Higher Auslander-Reiten sequences revisited
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929524728397824 |
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| author | He, Jian Yin, Hangyu Zhou, Panyue |
| author_facet | He, Jian Yin, Hangyu Zhou, Panyue |
| contents | Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category with enough projectives and enough injectives, and $\mathscr{X}$ be a cluster-tilting subcategory of $\mathscr{C}$. Liu and Zhou have shown that the quotient category $\mathscr{C}/\mathscr{X}$ is an $n$-abelian category. In this paper, we prove that if $\mathscr{C}$ has Auslander-Reiten $n$-exangles, then $\mathscr{C}/\mathscr{X}$ has Auslander-Reiten $n$-exact sequences. Moreover, we also show that if a Frobenius $n$-exangulated category $\mathscr{C}$ has Auslander-Reiten $n$-exangles, then the stable category $\overline{\mathscr{C}}$ of $\mathscr{C}$ has Auslander-Reiten $(n+2)$-angles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01834 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher Auslander-Reiten sequences revisited He, Jian Yin, Hangyu Zhou, Panyue Representation Theory Category Theory Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category with enough projectives and enough injectives, and $\mathscr{X}$ be a cluster-tilting subcategory of $\mathscr{C}$. Liu and Zhou have shown that the quotient category $\mathscr{C}/\mathscr{X}$ is an $n$-abelian category. In this paper, we prove that if $\mathscr{C}$ has Auslander-Reiten $n$-exangles, then $\mathscr{C}/\mathscr{X}$ has Auslander-Reiten $n$-exact sequences. Moreover, we also show that if a Frobenius $n$-exangulated category $\mathscr{C}$ has Auslander-Reiten $n$-exangles, then the stable category $\overline{\mathscr{C}}$ of $\mathscr{C}$ has Auslander-Reiten $(n+2)$-angles. |
| title | Higher Auslander-Reiten sequences revisited |
| topic | Representation Theory Category Theory |
| url | https://arxiv.org/abs/2410.01834 |