Extriangulated ideal quotients, with applications to cluster theory and gentle algebras

Abstract

We extend results of Brüstle-Yang on ideal quotients of 2-term subcategories of perfect derived categories of non-positive dg algebras to a relative setting. We find a new interpretation of such quotients: they appear as prototypical examples of a new construction of quotients of extriangulated categories by ideals generated by morphisms from injectives to projectives. We apply our results to Frobenius exact cluster categories and Higgs categories with suitable relative extriangulated structures, and to categories of walks related to gentle algebras. In all three cases, the extriangulated structures are well-behaved (they are 0-Auslander) and their quotients are equivalent to homotopy categories of two-term complexes of projectives over suitable finite-dimensional algebras.

Figures (7)

• Figure 1.1: On the left, a Higgs category in type $\sf{A}_3$ with principal coefficients. On the right, its quotient by the ideal generated by morphisms from injectives to projectives in an appropriate relative extriangulated structure, which is equivalent to ${\mathcal{K}}^{[-1,0]}_{}(\mathop{\mathrm{proj}}\nolimits k\vec{\sf{A}}_3)$. Empty circles are projective-injective objects; dashed horizontal lines represent almost split conflations; dotted vertical lines are identified after a reflection.
• Figure 1.2: On the left, the category of walks $\mathcal{W}_A$ of a gentle algebra of type $\sf{A}_3$ with a relation. On the right, the quotient of $\mathcal{W}_A$ by the ideal generated by morphisms from injectives to projectives, which is equivalent to ${\mathcal{K}}^{[-1,0]}_{}(\mathop{\mathrm{proj}}\nolimits A)$. Empty circles are projective-injective objects; dashed horizontal lines represent almost split conflations; curved dotted lines represent zero-relations that cannot be deduced from almost split conflations.
• Figure 2.1: The Auslander--Reiten quiver of the category ${\mathcal{K}}^{[-1,0]}(\mathop{\mathrm{proj}}\nolimits kQ/I)$, for $Q$ a $3$-cycle and $I$ generated by all paths of length $2$. Here $P_i^{\bullet}$ denotes the stalk complex with the indecomposable projective object $P_i \in \operatorname{mod} kQ/I$ in degree $0$, for $i = 1, 2, 3$.
• Figure 7.1: The Auslander--Reiten quiver of the cluster category ${\mathcal{C}}$ of type $\sf{A}_2$.
• Figure 7.3: The Auslander--Reiten quiver of the module category of the preprojective algebra of type $\sf{A}_2$. The indecomposable projective modules are denoted by $\Pi_i$ to avoid confusion with the projective modules of the algebra $A=\mathop{\mathrm{End}}\nolimits_{{\mathcal{E}}}(T)^\mathrm{op}$.

Theorems & Definitions (133)

• Theorem A: = Theorem \ref{['t:extriangulatedQuotient']}
• Theorem B: = Theorem \ref{['t:rel-2-term']}, Theorem \ref{['t:rel-2-term-functors']}, and Corollary \ref{['c:kerG']}
• Proposition 2.1
• Definition 2.2: cf. BTS
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Theorem 2.8
• proof