The Stacks project

Proof. Let $\Xi = (U, Z, y, x, \alpha )$ be an object of $\mathcal{H}_ d$. It follows from the remark following (97.15.0.1) that the pullback of $\Xi $ by $U' \to U$ belongs to $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ if and only if the base change of $x_\alpha $ is unramified and a local complete intersection morphism. Note that $Z \to U$ is finite locally free (hence flat, locally of finite presentation and universally closed) and that $X_ y \to U$ is flat and locally of finite presentation by our assumption on $F$. Then More on Morphisms of Spaces, Lemmas 76.49.1 and 76.49.7 imply exists an open subscheme $W \subset U$ such that a morphism $U' \to U$ factors through $W$ if and only if the base change of $x_\alpha $ via $U' \to U$ is unramified and a local complete intersection morphism. This implies that

is representable by $W$. Hence the final statement of the lemma holds. The first statement (that $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ is a stack in groupoids) follows from this and Algebraic Stacks, Lemma 94.15.5. $\square$

Proof. By More on Morphisms of Spaces, Lemma 76.48.12 the conormal sheaf $\mathcal{C}_{Z/X}$ of the unramified morphism $Z \to X$ is a finite locally free $\mathcal{O}_ Z$-module and by More on Morphisms of Spaces, Lemma 76.48.13 we have an exact sequence

of conormal sheaves. Since $Z$ is affine this sequence is split. Choose a splitting

Let $Z \subset Z''$ be the universal first order thickening of $Z$ over $X'$ (see More on Morphisms of Spaces, Section 76.15). Denote $\mathcal{I} \subset \mathcal{O}_{Z''}$ the quasi-coherent sheaf of ideals corresponding to $Z \subset Z''$. By definition we have $\mathcal{C}_{Z/X'}$ is $\mathcal{I}$ viewed as a sheaf on $Z$. Hence the splitting above determines a splitting

Let $Z' \subset Z''$ be the closed subscheme cut out by $\mathcal{C}_{Z/X} \subset \mathcal{I}$ viewed as a quasi-coherent sheaf of ideals on $Z''$. It is clear that $Z'$ is a first order thickening of $Z$ and that we obtain a commutative diagram of first order thickenings as in the statement of the lemma.

Since $X' \to U'$ is flat and since $X = U \times _{U'} X'$ we see that $\mathcal{C}_{X/X'}$ is the pullback of $\mathcal{C}_{U/U'}$ to $X$, see More on Morphisms of Spaces, Lemma 76.18.1. Note that by construction $\mathcal{C}_{Z/Z'} = i^*\mathcal{C}_{X/X'}$ hence we conclude that $\mathcal{C}_{Z/Z'}$ is isomorphic to the pullback of $\mathcal{C}_{U/U'}$ to $Z$. Applying More on Morphisms of Spaces, Lemma 76.18.1 once again (or its analogue for schemes, see More on Morphisms, Lemma 37.10.1) we conclude that $Z' \to U'$ is flat and that $Z = U \times _{U'} Z'$. Finally, More on Morphisms, Lemma 37.10.3 shows that $Z' \to U'$ is finite locally free of degree $d$. $\square$

Proof. We have to show the following: Given

1. an object $(U, Z, y, x, \alpha )$ of $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ over an affine scheme $U$,

2. a first order thickening $U \subset U'$, and

3. an object $y'$ of $\mathcal{Y}$ over $U'$ such that $y'|_ U = y$,

then there exists an object $(U', Z', y', x', \alpha ')$ of $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ over $U'$ with $Z = U \times _{U'} Z'$, with $x = x'|_ Z$, and with $\alpha = \alpha '|_ U$. Namely, the last two equalities will take care of the commutativity of (97.6.0.1).

Consider the morphism $x_\alpha : Z \to X_ y$ constructed in Equation (97.15.0.1). Denote similarly $X'_{y'}$ the algebraic space over $U'$ representing the $2$-fibre product $(\mathit{Sch}/U')_{fppf} \times _{y', \mathcal{Y}, F} \mathcal{X}$. By assumption the morphism $X'_{y'} \to U'$ is flat (and locally of finite presentation). As $y'|_ U = y$ we see that $X_ y = U \times _{U'} X'_{y'}$. Hence we may apply Lemma 97.15.2 to find $Z' \to U'$ finite locally free of degree $d$ with $Z = U \times _{U'} Z'$ and with $Z' \to X'_{y'}$ extending $x_\alpha $. By construction the morphism $Z' \to X'_{y'}$ corresponds to a pair $(x', \alpha ')$. It is clear that $(U', Z', y', x', \alpha ')$ is an object of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ over $U'$ with $Z = U \times _{U'} Z'$, with $x = x'|_ Z$, and with $\alpha = \alpha '|_ U$. As we've seen in Lemma 97.15.1 that $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \subset \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ is an “open substack” it follows that $(U', Z', y', x', \alpha ')$ is an object of $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ as desired. $\square$

Proof. It suffices to prove the following: For any field $k$ and object $y$ of $\mathcal{Y}$ over $\mathop{\mathrm{Spec}}(k)$ there exists an integer $d \geq 1$ and an object $(U, Z, y, x, \alpha )$ of $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ with $U = \mathop{\mathrm{Spec}}(k)$. Namely, in this case we see that $p$ is surjective on objects in the strong sense that an extension of the field is not needed.

Denote $X_ y$ the algebraic space over $U = \mathop{\mathrm{Spec}}(k)$ representing the $2$-fibre product $(\mathit{Sch}/U')_{fppf} \times _{y', \mathcal{Y}, F} \mathcal{X}$. By assumption the morphism $X_ y \to \mathop{\mathrm{Spec}}(k)$ is surjective and locally of finite presentation (and flat). In particular $X_ y$ is nonempty. Choose a nonempty affine scheme $V$ and an étale morphism $V \to X_ y$. Note that $V \to \mathop{\mathrm{Spec}}(k)$ is (flat), surjective, and locally of finite presentation (by Morphisms of Spaces, Definition 67.28.1). Pick a closed point $v \in V$ where $V \to \mathop{\mathrm{Spec}}(k)$ is Cohen-Macaulay (i.e., $V$ is Cohen-Macaulay at $v$), see More on Morphisms, Lemma 37.22.7. Applying More on Morphisms, Lemma 37.23.4 we find a regular immersion $Z \to V$ with $Z = \{ v\} $. This implies $Z \to V$ is a closed immersion. Moreover, it follows that $Z \to \mathop{\mathrm{Spec}}(k)$ is finite (for example by Algebra, Lemma 10.122.1). Hence $Z \to \mathop{\mathrm{Spec}}(k)$ is finite locally free of some degree $d$. Now $Z \to X_ y$ is unramified as the composition of a closed immersion followed by an étale morphism (see Morphisms of Spaces, Lemmas 67.38.3, 67.39.10, and 67.38.8). Finally, $Z \to X_ y$ is a local complete intersection morphism as a composition of a regular immersion of schemes and an étale morphism of algebraic spaces (see More on Morphisms, Lemma 37.62.9 and Morphisms of Spaces, Lemmas 67.39.6 and 67.37.8 and More on Morphisms of Spaces, Lemmas 76.48.6 and 76.48.5). The morphism $Z \to X_ y$ corresponds to an object $x$ of $\mathcal{X}$ over $Z$ together with an isomorphism $\alpha : y|_ Z \to F(x)$. We obtain an object $(U, Z, y, x, \alpha )$ of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$. By what was said above about the morphism $Z \to X_ y$ we see that it actually is an object of the subcategory $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ and we win. $\square$