Triangulated Categories Admitting Linear Generators

TL;DR

The paper addresses the problem of understanding infinite-rank triangulated categories that admit linear generators, focusing on the Paquette–Yıldırım completions $\overline{\mathcal{C}}_{n}$ of discrete cluster categories of Dynkin type $A_{\infty}$. It develops a DG-endomorphism framework: given a minimal linear generator $G$, the category $\mathcal{T}$ is additively equivalent to $\mathrm{Perf}(\Lambda^{G})$ for a DG algebra $\Lambda^{G}$ with underlying graded algebra $\chi^{G}$, and this equivalence preserves suspension and triangles with at least two indecomposable terms. The results yield explicit descriptions of indecomposables, triangles, and factorisations, and establish that the Rouquier dimension is at most one in these settings; moreover, any triangulated category with a linear generator embeds additively into some $\overline{\mathcal{C}}_{n}$. The constructions connect abstract triangulated-category theory to explicit graded-path-algebra models, enabling concrete descriptions via the graded quiver $Q$ and relations $I$. These findings provide a robust bridge between algebraic, combinatorial, and geometric viewpoints for infinite-rank cluster-like categories and their DG-endomorphism algebras.

Abstract

The main result of this paper is that there is an additive equivalence between $\overline{\mathcal{C}}_n$, the Paquette-Yildirim completion of the discrete cluster categories of Dynkin type $A_{\infty}$, and the perfect derived category of a certain DG algebra. This additive equivalence preserves some of the triangulated structure: it commutes with the suspension functor and preserves triangles with at least two indecomposable terms. In the process, we introduce the notion of a linear generator $G$ in a Krull-Schmidt, Hom-finite triangulated category. It turns out that the existence of a linear generator affords a large amount of control over $\mathcal{T}$. For example, it allows us to describe all indecomposable objects in $\mathcal{T}$ in terms of $G$, to determine all triangles of $\mathcal{T}$ with at least two indecomposable objects, and to show that the Rouquier dimension of $\mathcal{T}$ is at most one. Moreover, we prove that there is an additive equivalence (which preserves some of the triangulated structure) between $\mathcal{T}$ and the perfect derived category of a certain DG algebra. Finally, we show that any triangulated category with a linear generator is additively equivalent to a thick subcategory of $\overline{\mathcal{C}}_n$.

Figures (4)

• Figure 1: An admissible subset $\mathscr{M}$ of $S^1$. The marked points in $\mathscr{M}$ converge to the accumulation points represented as small circles, and each marked point $x$ has both a predecessor and a successor, labelled $x^-$ and $x^+$ respectively.
• Figure 2: Four arcs of $\overline{\mathscr{M}}_{}$, where $\ell_X$ is a short arc, $\ell_Y$ is a long arc, $\ell_W$ is a limit arc, and $\ell_Z$ is a double limit arc.
• Figure 3: The arcs corresponding to triangles coming from non-split extensions between indecomposable objects.
• Figure 4: A fan generator of $\overline{\EuScript{C}}_{n}$.

Theorems & Definitions (93)

• Theorem 2: Theorem \ref{['Thm1: F preserves 2 inds']}
• Proposition 3: Proposition \ref{['Prop: Endo Alg']}
• Theorem 4: Theorem \ref{['Thm: T is equivalent to perf']}
• Theorem 5: Theorem \ref{['Thm:path algebras']}
• Theorem 6: Theorem \ref{['Thm: Add Equiv']}
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.5
• proof