Lemma 81.14.4. Let $S$ be a scheme. Consider a diagram
\[ \xymatrix{ X \ar[d]_ f & U \ar[l] \ar[d]_{f|_ U} & A \ar[d] \ar[l] \\ Y & V \ar[l] & B \ar[l] } \]
of quasi-compact and quasi-separated algebraic spaces over $S$. Assume
$f$ is proper,
$V$ is a quasi-compact open of $Y$, $U = f^{-1}(V)$,
$B \subset V$ and $A \subset U$ are closed subspaces,
$f|_ A : A \to B$ is an isomorphism, and $f$ is étale at every point of $A$.
Then there exists a $V$-admissible blowing up $Y' \to Y$ such that the strict transform $f' : X' \to Y'$ satisfies: for every geometric point $\overline{a}$ of the closure of $|A|$ in $|X'|$ there exists a quotient $\mathcal{O}_{X', \overline{a}} \to \mathcal{O}$ such that $\mathcal{O}_{Y', f'(\overline{a})} \to \mathcal{O}$ is finite flat.
Proof.
Let $T' \subset |U|$ be the complement of the maximal open on which $f|_ U$ is étale. Then $T'$ is closed in $|U|$ and disjoint from $|A|$. Since $|U|$ is a spectral topological space (Properties of Spaces, Lemma 66.15.2) we can find constructible closed subsets $T_ c, T'_ c$ of $|U|$ with $|A| \subset T_ c$, $T' \subset T'_ c$ such that $T_ c \cap T'_ c = \emptyset $ (see proof of Lemma 81.14.3). By Lemma 81.14.2 there is a $U$-admissible blowing up $X_1 \to X$ such that $T_ c$ and $T'_ c$ have disjoint closures in $|X_1|$. Let $X_{1, 0}$ be the open subspace of $X_1$ corresponding to the open $|X_1| \setminus \overline{T}'_ c$ and set $U_0 = U \cap X_{1, 0}$. Observe that the scheme theoretic image $\overline{A}_1 \subset X_1$ of $A$ is contained in $X_{1, 0}$ by construction.
After replacing $Y$ by a $V$-admissible blowing up and taking strict transforms, we may assume $X_{1, 0} \to Y$ is flat, quasi-finite, and of finite presentation, see More on Morphisms of Spaces, Lemmas 76.39.1 and 76.37.3. Consider the commutative diagram
\[ \vcenter { \xymatrix{ X_1 \ar[rr] \ar[rd] & & X \ar[ld] \\ & Y } } \quad \text{and the diagram}\quad \vcenter { \xymatrix{ \overline{A}_1 \ar[rr] \ar[rd] & & \overline{A} \ar[ld] \\ & \overline{B} } } \]
of scheme theoretic images. The morphism $\overline{A}_1 \to \overline{A}$ is surjective because it is proper and hence the scheme theoretic image of $\overline{A}_1 \to \overline{A}$ must be equal to $\overline{A}$ and then we can use Morphisms of Spaces, Lemma 67.40.8. The statement on étale local rings follows by choosing a lift of the geometric point $\overline{a}$ to a geometric point $\overline{a}_1$ of $\overline{A}_1$ and setting $\mathcal{O} = \mathcal{O}_{X_1, \overline{a}_1}$. Namely, since $X_1 \to Y$ is flat and quasi-finite on $X_{1, 0} \supset \overline{A}_1$, the map $\mathcal{O}_{Y', f'(\overline{a})} \to \mathcal{O}_{X_1, \overline{a}_1}$ is finite flat, see Algebra, Lemmas 10.156.3 and 10.153.3.
$\square$
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