Nilpotence of $η$ in étale motivic spectra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911310053113856 |
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| author | Mattis, Klaus Tubach, Swann |
| author_facet | Mattis, Klaus Tubach, Swann |
| contents | We show that every object of the stable étale motivic homotopy category over any scheme is $η$-complete. In some cases we show that in fact the fourth power of $η$ is null, whereas the third power of $η$ is always nonvanishing, similar to the situation in topology. Moreover, we prove an étale version of May's nilpotence conjecture, that states that $H\mathbb{Z} \in \mathrm{Sp}$ detects the vanishing of $\mathbf{E}_\infty$-rings. We use this to show a version of Nishida's nilpotence theorem in $\mathrm{SH}_{\operatorname{\acute{e}t}}(S)$, i.e. that any positive degree self map of the unit is nilpotent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_09476 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nilpotence of $η$ in étale motivic spectra Mattis, Klaus Tubach, Swann Algebraic Geometry Algebraic Topology We show that every object of the stable étale motivic homotopy category over any scheme is $η$-complete. In some cases we show that in fact the fourth power of $η$ is null, whereas the third power of $η$ is always nonvanishing, similar to the situation in topology. Moreover, we prove an étale version of May's nilpotence conjecture, that states that $H\mathbb{Z} \in \mathrm{Sp}$ detects the vanishing of $\mathbf{E}_\infty$-rings. We use this to show a version of Nishida's nilpotence theorem in $\mathrm{SH}_{\operatorname{\acute{e}t}}(S)$, i.e. that any positive degree self map of the unit is nilpotent. |
| title | Nilpotence of $η$ in étale motivic spectra |
| topic | Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2511.09476 |