Adams-type maps are not stable under composition

TL;DR

The paper presents a counterexample demonstrating that Adams-type maps of ring spectra are not stable under composition, and shows that over a field, any map of $ ext{E}_ ext{infty}$-algebras can be constructed as a transfinite composition of such maps.

Contribution

It provides the first explicit counterexample to the stability of Adams-type maps under composition and characterizes their role in the structure of $ ext{E}_ ext{infty}$-algebra maps over a field.

Findings

Counterexample to stability under composition

Any $ ext{E}_ ext{infty}$-algebra map over a field is a transfinite composition of Adams-type maps

Demonstrates the extreme failure of stability in certain algebraic contexts

Abstract

We give a simple counterexample to the plausible conjecture that Adams-type maps of ring spectra are stable under composition. We then show that over a field, this failure is quite extreme, as any map of $\mathbb{E}_{\infty}$-algebras is a transfinite composition of Adams-type maps.