Local characterization of block-decomposability for multiparameter persistence modules
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912151133749248 |
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| author | Lebovici, Vadim Lerch, Jan-Paul Oudot, Steve |
| author_facet | Lebovici, Vadim Lerch, Jan-Paul Oudot, Steve |
| contents | Local conditions for the direct summands of a persistence module to belong to a certain class of indecomposables have been proposed in the 2-parameter setting, notably for the class of indecomposables called block modules, which plays a prominent role in levelset persistence. Here we generalize the local condition for decomposability into block modules to the n-parameter setting, and prove a corresponding structure theorem. Our result holds in the generality of pointwise finite-dimensional modules over finite products of arbitrary totally ordered sets. Our proof extends the one by Botnan and Crawley-Boevey from 2 to n parameters, which requires some crucial adaptations at places where their proof is fundamentally tied to the 2-parameter setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16624 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local characterization of block-decomposability for multiparameter persistence modules Lebovici, Vadim Lerch, Jan-Paul Oudot, Steve Representation Theory Algebraic Topology Local conditions for the direct summands of a persistence module to belong to a certain class of indecomposables have been proposed in the 2-parameter setting, notably for the class of indecomposables called block modules, which plays a prominent role in levelset persistence. Here we generalize the local condition for decomposability into block modules to the n-parameter setting, and prove a corresponding structure theorem. Our result holds in the generality of pointwise finite-dimensional modules over finite products of arbitrary totally ordered sets. Our proof extends the one by Botnan and Crawley-Boevey from 2 to n parameters, which requires some crucial adaptations at places where their proof is fundamentally tied to the 2-parameter setting. |
| title | Local characterization of block-decomposability for multiparameter persistence modules |
| topic | Representation Theory Algebraic Topology |
| url | https://arxiv.org/abs/2402.16624 |