I'm reading the proof of 7.5 from Mumford's & Oda's Algebraic Geometry II on page 198 and I'm a little confused about an argument.
Let $X$ be a noetherian integral scheme, $x \in X$ a formally unibranch point. Let $f : Y \to X$ be a morphism of finite type and let $y$ be an isolated point of $f^{−1}(x)$. Then we define $mult_y(f)$ as follows: Let $R = \hat{\mathcal{O}}_{x,X}/ \sqrt{(0)}$: By assumption this is an integral domain. Let $K$ be quotient field of $R$. Form the fibre product $Y':= Y \times_X Spec \text{ } R$. Let $y′ \in Y′$ be the unique point over $y$. define $mult_y(f):= \dim_K(\mathcal{O}_{y',Y'} \otimes_R K)$. let $Y_1':=Spec \text{ } \mathcal{O}_{y',Y'}$. because $y$ is a isolated point we can decompose disjointly $Y'=Y'_1 \cup Y' _2$.
Two futher quantities are introduced in the preparation and play in 7.5 & my question essential role:
${\text{mult}_y}^{\circ}(f)$ satisfying $mult_y(f) = [\mathbb{k}(y):\mathbb{k}(x)]_s{\text{mult}_y}^{\circ}(f)$ and let $\widetilde{\mathcal{O}}$ be the finite étale extension of $\hat{\mathcal{O}}_{x,X}$ with residue field $L$, as in Corollary IV.6.3. recall that $\mathbb{k}(x)$ is the residue field of $\mathcal{O}_{x,X}$. these all quantities occuring in
7.5. Assume $X$ is formally normal at $x$ and that all associated points of $Y$ lie over $η_X$. Then ${\text{mult}_y}^{\circ}(f)=1$ if and only if $f$ is étale at $y$.
the interesting part is ${\text{mult}_y}^{\circ}(f)=1$ implies $f$ is étale at $y$. beginning with this assumtion the proof shows that $\mathcal{O}_{y',Y'} \cong \widetilde{\mathcal{O}}$. this implies that that the expression $R \to \widetilde{\mathcal{O}}$ make sense. after that the author gives following chain of isomorphisms:
$$(\Omega_{Y/X})_y \otimes_{\mathcal{O}_{y,Y}} \mathbb{k}(y) \cong (\Omega_{Y_1 '/Spec \text{ } R}) \otimes_{\mathcal{O}_{y',Y'}} \mathbb{k}(y) \cong $$ $$(\Omega_{Spec \text{ } \widetilde{\mathcal{O}}/Spec \text{ } R})\otimes_{\widetilde{\mathcal{O}}} L = (0)$$
Q: the last two equalities I not understand. the first one is simply base change rule for differential: Suppose that we have ring maps $R\to R′$ and $R\to S$. Set $S′=S \otimes R′$, then $\Omega_{S/R} \otimes_RR′=\Omega_{S′/R′}$. the last one seems to use as essential ingredient that by construction $\widetilde{\mathcal{O}}$ is finite étale over $\hat{\mathcal{O}}_{x,X}$, is $\widetilde{\mathcal{O}}$ also étale over $R$? is that true?