Fix a completion $ mathbb {C} _p $ of an algebraic closure of $ mathbb {Q} _p $. Choose an inclination of it. The Told space (equivalence modulo Frobenius) is naturally parametrized by a regular, separate, netherian scheme of the Krull 1 dimension, noted $ X_ {FF} $. This scheme is naturally defined on $ mathbb {Q} _p $.

Choose an algebraic closure $ mathbb {Q} _p subset highlighting { mathbb {Q} _p} $. What I'm asking is, is it true that the basic change $ X_ {FF} times _ { mathrm {Spec} : mathbb {Q} _p} mathrm {Spec} : overline { mathbb {Q} _p} $ is a Noetherian scheme? The difficulty is of course that $ X_ {FF} $ is not of finite type on $ mathbb {Q} _p $. The discussion here is potentially relevant.

I do not ask this because of some applications of number theory, but rather to better understand the construction.