Timeline for What is the support of a coherent sheaf on $X\times Y$ if it is invariant by tensoring a very ample line bundle on $X$?

6 events

when toggle format	what		by	license	comment
Dec 5, 2016 at 6:05	comment	added	t3suji		I don't have a reference, but it is pretty straightforward: enough to consider the case $S=Spec(R)$, and then a coherent sheaf $F$ is given by a finitely generated graded $R[x_0,\dots,x_n]$-module $M=\bigoplus M_d$; for large enough $D$, $M_D$ generates $M_{\ge D}:=\bigoplus_{d\ge D} M_d$, so the annihilator ideal of $M_D$ in $R$ is contained in the annihilator ideal of $M_{\ge D}$.
Dec 5, 2016 at 2:41	comment	added	Zhaoting Wei		@t3suji Is it a standard result in algebraic geometry?
Dec 4, 2016 at 22:16	comment	added	t3suji		The claim is that for any (quasi-compact noetherian) scheme $S$ and any coherent sheaf $F$ on $S\times{\mathbb P}^n$, we have $p(supp(F))=supp(p_*(F(n)))$ for $n\gg 0$. Here $p$ is the projection onto $S$.
Dec 4, 2016 at 16:25	comment	added	Zhaoting Wei		@t3suji I'm sorry why this gives what I want? Could you give some more details?
Dec 4, 2016 at 7:25	comment	added	t3suji		Note that if $F$ has this property, then so does $F\otimes p_Y^* E$ for any sheaf $E$ on $Y$. In particular, you know that $dim(supp (p_(F\otimes p_Y^E)))=0$ for any $E$. Taking $E$ to be a high power of an ample line bundle gives you what you want.
Dec 4, 2016 at 3:53	history	asked	Zhaoting Wei	CC BY-SA 3.0