A rigidity theorem for $\ddot{I}{€}_{\ast }$-étale ${\mathbb{E}}_k$-algebras

Abstract

We prove a rigidity theorem for $\ddot{I}{€}_{\ast }$-étale ${\mathbb{E}}_k$-algebras over an ${\mathbb{E}}_{k+1}$-ring spectrum: the category of $\ddot{I}{€}_{\ast }$-étale extensions of an ${\mathbb{E}}_k$-algebra is identified with the ordinary category of étale 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 ${\mathbb{E}}_3$-$M{U}_{(p)}$-algebra realization of the Lubin-Tate theory, hence an ${\mathbb{E}}_4$-orientation $M{U}_{(p)}\to E_n$.

24 pages