Let $f : C\rightarrow S$ be a proper flat morphism whose geometric fibers are connected nodal curves.

Let $P\in C$ be a closed point with image $s\in S$, and suppose $P$ is a 'node' - that is, if $\widehat{\mathcal{O}}_s$ is the completed etale local ring of $S$ at $s$, then we have: $$\widehat{\mathcal{O}}_P = \widehat{\mathcal{O}}_s[[x,y]]/(xy-a)$$ where $a$ is in the maximal ideal $m_s$ of $\widehat{\mathcal{O}}_s$.

Let $\omega_C = \omega_{C/S}$ be the dualizing sheaf of $C/S$.

**Question 1** - How can we describe the completed stalk $\widehat{\omega_{C,P}}$?

If $S = \text{Spec }k$ for an algebraically closed field $k$, then if $\nu : C'\rightarrow C$ is the normalization map, then it's well known that

$$\omega_{C}(U) = \{w\in \Omega^1_{C'/S}(Q_1+Q_2)(\nu^{-1}(U)) : \text{res}_{Q_1}w + \text{res}_{Q_2}w = 0\}$$ where $Q_1,Q_2$ run over all distinct pairs of points in $C'$ which map to the same point $P$ in $C$.

This seems to imply that $\widehat{\omega_{C,P}}$ is the $\big(\widehat{\mathcal{O}_{P}} = k[[x,y]]/(xy)\big)$-submodule of $$\widehat{\Omega^1_{C'/S,Q_1}}(Q_1)\oplus\widehat{\Omega^1_{C'/S,Q_2}}(Q_2) = k[[x]]\frac{dx}{x}\oplus k[[y]]\frac{dy}{y}$$ generated by $w(x,y) := \left(\frac{dx}{x},-\frac{dy}{y}\right)$.

Does something like this still hold in the general case when $S$ isn't Spec $k$? What exactly would the description be?

Part of the reason for wanting to understand this is the following: In general there is always a canonical map $\theta : \Omega_{C/S}\rightarrow\omega_{C/S}$. If $e$ is an integer coprime invertible on $S$, then at the level of completed stalks, a paper I'm reading makes the claim: $$\theta(d(x^e+y^e)) = e(x^e-y^e)\tilde{w}(x,y)$$ where $\tilde{w}(x,y)$ is a generator of $\widehat{\omega_{C,P}}$ which reduces to $w(x,y)$ in the fiber above $s$ (actually the paper seems to call these completed stalks the 'etale local picture', so I guess they also want to assume that the residue field of $\widehat{\mathcal{O}_s}$ is separably closed, to parallel 'strict henselization')

Certainly, we have $$d(x^e+y^e) = e(x^{e-1}dx +y^{e-1}dy)$$ Thus, if we could write $\tilde{w}(x,y) = \text{"}\frac{dx}{x} - \frac{dy}{y}\text{"}$, in a way such that $x\frac{dy}{y} = y\frac{dx}{x} = 0$, then the claim would follow. This makes sense at the image of $P$ in the geometric fiber of $s$, but is it possible to make sense of this "computation" over $\widehat{\mathcal{O}}_P$?