Chai's conjecture on stabilizer-invariant ideals of the Lubin-Tate deformation ring

Theorem 1.1: for any open subgroup $U\subseteq A_n^\times$, the proper $U$-stable primes of $E_0$ are exactly $0,I_1,\ldots,I_n$ and the $U$-invariant radical ideals are exactly $0,I_1,\ldots,I_n,E_0$; the radical ideals invariant under the full extended group $G_n$ are the same list. This is Chai's Hope in the finite-residue-field formulation, at every prime and height, extended to open subgroups. Through Barthel-Heard-Naumann it yields the Hovey-Strickland conjecture (separate entry). It is stated for the Honda formal group over finite-field coefficients and does not treat other residue fields or the full scope of Chai's closed-fiber questions.

Theorem 1.1: for any open subgroup $U\subseteq A_n^\times$, the proper $U$-stable primes of $E_0$ are exactly $0,I_1,\ldots,I_n$ and the $U$-invariant radical ideals are exactly $0,I_1,\ldots,I_n,E_0$; the radical ideals invariant under the full extended group $G_n$ are the same list. This is Chai's Hope in the finite-residue-field formulation, at every prime and height, extended to open subgroups. Through Barthel-Heard-Naumann it yields the Hovey-Strickland conjecture (separate entry). It is stated for the Honda formal group over finite-field coefficients and does not treat other residue fields or the full scope of Chai's closed-fiber questions.