The Stacks project

Lemma 99.21.1. Let $\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram

\[ \xymatrix{ U \ar[d] & F \ar[l]^ p \ar[d] \\ \mathcal{X} & \mathop{\mathrm{Spec}}(k) \ar[l] } \]

where $U$ is an algebraic space, $k$ is a field, and $U \to \mathcal{X}$ is flat and locally of finite presentation. Let $f_1, \ldots , f_ r \in \Gamma (U, \mathcal{O}_ U)$ and $z \in |F|$ such that $f_1, \ldots , f_ r$ map to a regular sequence in the local ring $\mathcal{O}_{F, \overline{z}}$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism

\[ V(f_1, \ldots , f_ r) \longrightarrow \mathcal{X} \]

is flat and locally of finite presentation.

Proof. Choose a scheme $W$ and a surjective smooth morphism $W \to \mathcal{X}$. Choose an extension of fields $k \subset k'$ and a morphism $w : \mathop{\mathrm{Spec}}(k') \to W$ such that $\mathop{\mathrm{Spec}}(k') \to W \to \mathcal{X}$ is $2$-isomorphic to $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$. This is possible as $W \to \mathcal{X}$ is surjective. Consider the commutative diagram

\[ \xymatrix{ U \ar[d] & U \times _\mathcal {X} W \ar[l]^-{\text{pr}_0} \ar[d] & F' \ar[l]^-{p'} \ar[d] \\ \mathcal{X} & W \ar[l] & \mathop{\mathrm{Spec}}(k') \ar[l] } \]

both of whose squares are cartesian. By our choice of $w$ we see that $F' = F \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k')$. Thus $F' \to F$ is surjective and we can choose a point $z' \in |F'|$ mapping to $z$. Since $F' \to F$ is flat we see that $\mathcal{O}_{F, \overline{z}} \to \mathcal{O}_{F', \overline{z}'}$ is flat, see Morphisms of Spaces, Lemma 65.30.8. Hence $f_1, \ldots , f_ r$ map to a regular sequence in $\mathcal{O}_{F', \overline{z}'}$, see Algebra, Lemma 10.67.5. Note that $U \times _\mathcal {X} W \to W$ is a morphism of algebraic spaces which is flat and locally of finite presentation. Hence by More on Morphisms of Spaces, Lemma 74.28.1 we see that there exists an open subspace $U'$ of $U \times _\mathcal {X} W$ containing $p(z')$ such that the intersection $U' \cap (V(f_1, \ldots , f_ r) \times _\mathcal {X} W)$ is flat and locally of finite presentation over $W$. Note that $\text{pr}_0(U')$ is an open subspace of $U$ containing $p(z)$ as $\text{pr}_0$ is smooth hence open. Now we see that $U' \cap (V(f_1, \ldots , f_ r) \times _\mathcal {X} W) \to \mathcal{X}$ is flat and locally of finite presentation as the composition

\[ U' \cap (V(f_1, \ldots , f_ r) \times _\mathcal {X} W) \to W \to \mathcal{X}. \]

Hence Properties of Stacks, Lemma 98.3.5 implies $\text{pr}_0(U') \cap V(f_1, \ldots , f_ r) \to \mathcal{X}$ is flat and locally of finite presentation as desired. $\square$


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