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

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 $\ddot{I}{€}_{\ast }T$ is a graded field. We show that if $2\in \ddot{I}{€}_{0}A$ 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 $\ddot{I}{€}_{0}T$ is a field.

27 pages, comments welcome