fan that stems from silting theory, and which can be computed from the Dynkin diagram combinatorics of the minimal models found in the work of Iyama--Wemyss. In the isolated case, we moreover find an explicit description of the deformation theory of the stable modules in terms of factors of the endomorphism algebras of 2-term tilting complexes. To obtain these results we generalise a correspondence between 2-term silting theory and stability, which is known to hold for finite dimensional algebras, to the much broader setting of algebras over a complete local Noetherian base ring.