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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