The Homotopy Category of Strongly flat modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913775424110592 |
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| author | Asadollahi, Javad Sadeghi, Somayeh |
| author_facet | Asadollahi, Javad Sadeghi, Somayeh |
| contents | In this paper, we plan to build upon significant results by Amnon Neeman regarding the homotopy category of flat modules to study ${\mathbb{K}}({S\rm{SF}}\mbox{-}R)$, the homotopy category of $S$-strongly flat modules, where $S$ is a multiplicatively closed subset of a commutative ring $R$. The category ${\mathbb{K}}({S\rm{SF}}\mbox{-}R)$ is an intermediate triangulated category that includes ${\mathbb{K}}({\rm{Prj}\mbox{-}} R)$, the homotopy category of projective $R$-modules, which is always well generated by a result of Neeman, and is included in ${\mathbb{K}}({\rm{Flat}}\mbox{-} R)$, the homotopy category of flat $R$-modules, which is well generated if and only if $R$ is perfect, by a result of Štovíček. We analyze corresponding inclusion functors and the existence of their adjoints. In this way, we provide a new, fully faithful embedding of the homotopy category of projectives to the homotopy category of $S$-strongly flat modules. We introduce the notion of $S$-almost well generated triangulated categories. If $R$ is an $S$-almost perfect ring, ${\mathbb{K}}({\rm{Flat}}\mbox{-} R)$ is $S$-almost well generated. We show that the converse is true under certain conditions on the ring $R$. We hope that this approach provides insights into the largely mysterious class of $S$-strongly flat modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02601 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Homotopy Category of Strongly flat modules Asadollahi, Javad Sadeghi, Somayeh Commutative Algebra Representation Theory 13B30, 13D05, 18N55, 18A40, 18G80, 16G20 In this paper, we plan to build upon significant results by Amnon Neeman regarding the homotopy category of flat modules to study ${\mathbb{K}}({S\rm{SF}}\mbox{-}R)$, the homotopy category of $S$-strongly flat modules, where $S$ is a multiplicatively closed subset of a commutative ring $R$. The category ${\mathbb{K}}({S\rm{SF}}\mbox{-}R)$ is an intermediate triangulated category that includes ${\mathbb{K}}({\rm{Prj}\mbox{-}} R)$, the homotopy category of projective $R$-modules, which is always well generated by a result of Neeman, and is included in ${\mathbb{K}}({\rm{Flat}}\mbox{-} R)$, the homotopy category of flat $R$-modules, which is well generated if and only if $R$ is perfect, by a result of Štovíček. We analyze corresponding inclusion functors and the existence of their adjoints. In this way, we provide a new, fully faithful embedding of the homotopy category of projectives to the homotopy category of $S$-strongly flat modules. We introduce the notion of $S$-almost well generated triangulated categories. If $R$ is an $S$-almost perfect ring, ${\mathbb{K}}({\rm{Flat}}\mbox{-} R)$ is $S$-almost well generated. We show that the converse is true under certain conditions on the ring $R$. We hope that this approach provides insights into the largely mysterious class of $S$-strongly flat modules. |
| title | The Homotopy Category of Strongly flat modules |
| topic | Commutative Algebra Representation Theory 13B30, 13D05, 18N55, 18A40, 18G80, 16G20 |
| url | https://arxiv.org/abs/2504.02601 |