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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.23767 |
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| _version_ | 1866918233117818880 |
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| author | De Deyn, Timothy Miller, Sam K. |
| author_facet | De Deyn, Timothy Miller, Sam K. |
| contents | We prove that, given the Balmer spectrum of any essentially small monoidal-triangulated category, one has a classification of semiprime thick tensor-ideals arising in terms of a "pseudo-Hochster-dual" of the noncommutative Balmer spectrum. This extends Balmer's classification of radical thick tensor-ideals to noncommutative tensor-triangular geometry. To achieve this, we utilize the notion of support data for lattices and frames, under which the classification follows via Stone duality. We also give a characterization for when the noncommutative Balmer spectrum behaves as it does in tensor-triangular geometry, that is, when it is a spectral space with quasi-compact opens given by complements of supports. Finally, we show that rigid centrally generated monoidal-triangulated categories satisfy this property, and we answer a question posed by Negron--Pevtsova regarding classification of one-sided tensor-ideals via cohomological support. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23767 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Re-framing the classification of ideals in noncommutative tensor-triangular geometry De Deyn, Timothy Miller, Sam K. Category Theory Representation Theory 18G80, 06D22, 18F70, 18M05 We prove that, given the Balmer spectrum of any essentially small monoidal-triangulated category, one has a classification of semiprime thick tensor-ideals arising in terms of a "pseudo-Hochster-dual" of the noncommutative Balmer spectrum. This extends Balmer's classification of radical thick tensor-ideals to noncommutative tensor-triangular geometry. To achieve this, we utilize the notion of support data for lattices and frames, under which the classification follows via Stone duality. We also give a characterization for when the noncommutative Balmer spectrum behaves as it does in tensor-triangular geometry, that is, when it is a spectral space with quasi-compact opens given by complements of supports. Finally, we show that rigid centrally generated monoidal-triangulated categories satisfy this property, and we answer a question posed by Negron--Pevtsova regarding classification of one-sided tensor-ideals via cohomological support. |
| title | Re-framing the classification of ideals in noncommutative tensor-triangular geometry |
| topic | Category Theory Representation Theory 18G80, 06D22, 18F70, 18M05 |
| url | https://arxiv.org/abs/2510.23767 |