For complex orientations preserving power operations, p-typicality is atypical

Abstract

We show, for primes p ⩽ 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 ∞ 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 ∞ ring spectra. We conjecture that similar results hold for all primes.