Canonical Cohen rings for norm fields

Fix $K/\mathbf{Q}_p$ a finite extension and let $L/K$ be an infinite, strictly APF extension in the sense of Fontaine--Wintenberger. Let $X_K(L)$ denote its associated norm field. The goal of this paper is to associate to $L/K$, in a canonical and functorial way, a $p$-adically complete subring $\mathbf{A}_{L/K}^+ \subset \widetilde{\mathbf{A}}^+$ whose reduction modulo~$p$ is contained in the valuation ring of $X_K(L)$. When the extension $L/K$ is of a special form, which we call a $\varphi$-iterate extension, we prove that $X_K(L)$ is (at worst) a finite purely inseparable extension of the fraction field of $\mathbf{A}_{L/K}^+/(p)$. The class of $\varphi$-iterate extensions includes all Lubin--Tate extensions, as well as many other extensions such as the non-Galois ``Kummer" extension occurring in work of Faltings, Breuil, and Kisin. In particular, our work provides a canonical and functorial construction of every characteristic zero lift of the norm fields that have thus far played a foundational role in (integral) $p$-adic Hodge theory, as well as many other cases which have yet to be studied.