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ℙ1 Homotopy Theory of Schemes

  • Dipankar Maity
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Abstract

We study the ℙ1-local motivic homotopy theory under the Nisnevich topology on SmS. We show that although the homotopy category ℋℙ1(−) admits standard schematic functorialities, it fails six-functor gluing, thereby obstructing a full six-functor formalism. On the unstable side, we prove that the unstable ℙ1-motivic localization preserves Nisnevich connectivity, despite the absence of an interval structure on ℙ1. Over perfect fields, we then establish the ℙ1-invariance of the higher unstable homotopy group sheaves πiℙ1 (i ≥ 1) using the Bloch-Ogus-Gabber theorem. On the stable side, we prove the stable ℙ1-connectivity theorem over fields. As a consequence, we obtain a homotopy t-structure on the stable ∞-category of ℙ1-invariant Nisnevich spectra, and deduce the strict ℙ1-invariance of stable ℙ1-homotopy group sheaves.

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Cite

BibTeXRIS
BibTeX
@misc{maity2026ℙ1,
  title = {{$\mathbbP^1$ Homotopy Theory of Schemes}},
  author = {Dipankar Maity},
  howpublished = {HAL},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2609.26260},
  eprint = {2609.26260},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05736606v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - ℙ^1 Homotopy Theory of Schemes
AU  - Dipankar Maity
PY  - 2026
DA  - 2026/08/22
PB  - HAL
DO  - 10.48550/arxiv.2609.26260
UR  - https://hal.science/hal-05736606v1
AB  - We study the ℙ^1-local motivic homotopy theory under the Nisnevich topology on Sm_S. We show that although the homotopy category ℋ^ℙ^1(−) admits standard schematic functorialities, it fails six-functor gluing, thereby obstructing a full six-functor formalism. On the unstable side, we prove that the unstable ℙ^1-motivic localization preserves Nisnevich connectivity, despite the absence of an interval structure on ℙ^1. Over perfect fields, we then establish the ℙ^1-invariance of the higher unstable homotopy group sheaves π_i^ℙ^1 (i ≥ 1) using the Bloch-Ogus-Gabber theorem. On the stable side, we prove the stable ℙ^1-connectivity theorem over fields. As a consequence, we obtain a homotopy t-structure on the stable ∞-category of ℙ^1-invariant Nisnevich spectra, and deduce the strict ℙ^1-invariance of stable ℙ^1-homotopy group sheaves.
ER  -