Functoriality of the Blow-Up - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T04:07:46Z http://mathoverflow.net/feeds/question/74009 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74009/functoriality-of-the-blow-up Functoriality of the Blow-Up Jesko Hüttenhain 2011-08-29T22:31:33Z 2011-08-30T07:30:21Z <p>I have a very simple question, because I basically just need to know if a certain train of thought I've had is correct. My reference is Liu's book "Algebraic Geometry and Arithmetic Curves", in particular Proposition 8.1.15, and of course Hartshorne. Consider the following situation:</p> <p>Let $f:W\to X$ be a morphism of locally Noetherian schemes. Let $\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$. Now, I will only require </p> <p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\mathcal{K}\supseteq(f^{-1}\mathcal{I})\mathcal{O}_W=:\mathcal{J}$</p> <p>to be a quasi-coherent sheaf of ideals on $W$ which <b>contains</b> the inverse image ideal sheaf. Let $\pi:\widetilde{X}\to X$ and $\rho:\widetilde{W}\to W$ denote the blowing-ups of $X$ and $W$ with respective centers $\mathcal{I}$ and $\mathcal{K}$. Then there exists a map $\widetilde{f}:\widetilde{W}\to\widetilde{X}$ such that </p> <p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<code>$\begin{matrix} \widetilde{W} &amp; \xrightarrow{\quad\widetilde{f}\quad} &amp; \widetilde{X} \\ \hphantom{\scriptstyle\rho} \downarrow {\scriptstyle\rho} &amp; {\scriptstyle\circlearrowleft} &amp; \hphantom{\scriptstyle\pi} \downarrow {\scriptstyle\pi} \\ W &amp; \xrightarrow{\quad f\quad} &amp; X \end{matrix}$</code> </p> <p>This can be shown exactly as in Liu's book, but the more important point is this: It would seem to me that <code>$(\rho^{-1}\mathcal{J})\mathcal{O}_{\widetilde{W}}$</code> is an invertible sheaf on $\widetilde{W}$, so I would also get uniqueness of $\widetilde{f}$.</p> <p>My question is very simple: Have I missed anything or made some mistake? I am asking because both Hartshorne and Liu require $\mathcal{K}=\mathcal{J}$ in their respective propositions, but I see no reason why it could not be weakened to $\mathcal{K}\supseteq\mathcal{J}$.</p> http://mathoverflow.net/questions/74009/functoriality-of-the-blow-up/74042#74042 Answer by Damian Rössler for Functoriality of the Blow-Up Damian Rössler 2011-08-30T07:30:21Z 2011-08-30T07:30:21Z <p>Suppose <code>$f={\rm Id}_X$</code>, <code>$X={\bf A}^3_{\bf C}$</code> (affine space of dimension $3$ over the complex numbers). Suppose that $\cal I$ is the sheaf of ideals of a smooth curve going through $0$ and that $\cal K$ is the sheaf of ideals of the point $0$ in ${\bf A}^3_{\bf C}$. Then the pull-back of ${\cal J}={\cal I}$ to $\widetilde{W}$ defines a subscheme $Z$ of $\widetilde{W}$ and the dimension of the intersection of $Z$ with the complement of the exceptional divisor of $\widetilde{W}$ is of dimension $1$ and thus $Z$ is not a Cartier divisor (ie $(\rho^{-1}J){\cal O}_{\widetilde{W}}$ is not an invertible sheaf). </p>