Strict units of commutative ring spectra

TL;DR

This paper develops computational tools, including a spectral sequence, to calculate the strict units of commutative ring spectra, with explicit computations for polynomial rings and insights into other rings.

Contribution

It introduces a spectral sequence for computing strict units of $E_$-$H$-algebras and applies it to polynomial and truncated polynomial rings.

Findings

Computed strict units for polynomial rings and analyzed their Postnikov towers.

Developed a spectral sequence for strict units of $E_$-$H$-algebras.

Outlined methods for calculating strict units of rings like $S^0$.

Abstract

We provide computational tools to calculate the strict units of commutative ring spectra. We describe the Goerss-Hopkins-Miller spectral sequence for computing strict units of $E_\infty$-$H\mathbb{F}_p$-algebras, and use it to compute the strict units of polynomial and truncated polynomial rings, whose Postnikov towers we also analyze. We then sketch the calculation of the strict units of other rings, such as $S^0$, using the symmetric product of spheres filtration and transfers.