Vol. 310, No. 1, 2021

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Properties of triangulated and quotient categories arising from $n$-Calabi–Yau triples

Francesca Fedele

Vol. 310 (2021), No. 1, 1–21
Abstract

The original definition of cluster algebras by Fomin and Zelevinsky has been categorified and generalised in several ways over the course of the past 20 years, giving rise to cluster theory. This study lead to Iyama and Yang’s generalised cluster categories 𝒯 𝒯fd coming from n-Calabi–Yau triples (𝒯 ,𝒯fd,). In this paper, we use some classic tools of homological algebra to give a deeper understanding of such categories 𝒯 𝒯fd.

Let k be a field, n 3 an integer and 𝒯 a k-linear triangulated category with a triangulated subcategory 𝒯fd and a subcategory = add(M) such that (𝒯 ,𝒯fd,) is an n-Calabi–Yau triple. For every integer m and every object X in 𝒯, there is a unique, up to isomorphism, truncation triangle of the form

Xm X Xm+1 ΣXm,

with respect to the t-structure (Σ<m)𝒯,(Σ>m)𝒯. In this paper, we prove some properties of the triangulated categories 𝒯 and 𝒯 𝒯fd. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a gap theorem in 𝒯, showing when the truncation triangles split.

Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg k-algebras A and subcategories of the derived category of dg A-modules. This proves that 𝒯 𝒯fd is Hom-finite and (n1)-Calabi–Yau, its object M is (n1)-cluster tilting and the endomorphism algebras of M over 𝒯 and over 𝒯 𝒯fd are isomorphic. Note that these properties make 𝒯 𝒯fd a generalisation of the cluster category.

Keywords
Hom-spaces, limits and colimits, $(n{-}1)$-Calabi–Yau, $(n{-}1)$-cluster tilting, $n$-Calabi–Yau triple, quotient categories, truncation triangles
Mathematical Subject Classification
Primary: 16E45, 18A30, 18G80
Milestones
Received: 13 July 2020
Revised: 25 November 2020
Accepted: 28 November 2020
Published: 26 January 2021
Authors
Francesca Fedele
School of Mathematics, Statistics and Physics
Newcastle University
Newcastle upon Tyne
United Kingdom