Construction of $d$-abelian categories via derived categories

Abstract

In this work, we provide a simple way to construct $d$-abelian categories via bounded derived categories for certain values of $d$. Namely, let ${\mathcal C}$ be an abelian category, and let ${\mathcal C}[0,m]$ denote the full subcategory of the bounded derived category of ${\mathcal C}$ whose objects $X$ satisfy that $H_*(X)$ is concentrated in degrees $j$ where $0 \leq j \leq m$. We prove that if ${\mathcal C}$ is hereditary, then ${\mathcal C}[0,m]$ is a $d$-abelian category where $d = 3m + 1$. Beyond offering a uniform method for constructing $d$-abelian categories, this construction allows us to create $d$-abelian categories that exhibit some unexpected properties depending on the choice of the category ${\mathcal C}$. For instance, if ${\mathcal C}$ is the category of abelian groups, then ${\mathcal C}[0,m]$ is a $d$-abelian category which is not $\mathbb{K}$-linear over a field $\mathbb{K}$ but has set indexed products and coproducts. Similarly, if ${\mathcal C}$ is the category of coherent sheaves over certain algebraic curves, then ${\mathcal C}[0,m]$ is a $d$-abelian category without enough injectives. We extend our results to $(n+2)$-angulated categories. Namely, let $M$ be an $n$-cluster tilting object over an $n$-representation finite algebra and let ${\mathcal T}$ be the corresponding $(n+2)$-angulated category with $n$-suspension functor $\Sigma_n$. We prove that the full subcategory ${\mathcal T}[0,m] = \mathrm{add} \bigoplus^{m}_{j=0}\Sigma^j_n M$ is a $d$-abelian category where $d = (n+2)(m+1)-2$. Furthermore, we show that there is a bijection between the functorially finite wide subcategories of $\mathrm{add}\,M$ and the functorially finite repetitive wide subcategories of ${\mathcal T}[0,m]$.