Pure minimal injective resolutions and perfect modules for lattices
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914176897646592 |
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| author | Gottesman, Tal Klász, Viktória Kleinau, Markus Marczinzik, Rene |
| author_facet | Gottesman, Tal Klász, Viktória Kleinau, Markus Marczinzik, Rene |
| contents | In a recent article, Iyama and Marczinzik showed that a lattice is distributive if and only if the incidence algebra is Auslander regular, giving a new connection between homological algebra and lattice theory. In this article we study when a distributive lattice has a pure minimal injective coresolution, a notion first introduced and studied in a work of Ajitabh, Smith and Zhang. We will see that this problem naturally leads to studying when certain antichain modules are perfect modules. We give a classification of perfect antichain modules under the assumption that their canonical antichain resolution is minimal and use this to give a completion classification in lattice theoretic terms of incidence algebras of distributive lattices with pure minimal injective coresolution. We use our results to answer a question raised by Ajitabh, Smith and Zhang by showing that there exist Auslander-Gorenstein polynomial identity rings without a pure injective coresolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03385 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pure minimal injective resolutions and perfect modules for lattices Gottesman, Tal Klász, Viktória Kleinau, Markus Marczinzik, Rene Representation Theory Combinatorics 16G10, 16E10, 06A11 In a recent article, Iyama and Marczinzik showed that a lattice is distributive if and only if the incidence algebra is Auslander regular, giving a new connection between homological algebra and lattice theory. In this article we study when a distributive lattice has a pure minimal injective coresolution, a notion first introduced and studied in a work of Ajitabh, Smith and Zhang. We will see that this problem naturally leads to studying when certain antichain modules are perfect modules. We give a classification of perfect antichain modules under the assumption that their canonical antichain resolution is minimal and use this to give a completion classification in lattice theoretic terms of incidence algebras of distributive lattices with pure minimal injective coresolution. We use our results to answer a question raised by Ajitabh, Smith and Zhang by showing that there exist Auslander-Gorenstein polynomial identity rings without a pure injective coresolution. |
| title | Pure minimal injective resolutions and perfect modules for lattices |
| topic | Representation Theory Combinatorics 16G10, 16E10, 06A11 |
| url | https://arxiv.org/abs/2511.03385 |