Let $X=\mathrm{Spec}(A)$ be an affine variety, $Z\subseteq X$ a closed, reduced subscheme. Let 


$$\beta:Y=\mathrm{Bl}\_Z(X)\to X$$


be the blow-up of $X$ in $Z$. In other words, $Y=\mathrm{Proj}(A[IT])$ for $I:=I(Z)$. Let $E:=\beta^{-1}(Z)$ be the exceptional divisor. For a point $Q\in E$, I am now wondering how the completion $\hat{\mathcal{O}}\_{Y,Q}$ looks like. I would like to understand its relation to $\hat{\mathcal{O}}\_{X,\beta(Q)}$, in particular.


**Edit:** Some remarks and my thoughts on the matter.


If the base field $k$ is algebraically closed and both $X$ and $Y$ are nonsingular, then 
$$\hat{\mathcal{O}}\_{X,\beta(Q)}\cong k[[x\_1,\ldots,x\_d]]\cong\hat{\mathcal{O}}\_{Y,Q}.$$


It becomes tricky when they are (possibly) nonsingular. I thought it would be nice to know something about the local rings. Since $X$ and $Y$ share the same function field $K$ and $\beta$ is dominant, 


$$\mathcal{O}\_{X,\beta(Q)}\hookrightarrow\mathcal{O}\_{Y,Q}\hookrightarrow K.$$


We have adjoined the fractions $a/b$ for $a,b\in I\_{\beta(Q)}$ and $bT\notin Q$. 


Clearly, $\mathfrak{m}\_{\beta(Q)}\cdot\mathcal{O}\_{Y,Q}\ne\mathfrak{m}\_Q$ which denies us access to the exactness property of the completion. We can write $\mathfrak{m}\_{\beta(Q)}=(x\_1,\ldots,x\_d)$ with $x\_i\in I\_{\beta(Q)}$ iff $i\le r$. Then, I feel $\mathfrak{m}\_Q$ should be equal to $$(z\_1,\ldots,z\_r,x\_{r+1},\ldots,x\_d)$$ with $z\_i=x\_i/b$ for some appropriate $b$, at least under certain conditions. However, I don't know if that is correct and if it even helps.



Completion of local rings in the exceptional divisor of a blow-up
We can write $A = K[X\_1, \ldots, X\_n]/\mathfrak a$.
Suppose that $I = (a\_1, \ldots, a\_m)$; then
$A[It] \cong A[T\_1, \ldots, T\_m]/\mathfrak b$ for some ideal
$\mathfrak b$. A description for $\mathfrak b$ is given in Section 1.1 of
W. Vasconcelos, \textit{Integral Closure, Rees algebras, multiplicities and
Algorithms}, (Springer).


Let us suppose, for simplicity, that $P := \beta(Q)$ is defined by $(X\_1,
\ldots, X\_n)$ and that $Q$ is defined by $(X\_1, \ldots, X\_n, T\_1, \ldots,
T\_m)$. Let us write $R$ and $S$ for the local rings at $P$ and $Q$, respectively.
 Since we are interested in completions, we may consider the affine
open subset of $Y$ defined by $T\_1 \neq 0$. This can be thought of as the
spectrum of $A[\frac{a\_2}{a\_1}, \ldots, \frac{a\_m}{a\_1}] \cong 
A[Y\_2, \ldots, Y\_m]/\mathfrak c$ for some ideal $\mathfrak c$.
Then $\mathfrak c$
can be obtained from $\mathfrak b$ by `dehomogenizing' with respect to
$T\_1$; see, \textit{e.g.}, Section 5.5 of \textit{Integral Closure of
Ideals, Rings, and Modules} by I. Swanson and C. Huneke, (LMS Lecture Note
Series 336). It contains the relations $a\_1Y\_i = a\_i$ for all $1 \leq i
\leq m$, but, in general, could have more.
Hence
$\widehat R \cong K[[X\_1, \ldots, X\_n]]/\mathfrak a$
and
$\widehat S \cong\widehat R[[Y\_1, \ldots, Y\_m]]/\mathfrak c$.
(By abuse of notation, we write $\mathfrak a$ for an ideal of $K[X\_1,
\ldots, X\_n]$ and the ideal of $K[[X\_1, \ldots, X\_n]]$ generated by it.)


Therefore, in the non-singular situation, with $A =
K[X\_1, \ldots, X\_n]$ and $I = (X\_1, \ldots, X\_n)$, we have
$\widehat R \simeq K[[X\_1, \ldots, X\_n]]$ and
$\widehat S \simeq K[[X\_1, Y\_2, \ldots, Y\_n]]$.
(We use the fact that since $X\_1, \ldots, X\_n$ is a regular sequence in
$A$, the ideal $\mathfrak b$ is generated by their `Koszul syzygies', i.e.,
by $X\_iT\_j - X\_jT\_i, 1 \leq i < j \leq n$. Hence $\mathfrak c$ is
generated by $X\_1Y\_j - X\_j, 2 \leq j \leq n$.)
They are (abstractly) isomorphic as formal power series rings, but the
structure
morphism is not an isomorphism.


