Lemma 89.4.2. Let X, x_ i, U_ i \to X, u_ i be as in (89.3.0.1) and assume f : Y \to X corresponds to g_ i : Y_ i \to U_ i under F. Then there exists a factorization
Y = Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X
of f where Z_{j + 1} \to Z_ j is the blowing up of Z_ j at a closed point z_ j lying over \{ x_1, \ldots , x_ n\} if and only if for each i there exists a factorization
Y_ i = Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = U_ i
of g_ i where Z_{i, j + 1} \to Z_{i, j} is the blowing up of Z_{i, j} at a closed point z_{i, j} lying over u_ i.
Proof.
A blowing up is a representable morphism. Hence in either case we inductively see that Z_ j \to X or Z_{i, j} \to U_ i is representable. Whence each Z_ j or Z_{i, j} is a decent algebraic space by Decent Spaces, Lemma 68.6.5. This shows that the assertions make sense (since blowing up is only defined for decent spaces). To prove the equivalence, let's start with a sequence of blowups Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X. The first morphism Z_1 \to X is given by blowing up one of the x_ i, say x_1. Applying F to Z_1 \to X we find a blowup Z_{1, 1} \to U_1 at u_1 is the blowing up at u_1 and otherwise Z_{i, 0} = U_ i for i > 1. In the next step, we either blow up one of the x_ i, i \geq 2 on Z_1 or we pick a closed point z_1 of the fibre of Z_1 \to X over x_1. In the first case it is clear what to do and in the second case we use that (Z_1)_{x_1} \cong (Z_{1, 1})_{u_1} (Lemma 89.3.3) to get a closed point z_{1, 1} \in Z_{1, 1} corresponding to z_1. Then we set Z_{1, 2} \to Z_{1, 1} equal to the blowing up in z_{1, 1}. Continuing in this manner we construct the factorizations of each g_ i.
Conversely, given sequences of blowups Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = U_ i we construct the sequence of blowing ups of X in exactly the same manner.
\square
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