The Stacks project

Lemma 105.6.5. Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic stacks. Let $W$ be an algebraic space and let $W \to \mathcal{X}$ be a smooth morphism. There exists an étale covering $\{ W_ i \to W\} _{i \in I}$ and for each $i$ a cartesian diagram

\[ \xymatrix{ W_ i \ar[r] \ar[d] & W_ i' \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X}' } \]

with $W_ i' \to \mathcal{X}'$ smooth.

Proof. Choose a scheme $U'$ and a surjective smooth morphism $U' \to \mathcal{X}'$. As usual we set $U = \mathcal{X} \times _{\mathcal{X}'} U'$. Then $U \to \mathcal{X}$ is a surjective smooth morphism. Therefore the base change

\[ V = W \times _{\mathcal{X}} U \longrightarrow W \]

is a surjective smooth morphism of algebraic spaces. By Topologies on Spaces, Lemma 72.4.4 we can find an étale covering $\{ W_ i \to W\} $ such that $W_ i \to W$ factors through $V \to W$. After covering $W_ i$ by affines (Properties of Spaces, Lemma 65.6.1) we may assume each $W_ i$ is affine. We may and do replace $W$ by $W_ i$ which reduces us to the situation discussed in the next paragraph.

Assume $W$ is affine and the given morphism $W \to \mathcal{X}$ factors through $U$. Picture

\[ W \xrightarrow {i} U \to \mathcal{X} \]

Since $W$ and $U$ are smooth over $\mathcal{X}$ we see that $i$ is locally of finite type (Morphisms of Stacks, Lemma 100.17.8). After replacing $U$ by $\mathbf{A}^ n_ U$ we may assume that $i$ is an immersion, see Morphisms, Lemma 29.39.2. By Morphisms of Stacks, Lemma 100.44.4 the morphism $i$ is a local complete intersection. Hence $i$ is a Koszul-regular immersion (as defined in Divisors, Definition 31.21.1) by More on Morphisms, Lemma 37.60.3.

We may still replace $W$ by an affine open covering. For every point $w \in W$ we can choose an affine open $U'_ w \subset U'$ such that if $U_ w \subset U$ is the corresponding affine open, then $w \in i^{-1}(U_ w)$ and $i^{-1}(U_ w) \to U_ w$ is a closed immersion cut out by a Koszul-regular sequence $f_1, \ldots , f_ r \in \Gamma (U_ w, \mathcal{O}_{U_ w})$. This follows from the definition of Koszul-regular immersions and Divisors, Lemma 31.20.7. Set $W_ w = i^{-1}(U_ w)$; this is an affine open neighbourhood of $w \in W$. Choose lifts $f'_1, \ldots , f'_ r \in \Gamma (U'_ w, \mathcal{O}_{U'_ w})$ of $f_1, \ldots , f_ r$. This is possible as $U_ w \to U'_ w$ is a closed immersion of affine schemes. Let $W'_ w \subset U'_ w$ be the closed subscheme cut out by $f'_1, \ldots , f'_ r$. We claim that $W'_ w \to \mathcal{X}'$ is smooth. The claim finishes the proof as $W_ w = \mathcal{X} \times _{\mathcal{X}'} W'_ w$ by construction.

To check the claim it suffices to check that the base change $W'_ w \times _{\mathcal{X}'} X' \to X'$ is smooth for every affine scheme $X'$ smooth over $\mathcal{X}'$. Choose an étale morphism

\[ Y' \to U'_ w \times _{\mathcal{X}'} X' \]

with $Y'$ affine. Because $U'_ w \times _{\mathcal{X}'} X'$ is covered by the images of such morphisms, it is enough to show that the closed subscheme $Z'$ of $Y'$ cut out by $f'_1, \ldots , f'_ r$ is smooth over $X'$. Picture

\[ \xymatrix{ Z' \ar[r] \ar[d] & Y' \ar[d] \\ W'_ w \times _{\mathcal{X}'} X' \ar[d] \ar[r] & U'_ w \times _{\mathcal{X}'} X' \ar[d] \ar[r] & X' \\ W'_ w = V(f'_1, \ldots , f'_ r) \ar[r] & U'_ w } \]

Set $X = \mathcal{X} \times _{\mathcal{X}'} X'$, $Y = X \times _{X'} Y' = \mathcal{X} \times _{\mathcal{X}'} Y'$, and $Z = Y \times _{Y'} Z' = X \times _{X'} Z' = \mathcal{X} \times _{\mathcal{X}'} Z'$. Then $(Z \subset Z') \to (Y \subset Y') \subset (X \subset X')$ are (cartesian) morphisms of thickenings of affine schemes and we are given that $Z \to X$ and $Y' \to X'$ are smooth. Finally, the sequence of functions $f'_1, \ldots , f'_ r$ map to a Koszul-regular sequence in $\Gamma (Y', \mathcal{O}_{Y'})$ by More on Algebra, Lemma 15.30.5 because $Y' \to U'_ w$ is smooth and hence flat. By More on Algebra, Lemma 15.31.6 (and the fact that Koszul-regular sequences are quasi-regular sequences by More on Algebra, Lemmas 15.30.2, 15.30.3, and 15.30.6) we conclude that $Z' \to X'$ is smooth as desired. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CJV. Beware of the difference between the letter 'O' and the digit '0'.