Reduced points of $\mathbb{E}_{\infty}$-rings in positive characteristic

TL;DR

This paper explores the existence of reduced points in $ ext{E}_ ext{infinity}$-rings over fields of positive characteristic, revealing that such points exist when 2 is not invertible but fail at odd primes, with implications for module construction.

Contribution

It demonstrates the conditions under which reduced points exist in $ ext{E}_ ext{infinity}$-rings in positive characteristic, especially highlighting the failure at odd primes.

Findings

Reduced points exist when 2 is not invertible in $ ext{E}_ ext{infinity}$-rings.

Construction of non-zero $ ext{E}_ ext{infinity}$-rings over $ ext{F}_p$ with no maps to $ ext{E}_2$-algebras with field $ ext{pi}_0$ at odd primes.

Implication that free modules cannot be built from fewer cells in certain cases.

Abstract

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 $\pi_{\ast}T$ is a graded field. We show that if $2\in \pi_0A$ 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 $\pi_0T$ is a field.