Reduced points of $\mathbb{E}_{\infty}$-rings in positive characteristic
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911109361958912 |
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| author | Riedel, Florian |
| author_facet | Riedel, Florian |
| contents | We investigate whether an arbitrary non-zero $\mathbb{E}_\infty$-ring $A$ admits a reduced point, meaning an $\mathbb{E}_\infty$-map $A\to T$ such that $π_{\ast}T$ is a graded field. We show that if $2\in π_0A$ is not invertible, then $A$ admits a reduced point and as an application deduce that a free $A$-module on $n$ generators cannot be built from $n-1$ many cells. Perhaps surprisingly, the existence of reduced points completely fails at odd primes. More precisely, for any prime $p>2$, we construct a non-zero $\mathbb{E}_\infty$-ring over $\mathbb{F}_p$ which admits no map to an $\mathbb{E}_2$-algebra $T$ such that $π_0T$ is a field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12462 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reduced points of $\mathbb{E}_{\infty}$-rings in positive characteristic Riedel, Florian Algebraic Topology 55P43 We investigate whether an arbitrary non-zero $\mathbb{E}_\infty$-ring $A$ admits a reduced point, meaning an $\mathbb{E}_\infty$-map $A\to T$ such that $π_{\ast}T$ is a graded field. We show that if $2\in π_0A$ is not invertible, then $A$ admits a reduced point and as an application deduce that a free $A$-module on $n$ generators cannot be built from $n-1$ many cells. Perhaps surprisingly, the existence of reduced points completely fails at odd primes. More precisely, for any prime $p>2$, we construct a non-zero $\mathbb{E}_\infty$-ring over $\mathbb{F}_p$ which admits no map to an $\mathbb{E}_2$-algebra $T$ such that $π_0T$ is a field. |
| title | Reduced points of $\mathbb{E}_{\infty}$-rings in positive characteristic |
| topic | Algebraic Topology 55P43 |
| url | https://arxiv.org/abs/2508.12462 |