Gamma-fixed points of overline A_infty

Determine whether the Gamma_K-fixed subring of overline A_infty := /(mathfrak{q}_infty) equals W(k), where :=W(mathcal{O}_{K_}^\flat), mathfrak{q}_infty := { [a]x | a in mathfrak{m}_{K_}^\flat, x in }, and overline A_infty := / mathfrak{q}_infty. Specifically, ascertain if ( / mathfrak{q}_infty )^{Gamma_K} = W(k), beyond the already established equality ( / mathfrak{q}_infty )_{varphi\text{-fin}}^{Gamma_K} = W(k).

Background

In developing a classification of crystalline representations via crystalline (phi,Gamma)-modules over :=W(mathcal{O}{K}\flat), the paper introduces the ideal mathfrak{q}_infty subseteq and the quotient overline A_infty := / mathfrak{q}_infty. The authors need control over Gamma_K-fixed elements in this quotient to carry out their construction.

While they can prove that the varphi-finite Gamma_K-fixed part (overline A_infty)_{varphi\text{-fin}}{Gamma_K} is exactly W(k), they do not know whether the full fixed subring (without restricting to varphi-finite elements) also equals W(k). This uncertainty motivates working with the varphi-finite part in their arguments.

References

The reason why we think of $\varphi$-finite part is that we do not know whether $(/\mathfrak{q}\infty){\Gamma_K}=W(k)$ or not, but we know that $(/\mathfrak{q}\infty)_${\Gamma_K}=W(k)$ (see \cref{prop:G-fixed part of Bbar}). Note that H. Du claims the first equality in Theorem 4.9. (2), but it seems that there is a gap in the proof.

— On the $(\varphi,Γ)$-modules corresponding to crystalline representations (2405.19829 - Watanabe, 30 May 2024) in Introduction (discussion motivating condition (ii))