Logarithmic motives with compact support
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909148357066752 |
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| author | Opdan, Nikolai |
| author_facet | Opdan, Nikolai |
| contents | We develop a theory of motives with compact support for logarithmic schemes over a field. Starting from the notion of finite logarithmic correspondences with compact support, we define the logarithmic motive with compact support analogous to the classical case for schemes. We then establish an analog of a Gysin sequence and, assuming resolution of singularities, a K{ü}nneth formula. This implies that our theory is $\overline{\square}$-invariant, which presents a critical feature that is absent in the classical case. Further assuming resolution of singularities, we prove a duality theorem for log schemes which we use to establish a cancellation theorem for log schemes whose underlying scheme is proper. Moreover, we discuss new homology and cohomology theories for log smooth fs logarithmic schemes based on our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01099 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Logarithmic motives with compact support Opdan, Nikolai Algebraic Geometry K-Theory and Homology 14A21, 14F42 (Primary) 14C15, 19E15 (Secondary) We develop a theory of motives with compact support for logarithmic schemes over a field. Starting from the notion of finite logarithmic correspondences with compact support, we define the logarithmic motive with compact support analogous to the classical case for schemes. We then establish an analog of a Gysin sequence and, assuming resolution of singularities, a K{ü}nneth formula. This implies that our theory is $\overline{\square}$-invariant, which presents a critical feature that is absent in the classical case. Further assuming resolution of singularities, we prove a duality theorem for log schemes which we use to establish a cancellation theorem for log schemes whose underlying scheme is proper. Moreover, we discuss new homology and cohomology theories for log smooth fs logarithmic schemes based on our results. |
| title | Logarithmic motives with compact support |
| topic | Algebraic Geometry K-Theory and Homology 14A21, 14F42 (Primary) 14C15, 19E15 (Secondary) |
| url | https://arxiv.org/abs/2301.01099 |