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

Francesca Fedele

Vol. 310 (2021), No. 1, 1–21

DOI: 10.2140/pjm.2021.310.1

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 $\mathcal{𝒯}∕{\mathcal{𝒯}}^{fd}$ coming from $n$-Calabi–Yau triples $\left(\mathcal{𝒯},{\mathcal{𝒯}}^{fd},\mathcal{ℳ}\right)$. In this paper, we use some classic tools of homological algebra to give a deeper understanding of such categories $\mathcal{𝒯}∕{\mathcal{𝒯}}^{fd}$.

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

$$X^{\le m}\to X\to X^{\ge m+1}\to \Sigma X^{\le m},$$

with respect to the $t$-structure $\left(\right.{\left({\Sigma }^{<-m}\mathcal{ℳ}\right)}^{{\perp }_{\mathcal{𝒯}}},{\left({\Sigma }^{>-m}\mathcal{ℳ}\right)}^{{\perp }_{\mathcal{𝒯}}}\left)\right.$. In this paper, we prove some properties of the triangulated categories $\mathcal{𝒯}$ and $\mathcal{𝒯}∕{\mathcal{𝒯}}^{fd}$. 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 $\mathcal{𝒯}$, 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 $\mathcal{𝒯}∕{\mathcal{𝒯}}^{fd}$ is Hom-finite and $\left(n-1\right)$-Calabi–Yau, its object $M$ is $\left(n-1\right)$-cluster tilting and the endomorphism algebras of $M$ over $\mathcal{𝒯}$ and over $\mathcal{𝒯}∕{\mathcal{𝒯}}^{fd}$ are isomorphic. Note that these properties make $\mathcal{𝒯}∕{\mathcal{𝒯}}^{fd}$ 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