Timeline for Functoriality of the Blow-Up

5 events

when toggle format	what		by	license	comment
Sep 1, 2011 at 0:10	vote	accept	Jesko Hüttenhain
Aug 30, 2011 at 17:04	comment	added	Damian Rössler		(continued) ... In fact an example is simply the following. Let $f$ and $X$ be as in my answer and let $\cal K$ be the sheaf of ideals of a reduced surface in ${\bf A}^3_{\bf C}$; let $\cal I$ be any closed subscheme, with the same support as $\cal K$, which is not a Cartier divisor. Then $\rho$ is an isomorphism and by construction $(\rho^{-1}J){\cal O}_{\widetilde{W}}$ is not an invertible sheaf.
Aug 30, 2011 at 15:58	comment	added	Damian Rössler		@Jesko Hüttenhain : I don't think so. I don't have an example handy, but in the example I give, you could replace the smooth curve by some wild infinitesimal neighborhood of $0$ (one, which is not regularly embedded in $X$, for instance); there is no reason why the pull-back of such a neighborhood should give a Cartier divisor on $\widetilde{W}$. Remember that a codimension one closed subscheme in a regular scheme is not necessarily a Cartier divisor (but it will be if it is reduced - see Prop. 1.12A in Hartshorne).
Aug 30, 2011 at 13:13	comment	added	Jesko Hüttenhain		I see. My motivation was the following: I want to blow up $X$ in a subvariety $Y$ and $W$ in $f^{-1}(Y)$, so in my case $\cal K$ would be the radical of $\cal J$. Can I get it to work in this special case?
Aug 30, 2011 at 7:30	history	answered	Damian Rössler	CC BY-SA 3.0