Nonnoetherian homotopy dimer algebras and noncommutative crepant resolutions

TL;DR

This paper explores the relationship between nonnoetherian homotopy dimer algebras and noncommutative crepant resolutions, introducing a generalized framework and identifying conditions under which these algebras serve as noncommutative desingularizations.

Contribution

It generalizes NCCRs to nonnoetherian tiled matrix rings and establishes when nonnoetherian homotopy dimer algebras act as noncommutative desingularizations and NCCRs.

Findings

Nonnoetherian homotopy dimer algebras can be noncommutative desingularizations.

Certain conditions make these algebras serve as nonnoetherian NCCRs.

Contracting arrows relates nonnoetherian and noetherian dimer algebras.

Abstract

Noetherian dimer algebras form a prominent class of examples of noncommutative crepant resolutions (NCCRs). However, dimer algebras which are noetherian are quite rare, and we consider the question: how close are nonnoetherian homotopy dimer algebras to being NCCRs? To address this question, we introduce a generalization of NCCRs to nonnoetherian tiled matrix rings. We show that if a noetherian dimer algebra is obtained from a nonnoetherian homotopy dimer algebra $A$ by contracting each arrow whose head has indegree 1, then $A$ is a noncommutative desingularization of its nonnoetherian center. Furthermore, if any two arrows whose tails have indegree 1 are coprime, then $A$ is a nonnoetherian NCCR.