Flatness of sheaf of relative Kahler differentials - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:42:04Z http://mathoverflow.net/feeds/question/75410 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75410/flatness-of-sheaf-of-relative-kahler-differentials Flatness of sheaf of relative Kahler differentials Jordan 2011-09-14T15:11:02Z 2011-09-18T06:49:38Z <p>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.</p> <p>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).</p> <p>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?</p> <p>Thanks!</p> <p>Jordan </p> http://mathoverflow.net/questions/75410/flatness-of-sheaf-of-relative-kahler-differentials/75434#75434 Answer by inkspot for Flatness of sheaf of relative Kahler differentials inkspot 2011-09-14T19:19:40Z 2011-09-14T19:19:40Z <p>Suppose that $g:X\to S$ is a finite separable morphism of smooth curves over a field. Then $g$ is flat and is locally a family of hypersurfaces but $\Omega^1_{X/S}$ is supported on the ramification locus, so is not flat over $S$ unless $g$ is etale.</p> http://mathoverflow.net/questions/75410/flatness-of-sheaf-of-relative-kahler-differentials/75584#75584 Answer by Damian Rössler for Flatness of sheaf of relative Kahler differentials Damian Rössler 2011-09-16T09:04:56Z 2011-09-18T06:49:38Z <p>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). </p> <p>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 (?)). </p> <p>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 $Y$. </p> <p>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.</p>