A rigidity theorem for $\pi_*$-\'etale $\mathbb E_k$-algebras

Abstract

We prove a rigidity theorem for $\pi_*$-\'etale $\mathbb E_k$-algebras over an $\mathbb E_{k+1}$-ring spectrum: the category of $\pi_*$-\'etale extensions of an $\mathbb E_k$-algebra is identified with the ordinary category of \'etale Dirac algebras over its graded homotopy Dirac ring. The proof develops a relative Goerss-Hopkins type obstruction theory in synthetic spectra, including an $I$-complete version. As an application, the completed obstruction theory constructs the $I_n$-complete $\mathbb E_3$-$MU_{(p)}$-algebra realization of the Lubin-Tate theory, hence an $\mathbb E_4$-orientation $MU_{(p)}\to E_n$.