Suitability of Kedlaya–Liu’s proposed imperfect period rings

Determine whether the proposed definition of the ring \(\mathbf{A}_{\psi,X}^r\) in Kedlaya–Liu, Definition 5.2.1, gives suitable imperfect relative period rings in general for decompletion of relative \((\varphi,\Gamma)\)-modules.

Background

The paper discusses a decompletion formalism for passing from perfect relative period rings to imperfect relative period rings. It notes that several descent assertions in Kedlaya–Liu’s cited work appear incomplete and require further justification.

The unresolved issue is specifically whether the proposed ring defined by imposing conditions involving θ∘φ−n\theta\circ\varphi^{-n} is suitable in general for the intended decompletion of relative (φ,Γ)(\varphi,\Gamma)-modules. The paper avoids relying on that construction by introducing the more restrictive notions of decompleting data and strongly decompleting data.

References

In particular, it is unclear to us whether the proposed definition of ``\mathbf{A}{\psi, X}r'' in Definition~5.2.1 by \mathbf{A}{\psi, X}r\coloneqq { x \in \widetilde{\mathbf{A}r_{\psi,X} ~|~ \theta\circ \varphi{-n}(x)\in A_{\psi, X, n}\textrm{ for all }n -\log_p r} gives suitable imperfect period rings in general, in order to decomplete relative (\varphi, \Gamma)-modules.

— Relative $(\varphi, Γ)$-modules and $p$-adic differential equations  (2609.08179 - Diao et al., 8 Sep 2026) in Section 1, subsection “Related works”

It turns out \mathbf{N}{\mathrm{dR}}(\mathbb{L}) is a descent of \widetilde{\mathbf{N}}{\mathrm{dR}}(\mathbb{L}) along the arrow (11) in (\ref{diagram:big_fucking_diagram_of_rings}). However, it is unclear whether \mathbf{N}_{\mathrm{dR}}(\mathbb{L}) is the unique such descent (cf. Remark~\ref{remark:neither fully faithful nor essentially surjective}).

— A $p$-adic monodromy theorem for curves  (2609.11106 - Diao et al., 10 Sep 2026) in Remark “tilde N_dR--relative,” subsection “Construction of N_dR(L)”