Suppose we have a projective flat non-smooth morphism of Noetherian schemes $g: X \rightarrow S$. My question regards when the sheaf of relative Kahler differentials $\Omega_{X/S}$ is flat over $S$. In particular, I am wondering about the case where $S$ is the spectrum of a Dedekind domain, so we can just consider that case, if it makes things easier.
For example, if the geometric fibers are reduced curves with at most ordinary double points, then this morphism is a prestable curve and the sheaf $\Omega_{X/S}$ is flat over $S$ (Knudsen, projective of moduli space of stable curves).
I'm wondering about higher-dimensional analogues? How about if all the geometric fibers are reduced surfaces (even hypersurfaces in $\mathbb{P}^3$) with at most ordinary double points? If this works, would any isolated hypersurface singularities work? If it is an arbitrary local complete intersection morphism?
Thanks!
Jordan
Best Answer
Suppose that $Y$ is the spectrum of a smooth curve over perfect field $k$ and let $D\subseteq Y$ be a finite set of closed points. Let $f:X\to Y$ be a proper morphism and let $D\subseteq X$ be a normal crossings divisor. Suppose that $f$ is semi-stable relatively to $E,D$ and $k$, in the sense of Illusie in par. 1.4 of "Réduction semi-stable et décomposition...", Duke Math. J. 60 (1990).
The morphism $f$ is then flat and lci and its fibres are reduced normal crossings divisors. There is a relative residue sequence $$ 0\to \Omega_{X/Y}\to\Omega_{X/Y}({\rm log})\to F\to 0\ \ \ \ (*) $$ where $F$ is supported on the singular locus of the singular fibres of $f$, and $\Omega_{X/Y}({\rm log})$ is the locally free sheaf of differentials with (relative) logarithmic singularities along $D$. See for instance p. 23 in "Une conjecture sur la torsion..." by V. Maillot and D. Rössler (Publ. Res. Inst. Math. Sci. 46, no. 4 (2011) - for lack of a canonical reference (?)).
Now let $M$ be any quasi-coherent ${\cal O}_Y$-module. The tor-sequence corresponding to $\otimes_Y M$ when applied to (*) gives $$ \dots\to {\rm Tor}^1_Y(\Omega_{X/Y},M)\to{\rm Tor}^1_Y(\Omega_{X/Y}({\rm log}),M)\to{\rm Tor}^1_Y(F,M) $$ $$ \to \Omega_{X/Y}\otimes_Y M\to\Omega_{X/Y}({\rm log})\otimes_Y M\to F\otimes_Y M\to 0 $$ and since ${\rm Tor}^l_Y(\Omega_{X/Y}({\rm log}),M)=0$ for all $l>0$ (because $\Omega_{X/Y}({\rm log})$ is locally free and $f$ is flat) and ${\rm Tor}^l_Y(N,K)=0$ for any $l>1$ and any quasi-coherent ${\cal O}_Y$-modules $N,K$ (that is because $Y$ is the spectrum of a Dedekind domain and any finitely generated quasi-coherent ${\cal O}_Y$-module has a two-step projective resolution; the general case follows from compatibility of Tor with direct limits), we see that ${\rm Tor}^l_Y(\Omega_{X/Y},M)=0$, for all $l>0$, ie $\Omega_{X/Y}$ is flat over $Y$.
EDIT As remarked by Liu below, the sheaf $\Omega_{X/Y}$ can also be seen to be flat simply because it is the subsheaf of a torsion free sheaf.