76.48 Local complete intersection morphisms
This section is the analogue of More on Morphisms, Section 37.62 for morphisms of schemes. The reader is encouraged to read up on local complete intersection morphisms of schemes in that section first.
The property “being a local complete intersection morphism” of morphisms of schemes is étale local on the source-and-target. To see this use More on Morphisms, Lemmas 37.62.19 and 37.62.20 and Descent, Lemma 35.32.6. By Morphisms of Spaces, Lemma 67.22.1 we may define the notion of a local complete intersection morphism of algebraic spaces as follows and it agrees with the already existing notion defined in More on Morphisms, Section 37.62 when the algebraic spaces in question are representable.
Definition 76.48.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.
We say $f$ is a Koszul morphism, or that $f$ is a local complete intersection morphism if the equivalent conditions of Morphisms of Spaces, Lemma 67.22.1 hold with $\mathcal{P}(f) =$“$f$ is a local complete intersection morphism”.
Let $x \in |X|$. We say $f$ is Koszul at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is a local complete intersection morphism.
In some sense the defining property of a local complete intersection morphism is the result of the following lemma.
Lemma 76.48.2. Let $S$ be a scheme. Let $f : X \to Y$ be a local complete intersection morphism of algebraic spaces over $S$. Let $P$ be an algebraic space smooth over $Y$. Let $U \to X$ be an étale morphism of algebraic spaces and let $i : U \to P$ an immersion of algebraic spaces over $Y$. Picture:
\[ \xymatrix{ X \ar[rd] & U \ar[l] \ar[d] \ar[r]_ i & P \ar[ld] \\ & Y } \]
Then $i$ is a Koszul-regular immersion of algebraic spaces.
Proof.
Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $W$ and a surjective étale morphism $W \to P \times _ Y V$. Set $U' = U \times _ P W$, which is a scheme étale over $U$. We have to show that $U' \to W$ is a Koszul-regular immersion of schemes, see Definition 76.44.2. By Definition 76.48.1 above the morphism of schemes $U' \to V$ is a local complete intersection morphism. Hence the result follows from More on Morphisms, Lemma 37.62.3.
$\square$
It seems like a good idea to collect here some properties in common with all Koszul morphisms.
Lemma 76.48.3. Let $S$ be a scheme. Let $f : X \to Y$ be a local complete intersection morphism of algebraic spaces over $S$. Then
$f$ is locally of finite presentation,
$f$ is pseudo-coherent, and
$f$ is perfect.
Proof.
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.62.4.
$\square$
Beware that a base change of a Koszul morphism is not Koszul in general.
Lemma 76.48.4. A flat base change of a local complete intersection morphism is a local complete intersection morphism.
Proof.
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.62.6.
$\square$
Lemma 76.48.5. A composition of local complete intersection morphisms is a local complete intersection morphism.
Proof.
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.62.7.
$\square$
slogan
Lemma 76.48.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent
$f$ is flat and a local complete intersection morphism, and
$f$ is syntomic.
Proof.
Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.62.8.
$\square$
Lemma 76.48.7. Let $S$ be a scheme. A Koszul-regular immersion of algebraic spaces over $S$ is a local complete intersection morphism.
Proof.
Let $i : X \to Y$ be a Koszul-regular immersion of algebraic spaces over $S$. By definition there exists a surjective étale morphism $V \to Y$ where $V$ is a scheme such that $X \times _ Y V$ is a scheme and the base change $X \times _ Y V \to V$ is a Koszul-regular immersion of schemes. By More on Morphisms, Lemma 37.62.9 we see that $X \times _ Y V \to V$ is a local complete intersection morphism. From Definition 76.48.1 we conclude that $i$ is a local complete intersection morphism of algebraic spaces.
$\square$
Lemma 76.48.8. Let $S$ be a scheme. Let
\[ \xymatrix{ X \ar[rr]_ f \ar[rd] & & Y \ar[ld] \\ & Z } \]
be a commutative diagram of morphisms of algebraic spaces over $S$. Assume $Y \to Z$ is smooth and $X \to Z$ is a local complete intersection morphism. Then $f : X \to Y$ is a local complete intersection morphism.
Proof.
Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Choose a scheme $V$ and a surjective étale morphism $V \to W \times _ Z Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Then $U \to W$ is a local complete intersection morphism of schemes and $V \to W$ is a smooth morphism of schemes. By the result for schemes (More on Morphisms, Lemma 37.62.10) we conclude that $U \to V$ is a local complete intersection morphism. By definition this means that $f$ is a local complete intersection morphism.
$\square$
Lemma 76.48.9. The property $\mathcal{P}(f) =$“$f$ is a local complete intersection morphism” is fpqc local on the base.
Proof.
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{ Y_ i \to Y\} $ be an fpqc covering (Topologies on Spaces, Definition 73.9.1). Let $f_ i : X_ i \to Y_ i$ be the base change of $f$ by $Y_ i \to Y$. If $f$ is a local complete intersection morphism, then each $f_ i$ is a local complete intersection morphism by Lemma 76.48.4.
Conversely, assume each $f_ i$ is a local complete intersection morphism. We may replace the covering by a refinement (again because flat base change preserves the property of being a local complete intersection morphism). Hence we may assume $Y_ i$ is a scheme for each $i$, see Topologies on Spaces, Lemma 73.9.5. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. We have to show that $U \to V$ is a local complete intersection morphism of schemes. By Topologies on Spaces, Lemma 73.9.4 we have that $\{ Y_ i \times _ Y V \to V\} $ is an fpqc covering of schemes. By the case of schemes (More on Morphisms, Lemma 37.62.19) it suffices to prove the base change
\[ U \times _ Y Y_ i = U \times _ V (V \times _ Y Y_ i) \longrightarrow V \]
of $U \to V$ by $V \times _ Y Y_ i \to V$ is a local complete intersection morphism. We can write this as the composition
\[ U \times _ Y Y_ i \longrightarrow (V \times _ Y X) \times _ Y Y_ i = V \times _ Y X_ i \longrightarrow V \times _ Y Y_ i \]
The first arrow is an étale morphism of schemes (as a base change of $U \to V \times _ Y X$) and the second arrow is a local complete intersection morphism of schemes as a flat base change of $f_ i$. The result follows as being a local complete intersection morphism is syntomic local on the source and since étale morphisms are syntomic (More on Morphisms, Lemma 37.62.20 and Morphisms, Lemma 29.36.10).
$\square$
Lemma 76.48.10. The property $\mathcal{P}(f) =$“$f$ is a local complete intersection morphism” is syntomic local on the source.
Proof.
This follows from Descent on Spaces, Lemma 74.14.3 and More on Morphisms, Lemma 37.62.20.
$\square$
Lemma 76.48.11. Let $S$ be a scheme. Consider a commutative diagram
\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[ld]^ q \\ & Z } \]
of algebraic spaces over $S$. Assume that both $p$ and $q$ are flat and locally of finite presentation. Then there exists an open subspace $U(f) \subset X$ such that $|U(f)| \subset |X|$ is the set of points where $f$ is Koszul. Moreover, for any morphism of algebraic spaces $Z' \to Z$, if $f' : X' \to Y'$ is the base change of $f$ by $Z' \to Z$, then $U(f')$ is the inverse image of $U(f)$ under the projection $X' \to X$.
Proof.
This lemma is the analogue of More on Morphisms, Lemma 37.62.21 and in fact we will deduce the lemma from it. By Definition 76.48.1 the set $\{ x \in |X| : f \text{ is Koszul at }x\} $ is open in $|X|$ hence by Properties of Spaces, Lemma 66.4.8 it corresponds to an open subspace $U(f)$ of $X$. Hence we only need to prove the final statement.
Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Choose a scheme $V$ and a surjective étale morphism $V \to W \times _ Z Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Finally, choose a scheme $W'$ and a surjective étale morphism $W' \to W \times _ Z Z'$. Set $V' = W' \times _ W V$ and $U' = W' \times _ W U$, so that we obtain surjective étale morphisms $V' \to Y'$ and $U' \to X'$. We will use without further mention an étale morphism of algebraic spaces induces an open map of associated topological spaces (see Properties of Spaces, Lemma 66.16.7). Note that by definition $U(f)$ is the image in $|X|$ of the set $T$ of points in $U$ where the morphism of schemes $U \to V$ is Koszul. Similarly, $U(f')$ is the image in $|X'|$ of the set $T'$ of points in $U'$ where the morphism of schemes $U' \to V'$ is Koszul. Now, by construction the diagram
\[ \xymatrix{ U' \ar[r] \ar[d] & U \ar[d] \\ V' \ar[r] & V } \]
is cartesian (in the category of schemes). Hence the aforementioned More on Morphisms, Lemma 37.62.21 applies to show that $T'$ is the inverse image of $T$. Since $|U'| \to |X'|$ is surjective this implies the lemma.
$\square$
Lemma 76.48.12. Let $S$ be a scheme. Let $f : X \to Y$ be a local complete intersection morphism of algebraic spaces over $S$. Then $f$ is unramified if and only if $f$ is formally unramified and in this case the conormal sheaf $\mathcal{C}_{X/Y}$ is finite locally free on $X$.
Proof.
This follows from the corresponding result for morphisms of schemes, see More on Morphisms, Lemma 37.62.22, by étale localization, see Lemma 76.15.11. (Note that in the situation of this lemma the morphism $V \to U$ is unramified and a local complete intersection morphism by definition.)
$\square$
Lemma 76.48.13. Let $S$ be a scheme. Let $Z \to Y \to X$ be formally unramified morphisms of algebraic spaces over $S$. Assume that $Z \to Y$ is a local complete intersection morphism. The exact sequence
\[ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 \]
of Lemma 76.5.6 is short exact.
Proof.
Choose a scheme $U$ and a surjective étale morphism $U \to X$. Choose a scheme $V$ and a surjective étale morphism $V \to U \times _ X Y$. Choose a scheme $W$ and a surjective étale morphism $W \to V \times _ Y Z$. By Lemma 76.15.11 the morphisms $W \to V$ and $V \to U$ are formally unramified. Moreover the sequence $i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0$ restricts to the corresponding sequence $i^*\mathcal{C}_{V/U} \to \mathcal{C}_{W/U} \to \mathcal{C}_{W/V} \to 0$ for $W \to V \to U$. Hence the result follows from the result for schemes (More on Morphisms, Lemma 37.62.23) as by definition the morphism $W \to V$ is a local complete intersection morphism.
$\square$
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