Sequences of ICE-closed subcategories via preordered $τ^{-1}$-rigid modules
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913528814764032 |
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| author | Hanson, Eric J. |
| author_facet | Hanson, Eric J. |
| contents | Let $Λ$ be a finite-dimensional basic algebra. Sakai recently used certain sequences of image-cokernel-extension-closed (ICE-closed) subcategories of finitely generated $Λ$-modules to classify certain (generalized) intermediate $t$-structures in the bounded derived category. We classifying these "contravariantly finite ICE-sequences" using concepts from $τ$-tilting theory. More precisely, we introduce "cogen-preordered $τ^{-1}$-rigid modules" as a generalization of (the dual of) the "TF-ordered $τ$-rigid modules" of Mendoza and Treffinger. We then establish a bijection between the set of cogen-preordered $τ^{-1}$-rigid modules and certain sequences of intervals of torsion-free classes. Combined with the results of Sakai, this yields a bijection with the set of contravariantly finite ICE-sequences (of finite length), and thus also with the set of $(m+1)$-intermediate $t$-structures whose aisles are homology-determined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01963 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sequences of ICE-closed subcategories via preordered $τ^{-1}$-rigid modules Hanson, Eric J. Representation Theory 16G10, 16G20 (primary), 16E35, 18G80 (secondary) Let $Λ$ be a finite-dimensional basic algebra. Sakai recently used certain sequences of image-cokernel-extension-closed (ICE-closed) subcategories of finitely generated $Λ$-modules to classify certain (generalized) intermediate $t$-structures in the bounded derived category. We classifying these "contravariantly finite ICE-sequences" using concepts from $τ$-tilting theory. More precisely, we introduce "cogen-preordered $τ^{-1}$-rigid modules" as a generalization of (the dual of) the "TF-ordered $τ$-rigid modules" of Mendoza and Treffinger. We then establish a bijection between the set of cogen-preordered $τ^{-1}$-rigid modules and certain sequences of intervals of torsion-free classes. Combined with the results of Sakai, this yields a bijection with the set of contravariantly finite ICE-sequences (of finite length), and thus also with the set of $(m+1)$-intermediate $t$-structures whose aisles are homology-determined. |
| title | Sequences of ICE-closed subcategories via preordered $τ^{-1}$-rigid modules |
| topic | Representation Theory 16G10, 16G20 (primary), 16E35, 18G80 (secondary) |
| url | https://arxiv.org/abs/2410.01963 |