Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a 1-dimensional formal group defined over $\mathcal{O}_K$. Consider the field $K_\infty$ obtained by adjoining to $K$ all the solutions to $[p]^n(x) = 0$ with $x \in \mathbb{C}_p$ with $|x|<1$ and $n\geq1$. (These torsion points depend on a choice of coordinate for the formal group, but the field $K_\infty$ does not). Equivalently, if $V(G)$ is the Tate module of $G$, considered as a $\mathrm{Gal}(\overline{K}/K)$-representation, the field $K_\infty$ is the subfield of $\overline{K}$ which corresponds to its kernel via Galois theory.

Consider the $p$-adic completion $\widehat{K}_\infty$ of $K_\infty$. My Question is: is $\widehat{K}_\infty$ perfectoid?

At the very least, I know that $K_\infty$ must be infinitely ramified (the Tate module of $G$ has a nonzero Hodge-Tate weight). Also, the answer is yes when $G$ is a Lubin-Tate formal group.



1 Answer 1


Yes it is. The extension $K_\infty/K$ is deeply ramified. Now apply proposition 6.6.6 of Gabber-Ramero's "Almost ring theory".


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