Unstable étale motives
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912504534269952 |
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| author | Mattis, Klaus |
| author_facet | Mattis, Klaus |
| contents | We prove a rigidity result for certain $p$-complete étale $\mathbf{A}^{1}$-invariant sheaves
of anima over a qcqs finite-dimensional base scheme $S$ of bounded étale cohomological dimension with $p$ invertible on $S$.
This generalizes results of
Suslin--Voevodsky, Ayoub, Cisinski--Déglise, and Bachmann
to the unstable setting. Over a perfect field we exhibit a large class of sheaves to which our main theorem applies,
in particular the $p$-completion of the étale sheafification of any
$2$-effective $2$-connective motivic space,
as well as the $p$-completion of any $4$-connective $\mathbf{A}^{1}$-invariant étale sheaf.
We use this rigidity result to prove (a weaker version of) an étale analog of Morel's
theorem stating that for a Nisnevich sheaf of abelian groups, strong $\mathbf{A}^{1}$-invariance implies strict $\mathbf{A}^{1}$-invariance.
Moreover, this allows us to construct an unstable étale realization functor on $2$-effective $2$-connective motivic spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20320 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unstable étale motives Mattis, Klaus Algebraic Geometry Algebraic Topology We prove a rigidity result for certain $p$-complete étale $\mathbf{A}^{1}$-invariant sheaves of anima over a qcqs finite-dimensional base scheme $S$ of bounded étale cohomological dimension with $p$ invertible on $S$. This generalizes results of Suslin--Voevodsky, Ayoub, Cisinski--Déglise, and Bachmann to the unstable setting. Over a perfect field we exhibit a large class of sheaves to which our main theorem applies, in particular the $p$-completion of the étale sheafification of any $2$-effective $2$-connective motivic space, as well as the $p$-completion of any $4$-connective $\mathbf{A}^{1}$-invariant étale sheaf. We use this rigidity result to prove (a weaker version of) an étale analog of Morel's theorem stating that for a Nisnevich sheaf of abelian groups, strong $\mathbf{A}^{1}$-invariance implies strict $\mathbf{A}^{1}$-invariance. Moreover, this allows us to construct an unstable étale realization functor on $2$-effective $2$-connective motivic spaces. |
| title | Unstable étale motives |
| topic | Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2507.20320 |