Are Kahler differentials the same on the affine closure on a quasi-affine scheme? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:10:15Z http://mathoverflow.net/feeds/question/57412 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57412/are-kahler-differentials-the-same-on-the-affine-closure-on-a-quasi-affine-scheme Are Kahler differentials the same on the affine closure on a quasi-affine scheme? Greg Muller 2011-03-04T22:36:32Z 2011-03-05T15:39:31Z <p>Let $X$ be a quasi-affine scheme; that is, the natural map $$X\rightarrow \overline{X}:=Spec(\Gamma(X,\mathcal{O}_X))$$ is an inclusion. Each scheme has a quasi-coherent sheaf of Kahler differentials $\Omega$, and the above open inclusion induces a $\Gamma(X,\mathcal{O}_X)$-module map of global Kahler differentials</p> <p>$$\Gamma(\Omega_{\overline{X}})\rightarrow \Gamma(\Omega_{X})$$</p> <p>Is this map always an isomorphism?</p> <p><strong>Edit:</strong> Changed $\mathcal{O}_X$ to $\Gamma(X,\mathcal{O}_X)$.</p> http://mathoverflow.net/questions/57412/are-kahler-differentials-the-same-on-the-affine-closure-on-a-quasi-affine-scheme/57417#57417 Answer by Sándor Kovács for Are Kahler differentials the same on the affine closure on a quasi-affine scheme? Sándor Kovács 2011-03-04T23:15:12Z 2011-03-05T15:39:31Z <p>The assumption implies that the natural embedding induces an isomorphism $\Gamma(X,\mathscr O_X)\simeq \Gamma(\overline X,\mathscr O_{\overline X})$. Then this means that the complement of $X$ has at least codimension $2$. </p> <p>In addition assume that $X$ is noetherian and $S_2$ (for instance normal). </p> <p>In this case <em>if</em> $\Omega_{\overline X}$ is a reflexive sheaf, then the restriction $\Gamma(\overline X,\Omega_{\overline X})\to \Gamma(X,\Omega_{\overline X})=\Gamma(X,\Omega_{X})$ is an isomorphism.</p> <p>More generally, let $Z:=\overline X\setminus X$. If <code>$\mathrm{depth}_Z\Omega_{\overline X}\geq 2$</code> then the restriction $\Gamma(\overline X,\Omega_{\overline X})\to \Gamma(X,\Omega_{X})$ is an isomorphism. This certainly holds if $\Omega_{\overline X}$ is a reflexive sheaf, but obviously it could hold "by accident" even if one of the above conditions fail, so I am not claiming that these are necessary conditions, but at least they sure seem to provide a natural set of conditions under which the required map is an isomorphism.</p> <p><strong>Sketch</strong> that if $X$ is $S_2$, then a <em>reflexive</em> coherent sheaf $\mathscr F$ is also $S_2$: First observe that by the argument in <a href="http://mathoverflow.net/questions/45347/why-does-the-s2-property-of-a-ring-correspond-to-the-hartogs-phenomenon/45616#45616" rel="nofollow">this answer</a> to another MO question we may assume that $X$ is affine and it is enough to prove that $H^i_x(X,\mathscr F)=0$ for $i=0,1$ for all $x\in X$ and it also follows that $\mathrm{depth}_Z\mathscr F\geq 2$ even if $Z$ is not contained in an affine piece of $X$. To do that write $\mathscr F^\vee$ as the quotient of a (locally) free sheaf ($X$ is affine!). Then $\mathscr F$ is a submodule of the dual of this locally free sheaf, let's call it $\mathscr E$, and the quotient $\mathscr E/\mathscr F$ is torsion-free. Therefore none of them have torsion and so $H^0_x(X,\mathscr F)=0$ and $H^1_x(X,\mathscr F)$ embeds into $H^1_x(X,\mathscr E)$. But the latter is $0$ by the assumption that $X$ is $S_2$.</p> <p><strong>EDIT 1</strong> removed intro paragraph about the starting assumption.</p> <p><strong>EDIT 2</strong> added "more generally" paragraph.</p> <p><strong>EDIT 3</strong> added noetherian assumption. this is probably not necessary but without this one should possibly be more careful about the other conditions.</p> <p><strong>EDIT 4</strong> Added <strong>Sketch</strong> above.</p>