The Stacks project

95.15 LCI locus in the Hilbert stack

Please consult Examples of Stacks, Section 93.18 for notation. Fix a $1$-morphism $F : \mathcal{X} \longrightarrow \mathcal{Y}$ of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume that $F$ is representable by algebraic spaces. Fix $d \geq 1$. Consider an object $(U, Z, y, x, \alpha )$ of $\mathcal{H}_ d$. There is an induced $1$-morphism

\[ (\mathit{Sch}/Z)_{fppf} \longrightarrow (\mathit{Sch}/U)_{fppf} \times _{y, \mathcal{Y}, F} \mathcal{X} \]

(by the universal property of $2$-fibre products) which is representable by a morphism of algebraic spaces over $U$. Namely, since $F$ is representable by algebraic spaces, we may choose an algebraic space $X_ y$ over $U$ which represents the $2$-fibre product $(\mathit{Sch}/U)_{fppf} \times _{y, \mathcal{Y}, F} \mathcal{X}$. Since $\alpha : y|_ Z \to F(x)$ is an isomorphism we see that $\xi = (Z, Z \to U, x, \alpha )$ is an object of the $2$-fibre product $(\mathit{Sch}/U)_{fppf} \times _{y, \mathcal{Y}, F} \mathcal{X}$ over $Z$. Hence $\xi $ gives rise to a morphism $x_\alpha : Z \to X_ y$ of algebraic spaces over $U$ as $X_ y$ is the functor of isomorphisms classes of objects of $(\mathit{Sch}/U)_{fppf} \times _{y, \mathcal{Y}, F} \mathcal{X}$, see Algebraic Stacks, Lemma 92.8.2. Here is a picture

95.15.0.1
\begin{equation} \label{criteria-equation-relative-map} \vcenter { \xymatrix{ Z \ar[r]_{x_\alpha } \ar[rd] & X_ y \ar[d] \\ & U } } \quad \quad \vcenter { \xymatrix{ (\mathit{Sch}/Z)_{fppf} \ar[rd] \ar[r]_-{x, \alpha } & (\mathit{Sch}/U)_{fppf} \times _{y, \mathcal{Y}, F} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d]^ F \\ & (\mathit{Sch}/U)_{fppf} \ar[r]^ y & \mathcal{Y} } } \end{equation}

We remark that if $(f, g, b, a) : (U, Z, y, x, \alpha ) \to (U', Z', y', x', \alpha ')$ is a morphism between objects of $\mathcal{H}_ d$, then the morphism $x'_{\alpha '} : Z' \to X'_{y'}$ is the base change of the morphism $x_\alpha $ by the morphism $g : U' \to U$ (details omitted).

Now assume moreover that $F$ is flat and locally of finite presentation. In this situation we define a full subcategory

\[ \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \subset \mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \]

consisting of those objects $(U, Z, y, x, \alpha )$ of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ such that the corresponding morphism $x_\alpha : Z \to X_ y$ is unramified and a local complete intersection morphism (see Morphisms of Spaces, Definition 65.38.1 and More on Morphisms of Spaces, Definition 74.48.1 for definitions).

Lemma 95.15.1. Let $S$ be a scheme. Fix a $1$-morphism $F : \mathcal{X} \longrightarrow \mathcal{Y}$ of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $F$ is representable by algebraic spaces, flat, and locally of finite presentation. Then $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ is a stack in groupoids and the inclusion functor

\[ \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \]

is representable and an open immersion.

Proof. Let $\Xi = (U, Z, y, x, \alpha )$ be an object of $\mathcal{H}_ d$. It follows from the remark following (95.15.0.1) 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 74.49.1 and 74.49.7 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

\[ (\mathit{Sch}/U)_{fppf} \times _{\Xi , \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \]

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 92.15.5. $\square$

Local complete intersection morphisms are “locally unobstructed”. This holds in much greater generality than the special case that we need in this chapter here.

Lemma 95.15.2. Let $U \subset U'$ be a first order thickening of affine schemes. Let $X'$ be an algebraic space flat over $U'$. Set $X = U \times _{U'} X'$. Let $Z \to U$ be finite locally free of degree $d$. Finally, let $f : Z \to X$ be unramified and a local complete intersection morphism. Then there exists a commutative diagram

\[ \xymatrix{ (Z \subset Z') \ar[rd] \ar[rr]_{(f, f')} & & (X \subset X') \ar[ld] \\ & (U \subset U') } \]

of algebraic spaces over $U'$ such that $Z' \to U'$ is finite locally free of degree $d$ and $Z = U \times _{U'} Z'$.

Proof. By More on Morphisms of Spaces, Lemma 74.48.12 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 74.48.13 we have an exact sequence

\[ 0 \to i^*\mathcal{C}_{X/X'} \to \mathcal{C}_{Z/X'} \to \mathcal{C}_{Z/X} \to 0 \]

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

\[ \mathcal{C}_{Z/X'} = i^*\mathcal{C}_{X/X'} \oplus \mathcal{C}_{Z/X} \]

Let $Z \subset Z''$ be the universal first order thickening of $Z$ over $X'$ (see More on Morphisms of Spaces, Section 74.15). 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

\[ \mathcal{I} = i^*\mathcal{C}_{X/X'} \oplus \mathcal{C}_{Z/X} \]

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 74.18.1. 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 74.18.1 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 $d$. $\square$

Lemma 95.15.3. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $F$ is representable by algebraic spaces, flat, and locally of finite presentation. Then

\[ p : \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y} \]

is formally smooth on objects.

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 (95.6.0.1).

Consider the morphism $x_\alpha : Z \to X_ y$ constructed in Equation (95.15.0.1). 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 95.15.2 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 95.15.1 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. $\square$

Lemma 95.15.4. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $F$ is representable by algebraic spaces, flat, surjective, and locally of finite presentation. Then

\[ \coprod \nolimits _{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{Y} \]

is surjective on objects.

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 65.28.1). 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 37.20.7. Applying More on Morphisms, Lemma 37.21.4 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 10.121.1). 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 65.38.3, 65.39.10, and 65.38.8). 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 37.54.9 and Morphisms of Spaces, Lemmas 65.39.6 and 65.37.8 and More on Morphisms of Spaces, Lemmas 74.48.6 and 74.48.5). 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. $\square$


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