For Complex Orientations Preserving Power Operations, p-typicality is Atypical

TL;DR

The paper demonstrates that for primes up to 13, certain well-known MU_(p)-rings cannot admit commutative MU_(p)-algebra structures compatible with power operations, challenging assumptions about p-typical formal group laws.

Contribution

It shows that p-typical formal group laws are incompatible with power operations on complex cobordism for primes up to 13, revealing limitations in rigidifying orientations to E_ ring spectra.

Findings

p-typical formal group laws do not admit compatible power operations for primes ≤ 13

Standard p-typical orientations on BP, E(n), E_n do not lift to E_ ring spectra

Incompatibility extends conjecturally to all primes

Abstract

We show, for primes p less than or equal to 13, that a number of well-known MU_(p)-rings do not admit the structure of commutative MU_(p)-algebras. These spectra have complex orientations that factor through the Brown-Peterson spectrum and correspond to p-typical formal group laws. We provide computations showing that such a factorization is incompatible with the power operations on complex cobordism. This implies, for example, that if E is a Landweber exact MU_(p)-ring whose associated formal group law is p-typical of positive height, then the canonical map MU_(p) --> E is not a map of H_\infty ring spectra. It immediately follows that the standard p-typical orientations on BP, E(n), and E_n do not rigidify to maps of E_\infty ring spectra. We conjecture that similar results hold for all primes.