Lemma 106.9.1. Let f : \mathcal{X} \to Y be a morphism from an algebraic stack to an algebraic space. Let V \subset Y be an open subspace. Assume
Y is quasi-compact and quasi-separated,
f is of finite type and quasi-separated,
V is quasi-compact, and
\mathcal{X}_ V is flat and locally of finite presentation over V.
Then there exists a V-admissible blowup Y' \to Y and a closed substack \mathcal{X}' \subset \mathcal{X}_{Y'} with \mathcal{X}'_ V = \mathcal{X}_ V such that \mathcal{X}' \to Y' is flat and of finite presentation.
Proof.
Observe that \mathcal{X} is quasi-compact. Choose an affine scheme U and a surjective smooth morphism U \to \mathcal{X}. Let R = U \times _\mathcal {X} U so that we obtain a groupoid (U, R, s, t, c) in algebraic spaces over Y with \mathcal{X} = [U/R] (Algebraic Stacks, Lemma 94.16.2). We may apply More on Morphisms of Spaces, Lemma 76.39.1 to U \to Y and the open V \subset Y. Thus we obtain a V-admissible blowup Y' \to Y such that the strict transform U' \subset U_{Y'} is flat and of finite presentation over Y'. Let R' \subset R_{Y'} be the strict transform of R. Since s and t are smooth (and in particular flat) it follows from Divisors on Spaces, Lemma 71.18.4 that we have cartesian diagrams
\vcenter { \xymatrix{ R' \ar[r] \ar[d] & R_{Y'} \ar[d]^{s_{Y'}} \\ U' \ar[r] & U_{Y'} } } \quad \text{and}\quad \vcenter { \xymatrix{ R' \ar[r] \ar[d] & R_{Y'} \ar[d]^{t_{Y'}} \\ U' \ar[r] & U_{Y'} } }
In other words, U' is an R_{Y'}-invariant closed subspace of U_{Y'}. Thus U' defines a closed substack \mathcal{X}' \subset \mathcal{X}_{Y'} by Properties of Stacks, Lemma 100.9.11. The morphism \mathcal{X}' \to Y' is flat and locally of finite presentation because this is true for U' \to Y'. On the other hand, we already know \mathcal{X}' \to Y' is quasi-compact and quasi-separated (by our assumptions on f and because this is true for closed immersions). This finishes the proof.
\square
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