Classifying localizing subcategories of a Grothendieck category
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914170629259264 |
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| author | Sazeedeh, Reza |
| author_facet | Sazeedeh, Reza |
| contents | Let $\cA$ be a locally coherent Grothendieck category, $\fp\cA$ be the full subcategory of $\cA$ consisting of finitely presented objects and $\ASpec\cA$ be the atom spectrum of $\cA$. In this paper, we classify localizing subcategories of finite type of $\cA$ via open subsets of $\ASpec\cA$. We investigate $\ASpec\fp\cA$ and show that if $\ASpec\fp\cA=\ASpec\cA$, then $\cA$ is locally noetherian. As an application, we specialize our investigation to the case of commutative coherent rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classifying localizing subcategories of a Grothendieck category Sazeedeh, Reza Category Theory 13C05, 18E10, 18E35 Let $\cA$ be a locally coherent Grothendieck category, $\fp\cA$ be the full subcategory of $\cA$ consisting of finitely presented objects and $\ASpec\cA$ be the atom spectrum of $\cA$. In this paper, we classify localizing subcategories of finite type of $\cA$ via open subsets of $\ASpec\cA$. We investigate $\ASpec\fp\cA$ and show that if $\ASpec\fp\cA=\ASpec\cA$, then $\cA$ is locally noetherian. As an application, we specialize our investigation to the case of commutative coherent rings. |
| title | Classifying localizing subcategories of a Grothendieck category |
| topic | Category Theory 13C05, 18E10, 18E35 |
| url | https://arxiv.org/abs/2507.13749 |