Note

Units of ring spectra in a finite Postnikov range

Author: Wei Yang · Added: October 15, 2026

Abstract

Let $p$ be a prime and $n\geq 1$ . We give a proof of the natural equivalence between the Postnikov slices of a $p$ -local $E_\infty$ -ring spectrum and its spectrum of units in degrees $n\leq i<pn$ . The proof uses the Goodwillie tower of the reduced suspension spectrum of an infinite loop space. Tate vanishing splits the first $p-1$ stages and supplies a section of the truncated stabilization counit. Testing this section on truncated sphere spectra identifies its effect on homotopy groups, allowing a based space equivalence to be replaced by a spectrum equivalence in the stated range. We also record the factorial-invertibility variant and identify the induced comparison on finite complexes with the truncated logarithm. Counterexamples for spectra of units show that the upper endpoint cannot be included uniformly.