Triangulated categories arising from n-fold matrix factorizations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912730621935616 |
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| author | Zhang, Yixia Zhou, Panyue |
| author_facet | Zhang, Yixia Zhou, Panyue |
| contents | Let $\mathcal{A}$ be an additive category and let $T\colon \mathcal{A}\rightarrow \mathcal{A}$ be an additive functor equipped with a natural transformation $ω\colon \mathrm{Id}_{\mathcal{A}}\rightarrow T$. We prove that the homotopy category of $n$-fold matrix factorizations of $ω$, denoted ${\rm HFact}_{n}(\mathcal{A},T,ω)$, admits a natural structure of a right triangulated category. In particular, when $T$ is an automorphism, the homotopy category ${\rm HFact}_{n}(\mathcal{A},T,ω)$ becomes triangulated. Furthermore, if $\mathcal{A}$ is a Frobenius exact category and $T$ is an autoequivalence, we obtain that the category ${\rm Fact}_{n}(\mathcal{A},T,ω)$ of $n$-fold $(\mathcal{A},T)$-factorizations of $ω$ is a Frobenius exact category. Consequently, the stable category of the Frobenius exact category ${\rm Fact}_{n}(\mathcal{A},T,ω)$ is a triangulated category. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21379 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Triangulated categories arising from n-fold matrix factorizations Zhang, Yixia Zhou, Panyue Representation Theory Category Theory Let $\mathcal{A}$ be an additive category and let $T\colon \mathcal{A}\rightarrow \mathcal{A}$ be an additive functor equipped with a natural transformation $ω\colon \mathrm{Id}_{\mathcal{A}}\rightarrow T$. We prove that the homotopy category of $n$-fold matrix factorizations of $ω$, denoted ${\rm HFact}_{n}(\mathcal{A},T,ω)$, admits a natural structure of a right triangulated category. In particular, when $T$ is an automorphism, the homotopy category ${\rm HFact}_{n}(\mathcal{A},T,ω)$ becomes triangulated. Furthermore, if $\mathcal{A}$ is a Frobenius exact category and $T$ is an autoequivalence, we obtain that the category ${\rm Fact}_{n}(\mathcal{A},T,ω)$ of $n$-fold $(\mathcal{A},T)$-factorizations of $ω$ is a Frobenius exact category. Consequently, the stable category of the Frobenius exact category ${\rm Fact}_{n}(\mathcal{A},T,ω)$ is a triangulated category. |
| title | Triangulated categories arising from n-fold matrix factorizations |
| topic | Representation Theory Category Theory |
| url | https://arxiv.org/abs/2511.21379 |