Construction of $d$-abelian categories via derived categories
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arXiv
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| Format: | Preprint |
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2025
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| author | Jorgensen, Peter Sen, Emre |
| author_facet | Jorgensen, Peter Sen, Emre |
| contents | 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 $Σ_n$. We prove that the full subcategory ${\mathcal T}[0,m] = \mathrm{add} \bigoplus^{m}_{j=0}Σ^j_n M$ 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 ${\mathcal T}[0,m]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_17922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction of $d$-abelian categories via derived categories Jorgensen, Peter Sen, Emre Representation Theory 16E05, 16E10, 16G20 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 $Σ_n$. We prove that the full subcategory ${\mathcal T}[0,m] = \mathrm{add} \bigoplus^{m}_{j=0}Σ^j_n M$ 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 ${\mathcal T}[0,m]$. |
| title | Construction of $d$-abelian categories via derived categories |
| topic | Representation Theory 16E05, 16E10, 16G20 |
| url | https://arxiv.org/abs/2509.17922 |