An Euler Characteristic for Unbounded Chain Complexes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908966626263040 |
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| author | Huettemann, Thomas Kucerovsky, Dan |
| author_facet | Huettemann, Thomas Kucerovsky, Dan |
| contents | We propose a definition of an Euler characteristic for unbounded chain complexes by taking the (usual) Euler characteristics of successively longer parts of the complex, weighted inversely proportional to the length, and passing to the limit. This amounts to taking the limit of the sequence of ranks of homology modules with alternating signs in the sense of the Hölder summation method. We establish the structure of a category with cofibrations and weak equivalences on unbounded complexes for which the infinite Euler characteristic is defined, and show that its Grothendieck group is unusually large (viz., uncountable). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13874 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Euler Characteristic for Unbounded Chain Complexes Huettemann, Thomas Kucerovsky, Dan K-Theory and Homology 18G35 (Primary) 19A99, 19D99, 40G05 (Secondary) We propose a definition of an Euler characteristic for unbounded chain complexes by taking the (usual) Euler characteristics of successively longer parts of the complex, weighted inversely proportional to the length, and passing to the limit. This amounts to taking the limit of the sequence of ranks of homology modules with alternating signs in the sense of the Hölder summation method. We establish the structure of a category with cofibrations and weak equivalences on unbounded complexes for which the infinite Euler characteristic is defined, and show that its Grothendieck group is unusually large (viz., uncountable). |
| title | An Euler Characteristic for Unbounded Chain Complexes |
| topic | K-Theory and Homology 18G35 (Primary) 19A99, 19D99, 40G05 (Secondary) |
| url | https://arxiv.org/abs/2604.13874 |