A new dimension for Grothendieck categories via the atom spectrum
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866917218171748352 |
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| author | Alipour, Negar Sazeedeh, Reza |
| author_facet | Alipour, Negar Sazeedeh, Reza |
| contents | In this paper, we define a new dimension for objects in a Grothendieck category $\mathcal{A}$. We show that it serves as a lower bound for Gabriel-Krull dimension and under certain conditions, the two dimensions coincide. We carry out our investigation for a fully right bounded ring $A$. We introduce a new spectrum Comp$\,A$ via compressible right $A$-modules. In analogy with dimension theory for commutative rings, we show that the Krull dimension of right $A$-modules can be computed via the length of chain of prime ideals of $A$ and also the length of chain of elements of Comp$\,A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_03688 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A new dimension for Grothendieck categories via the atom spectrum Alipour, Negar Sazeedeh, Reza Category Theory 16P60, 18E10, 18E35 In this paper, we define a new dimension for objects in a Grothendieck category $\mathcal{A}$. We show that it serves as a lower bound for Gabriel-Krull dimension and under certain conditions, the two dimensions coincide. We carry out our investigation for a fully right bounded ring $A$. We introduce a new spectrum Comp$\,A$ via compressible right $A$-modules. In analogy with dimension theory for commutative rings, we show that the Krull dimension of right $A$-modules can be computed via the length of chain of prime ideals of $A$ and also the length of chain of elements of Comp$\,A$. |
| title | A new dimension for Grothendieck categories via the atom spectrum |
| topic | Category Theory 16P60, 18E10, 18E35 |
| url | https://arxiv.org/abs/2003.03688 |