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 $\widehat{\mathcal{O}}\_P$?



Understanding the completed stalk of the dualizing sheaf of a family of nodal curves at a node
So as it turns out the key fact is the beautiful description of dualizing sheaves in terms of 'generalized determinants' given in Knudsen-Mumford's "The Projectivity of the Moduli Space of Stable Curves" (I and II).


There, in general given a scheme $X$ and a perfect complex of $\mathcal{O}\_X$-modules $F^\bullet$, which on an open cover $\{U\}$ of $X$ is quasi-isomorphic to a bounded complex of finite-free $\mathcal{O}\_X$-modules $G^\bullet\_U$, one can define the generalized determinant bundle $\det(F^\bullet)$, which is locally given by the alternating tensor product of determinants of the free sheaves $G^i$.
$$\det(F^\bullet)|\_U = \bigotimes\_{i\ge 0} (\det(G^i))^{(-1)^i}$$
This generalized $\det$ commutes with arbitrary base change and is invariant under quasi-isomorphisms, so in practice to compute it on a coherent sheaf $F$, it suffices to let $G^\bullet$ be a bounded finite locally free resolution of $F$ and apply the above formula.


In our case, we want to take $F^\bullet$ to be the complex $\Omega^1\_{C/S}$ concentrated in degree zero, and we'd like to understand the etale local picture of $\det(\Omega^1\_{C/S})$. Since the fibers of $C$ have at worst ordinary double points, we may find an etale covering $\{u : U\rightarrow C\}$ such that each $U$ admits a regular immersion $j : U\hookrightarrow M$ with $M$ smooth of relative dimension 2 over $S$ (needs justification). In fact one should be able to take $M = \mathbb{A}^2\_S$.


Letting $I$ be the sheaf of ideals of $U\hookrightarrow M$, we then have an exact sequence (needs justification)
$$0\rightarrow I/I^2\rightarrow j^\*\Omega^1\_{M/S}\rightarrow\Omega^1\_{U/S}\rightarrow 0$$
where the first two terms are locally free. Since $\det$ commutes with base change, and $u$ is etale, we have
$$u^\*\omega\_{C/S} = u^\*\det(\Omega^1\_{C/S}) = \det(u^\*\Omega^1\_{C/S}) = \det(\Omega^1\_{U/S}) = (I/I^2)^\vee\otimes \big(\wedge^2 j^\*\Omega^1\_{M/S}\big)$$
Fixing a point $P\in C$, and passing to the limit over all etale neighborhoods of $P$, the exact sequence above becomes:
$$0\rightarrow \widehat{I}\_P/\widehat{I}\_P^2\rightarrow \widehat{j^\*\Omega^1\_{M/S,P}}\rightarrow \widehat{\Omega^1\_{C/S,P}}\rightarrow 0$$
or more concretely,
$$0\rightarrow \frac{(xy-a)}{(xy-a)^2}\rightarrow \left(\frac{\widehat{\mathcal{O}}\_s[[x,y]]}{(xy-a)}\right)(dx\oplus dy)\rightarrow\frac{\widehat{\mathcal{O}}\_s[[x,y]]}{(xy-a)}((dx\oplus dy)/(ydx+xdy)\rightarrow 0$$
where now the first two terms are free, and the map between them is given by
$$xy-a\mapsto d(xy-a) = ydx + xdy$$
Assuming $\det$ commutes with ind-etale localization (needs justification?), we get
$$\widehat{\omega\_{C/S,P}} = (\widehat{I}\_P/\widehat{I}\_P^2)^\vee\otimes\big(\wedge^2 \widehat{j^\*\Omega^1\_{M/S,P}}\big)$$
Let $z\in \widehat{I/I^2}\_P$ be a basis, with dual basis $z^\vee$. Then the canonical map $\theta : \widehat{\Omega^1\_{C/S,P}}\rightarrow\widehat{\omega\_{C/S,P}}$ is given by sending $w\in\widehat{\Omega^1\_{C/S,P}}$ to
$$\theta(w) = z^\vee\otimes (dz\wedge \tilde{w})$$
where $\tilde{w}$ is any lift of $w$ to $j^\*\Omega^1\_{M/S}$, and one may check that this map does not depend on the choice of the basis $z$ or on the lift $\tilde{w}$. In particular, we may choose $z = xy-a$, and we would get:
$$\theta(dx) = (xy-a)^\vee\otimes ((ydx+xdy)\wedge dx = (xy-a)^\vee\otimes x(dy\wedge dx)$$
$$\theta(dy) = (xy-a)^\vee\otimes ((ydx+xdy)\wedge dy = (xy-a)^\vee\otimes y(dx\wedge dy)$$
The choice of coordinates $x,y$ in the presentation of $\widehat{\mathcal{O}\_{P}}$ determines a basis of $\widehat{\omega\_{C/S,P}}$ given by
$$w(x,y) := (xy-a)^\vee\otimes(dy\wedge dx)$$ 
where we have
$$x\cdot w(x,y) = \theta(dx)\quad\text{and}\quad y\cdot w(x,y) = -\theta(dy)$$
Thus, $w(x,y)$ is basically "$dx/x$" (ie, $\frac{1}{x}\theta(dx)$), or equivalently "$-dy/y$". Thus, we may compute:
$$(x^e-y^e)w(x,y) = x^ew(x,y) - y^ew(x,y) = x^{e-1}\theta(dx) + y^{e-1}\theta(dy)$$
as desired.


