The Stacks project

105.9 Blowing up and flatness

This section quickly discusses what you can deduce from More on Morphisms of Spaces, Sections 75.38 and 75.39 for algebraic stacks over algebraic spaces.

Lemma 105.9.1. 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

  1. $Y$ is quasi-compact and quasi-separated,

  2. $f$ is of finite type and quasi-separated,

  3. $V$ is quasi-compact, and

  4. $\mathcal{X}_ V$ 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 93.16.2). We may apply More on Morphisms of Spaces, Lemma 75.39.1 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 70.18.4 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 99.9.11. 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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