Motivic homotopy theory with ramification filtrations
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915225078333440 |
|---|---|
| author | Koizumi, Junnosuke Miyazaki, Hiroyasu Saito, Shuji |
| author_facet | Koizumi, Junnosuke Miyazaki, Hiroyasu Saito, Shuji |
| contents | The aim of this paper is to connect two important and apparently unrelated theories: motivic homotopy theory and ramification theory. We construct motivic homotopy categories over a qcqs base scheme $S$, in which cohomology theories with ramification filtrations are representable. Every such cohomology theory enjoys basic properties such as the Nisnevich descent, the cube-invariance, the blow-up invariance, the smooth blow-up excision, the Gysin sequence, the projective bundle formula and the Thom isomorphism. In case $S$ is the spectrum of a perfect field, the cohomology of every reciprocity sheaf is upgraded to a cohomology theory with a ramification filtration represented in our categories. We also address relations of our theory with other non-$\mathbb{A}^1$-invariant motivic homotopy theories such as the logarithmic motivic homotopy theory of Binda, Park, and Østvær and the theory of motivic spectra of Annala-Iwasa. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Motivic homotopy theory with ramification filtrations Koizumi, Junnosuke Miyazaki, Hiroyasu Saito, Shuji Algebraic Geometry K-Theory and Homology 14F42 (13F35, 14F30, 19E15, 11S15, 14E22) The aim of this paper is to connect two important and apparently unrelated theories: motivic homotopy theory and ramification theory. We construct motivic homotopy categories over a qcqs base scheme $S$, in which cohomology theories with ramification filtrations are representable. Every such cohomology theory enjoys basic properties such as the Nisnevich descent, the cube-invariance, the blow-up invariance, the smooth blow-up excision, the Gysin sequence, the projective bundle formula and the Thom isomorphism. In case $S$ is the spectrum of a perfect field, the cohomology of every reciprocity sheaf is upgraded to a cohomology theory with a ramification filtration represented in our categories. We also address relations of our theory with other non-$\mathbb{A}^1$-invariant motivic homotopy theories such as the logarithmic motivic homotopy theory of Binda, Park, and Østvær and the theory of motivic spectra of Annala-Iwasa. |
| title | Motivic homotopy theory with ramification filtrations |
| topic | Algebraic Geometry K-Theory and Homology 14F42 (13F35, 14F30, 19E15, 11S15, 14E22) |
| url | https://arxiv.org/abs/2504.02223 |