subscheme structure of support - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T01:17:19Z http://mathoverflow.net/feeds/question/73634 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73634/subscheme-structure-of-support subscheme structure of support Descartes 2011-08-25T06:32:39Z 2011-08-25T12:21:10Z <p>One knows that the support $S$ of a coherent sheaf on a noetherian scheme is closed. E.g. on an affine scheme $X=Spec(A)$ and $F$ corresponding to a finitely generated $A$-module $M$, then the closed subset which corresponds to $S$ is just $V(Ann(M))$.</p> <p>One often says that $S$ is endowed with the structure of a closed subscheme by taking the sheaf of ideals $Ann(F)$ defined as the kernel of $\mathcal O_X \rightarrow Hom_{\mathcal O_X}(F,F)$.</p> <p>Now my question: this is not the (unique) reduced subscheme structure, isn't it? Can one anyhow describe the reduced structure?</p> <p>Thanks</p> http://mathoverflow.net/questions/73634/subscheme-structure-of-support/73637#73637 Answer by Georges Elencwajg for subscheme structure of support Georges Elencwajg 2011-08-25T07:15:12Z 2011-08-25T12:21:10Z <p>The answer to your second question is pleasantly general and simple. </p> <p>Given a completely general scheme $X$ (no noetherian, separation, ...hypothesis) and an arbitrary closed subspace $F\subset |X|$ of its underlying topological space, there is a <em>unique</em> closed reduced subscheme $Y\subset X$ whose underlying set is $|Y|=F$. Here is the proof:<br> i) If $X=Spec A$ is affine, $Y$ is given by the reduced ideal $I=\bigcap_{x\in F} j_x \;$<br> [as usual, for $x\in SpecA, j_x \subset A$ denotes the ideal corresponding to the point $x$],<br> ii) If $X$ is not affine, the reduced scheme $Y=V_{sch}(\mathcal I)$ is obtained by the unique ideal sheaf $\mathcal I\subset \mathcal O_X$ restricting on each open affine $U=Spec A$ to the ideal sheaf $\tilde I$ associated to the $I$ above.</p> <p><strong>Reference</strong> EGA 1, Chap.1 , §5, <em>Proposition</em> (5.2.1)</p> <p><strong>Addendum: the scheme structure on the support of a sheaf.</strong><br> For reference purpose, let me describe the schematic structure on the support of a sheaf in a fairly general setting.<br> The situation is that we have a completely arbitrary scheme $X$ (no noetherian assumption) and a quasi-coherent sheaf $\mathcal F$ of $\mathcal O_X$-Modules of finite type on $X$. ($\mathcal F$ needn't be coherent and so this applies to those strange schemes where $\mathcal O_X$ is not coherent!)<br> Then there exists a smallest closed subscheme $i:Y\hookrightarrow X$ with underlying set $|Y|=supp(\mathcal F)$ and a sheaf of finite type $\mathcal F'$ of $\mathcal O_Y$-Modules with support $|Y|$ such that $i_* \mathcal F'=\mathcal F$.<br> Of course if $X=SpecA$ then $\mathcal F=\tilde M$ for some finitely generated $A$-module $M$, then we have $Y=V_{sch}(annM)$ and $\mathcal F'=\widetilde {M^\prime}$ , where $M^\prime$ is $M$ seen as an $A/annM$-module.<br> Although no coherence is requested of $\mathcal F$, some finiteness condition is necessary, else $supp M$ wouldn't even be closed: just look at the $\mathbb Z$-module $\mathbb Q$ whose support is the non-closed generic point of $Spec(\mathbb Z)$</p> http://mathoverflow.net/questions/73634/subscheme-structure-of-support/73644#73644 Answer by Damian Rössler for subscheme structure of support Damian Rössler 2011-08-25T09:11:06Z 2011-08-25T09:11:06Z <p>No it isn't the reduced induced closed subscheme structure in general. For example, let $A={\bf Z}$, $M={\bf Z}/4{\bf Z}$. Then ${\rm Ann}(M)=4{\bf Z}$ and the prime ideal defining $S$ (with its reduced structure) is $2{\bf Z}={\rm rad}({\rm Ann}(M))$. So if $S$ is endowed with the reduced structure, it is isomorphic to ${\rm Spec}({\bf Z}/2{\bf Z})$ and if it is endowed with the structure given by the annihilator then it is isomorphic to ${\rm Spec}({\bf Z}/4{\bf Z})$.</p>