Lemma 89.4.3. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $T \subset |X|$ be a finite set of closed points $x$ such that (1) $X$ is regular at $x$ and (2) the local ring of $X$ at $x$ has dimension $2$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals such that $\mathcal{O}_ X/\mathcal{I}$ is supported on $T$. Then there exists a sequence
\[ X_ m \to X_{m - 1} \to \ldots \to X_1 \to X_0 = X \]
where $X_{j + 1} \to X_ j$ is the blowing up of $X_ j$ at a closed point $x_ j$ lying above a point of $T$ such that $\mathcal{I}\mathcal{O}_{X_ n}$ is an invertible ideal sheaf.
Proof.
Say $T = \{ x_1, \ldots , x_ r\} $. Pick elementary étale neighbourhoods $(U_ i, u_ i) \to (X, x_ i)$ as in Section 89.3. For each $i$ the restriction $\mathcal{I}_ i = \mathcal{I}|_{U_ i} \subset \mathcal{O}_{U_ i}$ is a quasi-coherent sheaf of ideals supported at $u_ i$. The local ring of $U_ i$ at $u_ i$ is regular and has dimension $2$. Thus we may apply Resolution of Surfaces, Lemma 54.4.1 to find a sequence
\[ X_{i, m_ i} \to X_{i, m_ i - 1} \to \ldots \to X_1 \to X_{i, 0} = U_ i \]
of blowing ups in closed points lying over $u_ i$ such that $\mathcal{I}_ i \mathcal{O}_{X_{i, m_ i}}$ is invertible. By Lemma 89.4.2 we find a sequence of blowing ups
\[ X_ m \to X_{m - 1} \to \ldots \to X_1 \to X_0 = X \]
as in the statement of the lemma whose base change to our $U_ i$ produces the given sequences. It follows that $\mathcal{I}\mathcal{O}_{X_ n}$ is an invertible ideal sheaf. Namely, we know this is true over $X \setminus \{ x_1, \ldots , x_ n\} $ and in an étale neighbourhood of the fibre of each $x_ i$ it is true by construction.
$\square$
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