Boundary Idempotents and $2$-precluster-tilting categories

TL;DR

This paper explores the relationship between 2-precluster-tilting subcategories of module categories over finite-dimensional algebras and their quotients, extending homological theories and applying to generalized dimer algebras on surfaces.

Contribution

It generalizes a criterion for relating 2-precluster-tilting subcategories to algebra quotients and applies this to the study of dimer algebras with boundary idempotents.

Findings

Established a correspondence between 2-precluster-tilting subcategories and algebra quotients.

Extended homological techniques to higher Auslander-Reiten theory.

Applied results to generalized dimer algebras on surfaces.

Abstract

The homological theory of Auslander-Platzeck-Todorov on idempotent ideals laid much of the groundwork for higher Auslander-Reiten theory, providing the key technical lemmas for both higher Auslander correspondence as well as the construction of higher Nakayama algebras, among other results. Given a finite-dimensional algebra $A$ and idempotent $e$, we expand on a criterion of Jasso-K\"ulshammer in order to determine a correspondence between the $2$-precluster-tilting subcategories of $\mathrm{mod}(A)$ and $\mathrm{mod}(A/\langle e\rangle)$. This is then applied in the context of generalising dimer algebras on surfaces with boundary idempotent.