Pull back on Stalks of Ideal Sheaf My question refers to the proof of Corollary 6.3.8 and a remark below in Liu's "Algebraic Geometry":

Firstly: Why $f^*(\mathcal{J}/ \mathcal{J}^2)_x = \mathcal{J}_y / \mathcal{J}^2 _y$ holds?
I intend to work with with the definition of $f^*$ on halms as at page 163:

So in this case I get $f^*(\mathcal{J}/ \mathcal{J}^2)_x = (\mathcal{J}/ \mathcal{J}^2) _{f(x)} \otimes _{\mathcal{O}_{Y, f(x)} } \mathcal{O}_{X, x}= \mathcal{J}_{f(x)}/ \mathcal{J}^2 _{f(x)} \otimes _{\mathcal{O}_{Y, f(x)} } \mathcal{O}_{X, x}$ but how to get rid of the $\mathcal{O}_{X, x}$ factor?
And the second question:
Why the fact that $x \to rank(\mathcal{J}/ \mathcal{J}^2)_x$ is locally constant imply that $f^*(\mathcal{J}/ \mathcal{J}^2)_x = (\mathcal{J}/ \mathcal{J}^2) $ is allready locally free of finite rank?
 A: $\mathcal J$ is the kernel of the surjective morphism of sheaves $i^{\#}: \mathcal O_{Y}|_V \rightarrow f_{\ast} \mathcal O_X$.  Passing to stalks, $\mathcal J_{f(x)}$ is the kernel of the surjective homomorphism $f^{\#}_{f(x)}: \mathcal O_{Y,f(x)} \rightarrow \mathcal O_{X,x}$.
As you say, $$f^{\ast}(\mathcal J/\mathcal J^2) = \mathcal J_{f(x)}/\mathcal J_{f(x)}^2 \otimes_{\mathcal O_{Y,f(x)}} \mathcal O_{X,x} =  \mathcal J_{f(x)}/\mathcal J_{f(x)}^2 \otimes_{\mathcal O_{Y,f(x)}} \mathcal O_{Y,f(x)}/\mathcal J_{f(x)}$$
Let $R$ be a ring, and $I$ an ideal of $R$.  Then as $R$-modules
$$I/I^2 \otimes_R R/I \cong (I \otimes_R R)/P$$
where $P$ is the submodule of $I \otimes_R R$ generated by things of the form $x \otimes y, z \otimes w$ where $x \in I^2, y \in R, z, w \in I$.  You have $I \otimes_R R \cong I$, and under this isomorphism, $P$ becomes $I^2$.  Thus $I/I^2 \otimes_R R/I \cong I/I^2$.
For your second question, it follows from the definition of regular immersion that $f^{\ast}(\mathcal J/\mathcal J^2)_x \cong \mathcal J_{f(x)}/\mathcal J_{f(x)}^2$ is a free $\mathcal O_{Y,f(x)}$-module of finite rank: the hypothesis is that for each $x \in X$, $\mathcal J_{f(x)}$ is an ideal of $\mathcal O_{Y,f(x)}$ which is generated by a regular sequence, so free of finite rank is Lemma 3.6.
