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Mod p identification A_K^+/p ≅ E_K^+ for K ≠ K_0

Determine whether the natural reduction map identifies A_K^+/p with E_K^+ when K is ramified (K ≠ K_0), where A_K^+ is the Wach-module base ring and E_K^+ denotes the valuation ring of the field of norms associated to the cyclotomic extension K_\infty/K.

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Background

In relating prismatic F-crystals and (phi,Gamma)-modules, existing arguments (e.g., in Wu 2021) rely on certain identifications involving A_K+ and its mod p reduction. The paper points out a potential gap when K is ramified: the author cannot verify the isomorphism A_K+/p ≅ E_K+ used in those arguments.

To avoid this issue, the paper uses the prism (, varphi(ξ)) instead, which allows proving the desired equivalence without assuming the contested identification.

References

It seems that Lemma 2.5. has some gaps when $K\neq K_0$. To be more precise, the author does not know whether $\mathbb{A}_K+/p \cong \mathbb{E}_K+$ or not.

— On the $(\varphi,Γ)$-modules corresponding to crystalline representations (2405.19829 - Watanabe, 30 May 2024) in Remark following Theorem 5.2 (Subsection: Summary of prisms)