Remark 37.17.2. We fix a ring R and we set S = \mathop{\mathrm{Spec}}(R). Fix integers 0 \leq m and 1 \leq n. Consider the closed immersion
Z = \mathbf{A}^ m_ S \longrightarrow \mathbf{A}^{m + n}_ S = X,\quad (a_1, \ldots , a_ m) \mapsto (a_1, \ldots , a_ m, 0, \ldots 0).
We are going to consider the blowing up X' of X along the closed subscheme Z. Write
X = \mathop{\mathrm{Spec}}(A) \quad \text{with}\quad A = R[x_1, \ldots , x_ m, y_1, \ldots , y_ n]
Then X' is the Proj of the Rees algebra of A with respect to the ideal (y_1, \ldots , y_ n). This Rees algebra is equal to B = A[T_1, \ldots , T_ n]/(y_ iT_ j - y_ jT_ i); details omitted. Hence X' = \text{Proj}(B) is smooth over S as it is covered by the affine opens
\begin{align*} D_+(T_ i) & = \mathop{\mathrm{Spec}}(B_{(T_ i)}) \\ & = \mathop{\mathrm{Spec}}(A[t_1, \ldots , \hat t_ i, \ldots t_ n]/(y_ j - y_ i t_ j)) \\ & = \mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ m, y_ i, t_1, \ldots , \hat t_ i, \ldots , t_ n]) \end{align*}
which are isomorphic to \mathbf{A}^{n + m}_ S. In this chart the exceptional divisor is cut out by setting y_ i = 0 hence the exceptional divisor is smooth over S as well.
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