Functoriality of the Blow-Up

2011-08-29T22:31:33Z

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:

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

$\mathcal{K}\supseteq(f^{-1}\mathcal{I})\mathcal{O}_W=:\mathcal{J}$

to be a quasi-coherent sheaf of ideals on $W$ which contains 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

$\begin{matrix} \widetilde{W} & \xrightarrow{\quad\widetilde{f}\quad} & \widetilde{X} \\ \hphantom{\scriptstyle\rho} \downarrow {\scriptstyle\rho} & {\scriptstyle\circlearrowleft} & \hphantom{\scriptstyle\pi} \downarrow {\scriptstyle\pi} \\ W & \xrightarrow{\quad f\quad} & X \end{matrix}$

This can be shown exactly as in Liu's book, but the more important point is this: It would seem to me that $(\rho^{-1}\mathcal{J})\mathcal{O}_{\widetilde{W}}$ is an invertible sheaf on $\widetilde{W}$, so I would also get uniqueness of $\widetilde{f}$.

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 $\mathcal{K}\supseteq\mathcal{J}$.

Answer by Damian Rössler for Functoriality of the Blow-Up

2011-08-30T07:30:21Z

Suppose $f={\rm Id}_X$ , $X={\bf A}^3_{\bf C}$ (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).

Functoriality of the Blow-Up - MathOverflow

Functoriality of the Blow-Up

Answer by Damian Rössler for Functoriality of the Blow-Up