Whyt he pullback of the maximal ideal sheaf of the origin in $\mathbb{C}^2$ under blow-up of the origin is not torsion-free? Maybe it is a silly question but i don't uderstand why the following statement is true:
"Let X be a complex space and  $\pi :Y \longrightarrow X$ be a proper modification of $X$. The pull back  $\pi^∗S$ of a torsion-free coherent  sheaf of $\mathcal{O}_X$-module $S$ is not torsion-free in general. For a counterexample, see the example in Hans Grauert and Oswald Riemenschneider, Verschwindungssa ̈tze fu ̈r analytische Kohomologiegruppen auf komplexen Ra ̈umen (it is in german),
i. e. the pullback of the maximal ideal sheaf of the origin in $\mathbb{C}^2$ under blow-up of the origin is not torsion-free. One can say more or less that $\pi^∗S$ is torsion-free in a point $y \in Y$ if and only if $S$ is locally free in $π(y)$.''
I don't understand this example, if i take $\pi: \mathrm{BL}(0,\mathbb{C}^2) \longrightarrow X$ to be the blow up of the origin of $\mathbb{C}^2$ and i look in local coordinates $(x,y)=(x',x'y')$, the pull back of the function $f(x,y)=\alpha x+  \beta y$ is the function  $f \circ \pi(x',y')=x'(\alpha+ \beta y')$ how can it be a torsion element ? by what element of $\mathcal{O}_{\mathrm{BL}(0,\mathbb{C}^2)}$ should i multiply it to get zero ?
Thanks in advance for your help
 A: To compute the pullback one can use a locally free resolution:
$$
0 \to \mathcal{O}_X \to \mathcal{O}_X \oplus \mathcal{O}_X \to \mathfrak{m} \to 0,\tag{*}
$$
where the second arrow is induced by $(x,y)$ and the first by $(y,-x)$. Note that the pullbacks of these two functions to the blowup $\tilde{X}$ vanish along the exceptional divisor $E \subset \tilde{X}$, therefore they can be written as
$$
x = x' \cdot e,
\qquad
y = y' \cdot e,
$$
where $e$ is a section of the line bundle $\mathcal{O}_{\tilde{X}}(E)$ vanishing on $E$ and $x',y'$ are sections of $\mathcal{O}_{\tilde{X}}(-E)$. Therefore, the pullback of $(*)$ looks as
$$
\mathcal{O}_{\tilde{X}} \to \mathcal{O}_{\tilde{X}} \oplus \mathcal{O}_{\tilde{X}} \to \pi^*\mathfrak{m} \to 0,
$$
where the first arrow is given by $(y'\cdot e, -x'\cdot e)$, hence it factors as
$$
\mathcal{O}_{\tilde{X}} \stackrel{e}\to \mathcal{O}_{\tilde{X}}(E) \stackrel{(y',-x')}\to \mathcal{O}_{\tilde{X}} \oplus \mathcal{O}_{\tilde{X}}.
$$
It follows that there is an exact sequence
$$
0 \to \mathrm{Coker}(e) \to \pi^*\mathfrak{m} \to \mathrm{Coker}(y',-x') \to 0
$$
so it remains to note that
$$
\mathrm{Coker}(e) \cong \mathcal{O}_E(E)
$$
is a torsion sheaf.
