Spectrum of an abelian category via premonoform objects
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909492934868992 |
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| author | Sazeedeh, Reza |
| author_facet | Sazeedeh, Reza |
| contents | Let $\cA$ be an abelian category. In this paper, we study ${\rm (n)PSpec}\cA$, a topological space formed by equivalence classes derived from an equivalence relation on (noetherian) premonoform objects. We classify torsion classes of $\cA$ via closed subclasses of $\nPSpec\cA$. We introduce a new topology on $\PSpec\cA$ and we classify Serre subcategories of $\noeth A$ and localizing subcategories of $\cA$ using this topology. If $A$ is a commutative noetherian ring, we show that $\nPSpec A$ is homeomorphic to $\Spec A$. Moreover, there is a one-to-one correspondence between the closed subsets of $\nPSpec A$ and the open subsets of $\ASpec A$, the atom spectrum of $A$. Finally, we explore the relationships between the new subctegories of $\Mod A$ and subsets of $\nPSpec A$ introduced in this paper, and the known subcategories of $\Mod A$ and subsets of other spectra of $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02659 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectrum of an abelian category via premonoform objects Sazeedeh, Reza Category Theory 18E10, 13C60, 16D20 Let $\cA$ be an abelian category. In this paper, we study ${\rm (n)PSpec}\cA$, a topological space formed by equivalence classes derived from an equivalence relation on (noetherian) premonoform objects. We classify torsion classes of $\cA$ via closed subclasses of $\nPSpec\cA$. We introduce a new topology on $\PSpec\cA$ and we classify Serre subcategories of $\noeth A$ and localizing subcategories of $\cA$ using this topology. If $A$ is a commutative noetherian ring, we show that $\nPSpec A$ is homeomorphic to $\Spec A$. Moreover, there is a one-to-one correspondence between the closed subsets of $\nPSpec A$ and the open subsets of $\ASpec A$, the atom spectrum of $A$. Finally, we explore the relationships between the new subctegories of $\Mod A$ and subsets of $\nPSpec A$ introduced in this paper, and the known subcategories of $\Mod A$ and subsets of other spectra of $A$. |
| title | Spectrum of an abelian category via premonoform objects |
| topic | Category Theory 18E10, 13C60, 16D20 |
| url | https://arxiv.org/abs/2501.02659 |