What in the meaning of "schematic support"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:08:00Z http://mathoverflow.net/feeds/question/34621 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34621/what-in-the-meaning-of-schematic-support What in the meaning of "schematic support"? jonn 2010-08-05T13:00:59Z 2010-08-05T15:06:13Z <p>In the book FGA of AMS page 109 said, "the schematic suppor of F is... " (F is a sheaf) What that means? Thank you for help.</p> http://mathoverflow.net/questions/34621/what-in-the-meaning-of-schematic-support/34624#34624 Answer by Charles Siegel for What in the meaning of "schematic support"? Charles Siegel 2010-08-05T13:18:39Z 2010-08-05T13:18:39Z <p>A sheaf $\mathcal{F}$ doesn't just come with topological support on a scheme. If you look at the definition of support (Which should be available in any good book, none of which I have at hand at the moment) it should describe to you that the support is a subscheme of the ambient scheme, not just a subspace, and may really have non-variety structure on it.</p> http://mathoverflow.net/questions/34621/what-in-the-meaning-of-schematic-support/34632#34632 Answer by Andrea Ferretti for What in the meaning of "schematic support"? Andrea Ferretti 2010-08-05T14:00:36Z 2010-08-05T14:00:36Z <p>The scheme structure on the support of the coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is defined by the sheaf of ideals $Ann(\mathcal{F}) \subset \mathcal{O}_X$.</p> <p>It is easy to see that the (topological space underlying the) locus defined by this sheaf of ideals is the same as the locus supporting $\mathcal{F}$. Indeed this is a local assertion, which we prove as follows.</p> <p>Let $A$ be a ring, $M$ a finitely generated $A$-module, $P \subset A$ a prime. Then we must check that $M_P \neq 0$ if and only if $Ann(M) \subset P$.</p> <p>Note that an element $m/1 \in M_P$ is $0$ if and only if there exists $a \notin P$ such that $a m = 0$, that is $Ann(m) \not \subset P$, and $M_P$ will be trivial if and only if $m/1 = 0$ for all $m$. Now use the fact that $$Ann(M) = \bigcap_{m \in M} Ann(m),$$ which can be written as a finite intersection, taking only the generators. Since $P$ is prime, it contains this intersection if and only if if contains at least one term $Ann(m)$, which means exactly that $M_P \neq 0$.</p> http://mathoverflow.net/questions/34621/what-in-the-meaning-of-schematic-support/34639#34639 Answer by MartinG for What in the meaning of "schematic support"? MartinG 2010-08-05T15:06:13Z 2010-08-05T15:06:13Z <p>Assume $\mathcal{F}$ is coherent on $X$. To complement the answers you have already got, there are two useful definitions of its schematic support: either use the annihilator ideal or the Fitting ideal. They have the same underlying reduced scheme, but in general different scheme structures. The annihilator is usually (always, I think) meant if nothing else is said.</p> <p>The annihilator support $Z = V(\mathrm{Ann}(\mathcal{F}))$ can be viewed as the minimal closed subscheme $i\colon Z \subset X$ such that the natural map <code>$\mathcal{F} \to i_*i^*\mathcal{F}$</code> is an isomorphism (more or less a tautology).</p> <p>The Fitting ideal (locally the maximal minors of a free presentation) gives a subscheme that contains $V(\mathrm{Ann}(\mathcal{F}))$, but with a possibly thicker scheme structure. A feature is that it is compatible with pullback; the annihilator construction is not.</p> <p>Example: a rank $r$ vector bundle $i_*E$ on a divisor $i\colon D\subset X$ has $D$ as annihilator support, and $rD$ as Fitting support.</p>