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

  • 1
    $\begingroup$ Your description of the completed local ring is incorrect since $P$ may not have the same residue field as $s$ (though $k(P)/k(s)$ is always separable; not obvious!). Is there a reason you use completions instead of an etale neighborhood? Using Artin approximation, the book of Freitag & Kiehl on the Weil conjectures shows (Prop. 2.7, 2.8 in Ch. III) that there exists a residually trivial local-etale neighborhood ${\rm{Spec}}(A) \rightarrow {\rm{Spec}}(O_{S,s})$ and $a \in \mathfrak{m}_A$ such that $({\rm{Spec}}(A[x,y]/(xy-a), (\mathfrak{m}_A, (x,y)))$ shares an etale neighborhood with $(C,P)$. $\endgroup$
    – nfdc23
    Commented Feb 18, 2017 at 2:08
  • $\begingroup$ @nfdc23 That is a good point, and I suppose this implies that by 'completion' I should always mean the 'completion of the strict henselization of the local ring' - ie, the completion of the etale local ring. I use completions because this paper always uses completions (this is Chapter 4 of Bertin-Romagny's "Champs de Hurwitz"), though they seem to speak of completions and etale local rings interchangeably, which is a bit annoying for me, though I suppose it is somewhat justified by your comment. Perhaps they use completions because it is easier to write power series than etale local functions? $\endgroup$ Commented Feb 18, 2017 at 20:54
  • $\begingroup$ The completion has the merit of living over the ring whereas the standard model I described only has a pointed etale neighborhood in common. It is indeed convenient to be able to go back and forth (usually with justification by Artin approximation if a more elementary method doesn't come to mind). In deJong's paper on alterations he goes between these etale-local situations and completions, depending on the context. The formation of the dualizing sheaf is compatible with etale localization on the source; that should reduce whatever you need to computing with the "standard model" I mentioned. $\endgroup$
    – nfdc23
    Commented Feb 18, 2017 at 21:45
  • $\begingroup$ @nfdc23 Do you have a reference for the statement "formation of the dualizing sheaf is compatible with etale localization on the source"? Does this mean that if $u : U\rightarrow C$ is etale, then $u^*\omega_{C/S}$ is canonically isomorphic to $\omega_{U/S}$? $\endgroup$ Commented Feb 18, 2017 at 22:57

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.