Remark 15.1.1.
Recall that a morphism \(R \to S\) of rings is smooth if and only if locally on \(\Spec(S)\text{,}\) it can be written in the form \(R \to R[x_1,\dots,x_r] \to S\) where the second map is étale (see [117], tag 054L). Similarly, if \(R \to S\) is a \(p\)-completely smooth map, then locally on \(\Spec(S/p)\) it can be written in the form \(R \to R \langle x_1,\dots,x_r\rangle \to S\) where the second map is \(p\)-completely étale (use Proposition 6.5.3 to reduce to the previous statement).
If \(\overline{A} \to R\) is \(p\)-completely smooth, then \(\widehat{\Omega}^1_{R/\overline{A}}\) is a finite projective \(R\)-module (again by Proposition 6.5.3 to reduce to the corresponding statement about differentials for a smooth morphism). Consequently, if \(R \to S\) is \(p\)-completely étale, then \(\widehat{\Omega}^i_{R/\overline{A}} \widehat{\otimes}_R S \cong \widehat{\Omega}^i_{S/\overline{A}}\) for all \(i\text{.}\)
This suggests the strategy of proving the Hodge-Tate comparison for a general \(p\)-completely smooth \(\overline{A}\)-algebra \(R\) by proving the corresponding compatibility with étale maps, and then using this to reduce to the case \(R = \overline{A} \langle X_1,\dots,X_r \rangle\text{.}\) The first step in this program is executed by Lemma 15.1.2.