Triangulated Categories Admitting Linear Generators

Abstract

We introduce the notion of a $w$-linear generator in a $k$-linear, Hom-finite, Krull-Schmidt algebraic triangulated category $\mathcal{T}$ over a field $k$, for some positive integer $w$. We show that the existence of a $w$-linear generator gives us a complete classification of indecomposable objects and morphisms spaces in $\mathcal{T}$, as well as the factorisation of morphisms. We compute the graded endomorphism ring $\Lambda$ of a $w$-linear generator viewed as a differential graded algebra with trivial differential, and the Hochschild cohomology of $\Lambda$. It turns out that the graded endomorphism ring of a $w$-linear generator is intrinsically formal, and so $\mathcal{T}$ is triangle equivalent to the perfect derived category of $\Lambda$. As a application of these results, we find a description of the Paquette--Yıldırım completion of discrete cluster categories of type $A_{\infty }$ as a perfect derived category. Further, we confirm a conjecture of Franchini in her PhD thesis, and show that the $(w+1)$-Calabi--Yau candidate categories she introduces are in fact triangulated categories.

arXiv comment41 pages, comments welcome. v2 has significant updates on v1, including a generalisation to w-linear generators, and constructing a triangulated rather than additive equivalence