## 75.44 Regular immersions

This section is the analogue of Divisors, Section 31.21 for morphisms of algebraic spaces. The reader is encouraged to read up on regular immersions of schemes in that section first.

In Divisors, Section 31.21 we defined four types of regular immersions for morphisms of schemes. Of these only three are (as far as we know) local on the target for the étale topology; as usual plain old regular immersions aren't. This is why for morphisms of algebraic spaces we cannot actually define regular immersions. (These kinds of annoyances prompted Grothendieck and his school to replace original notion of a regular immersion by a Koszul-regular immersions, see [Exposee VII, Definition 1.4, SGA6].) But we can define Koszul-regular, $H_1$-regular, and quasi-regular immersions. Another remark is that since Koszul-regular immersions are not preserved by arbitrary base change, we cannot use the strategy of Morphisms of Spaces, Section 66.3 to define them. Similarly, as Koszul-regular immersions are not étale local on the source, we cannot use Morphisms of Spaces, Lemma 66.22.1 to define them either. We replace this lemma instead by the following.

Lemma 75.44.1. Let $\mathcal{P}$ be a property of morphisms of schemes which is étale local on the target. Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Consider commutative diagrams

$\xymatrix{ X \times _ Y V \ar[d] \ar[r] & V \ar[d] \\ X \ar[r]^ f & Y }$

where $V$ is a scheme and $V \to Y$ is étale. The following are equivalent

1. for any diagram as above the projection $X \times _ Y V \to V$ has property $\mathcal{P}$, and

2. for some diagram as above with $V \to Y$ surjective the projection $X \times _ Y V \to V$ has property $\mathcal{P}$.

If $X$ and $Y$ are representable, then this is also equivalent to $f$ (as a morphism of schemes) having property $\mathcal{P}$.

Proof. Let us prove the equivalence of (1) and (2). The implication (1) $\Rightarrow$ (2) is immediate. Assume

$\xymatrix{ X \times _ Y V \ar[d] \ar[r] & V \ar[d] \\ X \ar[r]^ f & Y } \quad \quad \xymatrix{ X \times _ Y V' \ar[d] \ar[r] & V' \ar[d] \\ X \ar[r]^ f & Y }$

are two diagrams as in the lemma. Assume $V \to Y$ is surjective and $X \times _ Y V \to V$ has property $\mathcal{P}$. To show that (2) implies (1) we have to prove that $X \times _ Y V' \to V'$ has $\mathcal{P}$. To do this consider the diagram

$\xymatrix{ X \times _ Y V \ar[d] & (X \times _ Y V) \times _ X (X \times _ Y V') \ar[l] \ar[d] \ar[r] & X \times _ Y V' \ar[d] \\ V & V \times _ Y V' \ar[l] \ar[r] & V' }$

By our assumption that $\mathcal{P}$ is étale local on the source, we see that $\mathcal{P}$ is preserved under étale base change, see Descent, Lemma 35.22.2. Hence if the left vertical arrow has $\mathcal{P}$ the so does the middle vertical arrow. Since $U \times _ X U' \to U'$ is surjective and étale (hence defines an étale covering of $U'$) this implies (as $\mathcal{P}$ is assumed local for the étale topology on the target) that the left vertical arrow has $\mathcal{P}$.

If $X$ and $Y$ are representable, then we can take $\text{id}_ Y : Y \to Y$ as our étale covering to see the final statement of the lemma is true. $\square$

Note that “being a Koszul-regular (resp. $H_1$-regular, resp. quasi-regular) immersion” is a property of morphisms of schemes which is fpqc local on the target, see Descent, Lemma 35.23.32. Hence the following definition now makes sense.

Definition 75.44.2. Let $S$ be a scheme. Let $i : X \to Y$ be a morphism of algebraic spaces over $S$.

1. We say $i$ is a Koszul-regular immersion if $i$ is representable and the equivalent conditions of Lemma 75.44.1 hold with $\mathcal{P}(f) =$“$f$ is a Koszul-regular immersion”.

2. We say $i$ is an $H_1$-regular immersion if $i$ is representable and the equivalent conditions of Lemma 75.44.1 hold with $\mathcal{P}(f) =$“$f$ is an $H_1$-regular immersion”.

3. We say $i$ is a quasi-regular immersion if $i$ is representable and the equivalent conditions of Lemma 75.44.1 hold with $\mathcal{P}(f) =$“$f$ is a quasi-regular immersion”.

Lemma 75.44.3. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. We have the following implications: $i$ is Koszul-regular $\Rightarrow$ $i$ is $H_1$-regular $\Rightarrow$ $i$ is quasi-regular.

Proof. Via the definition this lemma immediately reduces to Divisors, Lemma 31.21.2. $\square$

Lemma 75.44.4. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Assume $X$ is locally Noetherian. Then $i$ is Koszul-regular $\Leftrightarrow$ $i$ is $H_1$-regular $\Leftrightarrow$ $i$ is quasi-regular.

Proof. Via Definition 75.44.2 (and the definition of a locally Noetherian algebraic space in Properties of Spaces, Section 65.7) this immediately translates to the case of schemes which is Divisors, Lemma 31.21.3. $\square$

Lemma 75.44.5. Let $S$ be a scheme. Let $i : Z \to X$ be a Koszul-regular, $H_1$-regular, or quasi-regular immersion of algebraic spaces over $S$. Let $X' \to X$ be a flat morphism of algebraic spaces over $S$. Then the base change $i' : Z \times _ X X' \to X'$ is a Koszul-regular, $H_1$-regular, or quasi-regular immersion.

Proof. Via Definition 75.44.2 (and the definition of a flat morphism of algebraic spaces in Morphisms of Spaces, Section 66.30) this lemma reduces to the case of schemes, see Divisors, Lemma 31.21.4. $\square$

Lemma 75.44.6. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Then $i$ is a quasi-regular immersion if and only if the following conditions are satisfied

1. $i$ is locally of finite presentation,

2. the conormal sheaf $\mathcal{C}_{Z/X}$ is finite locally free, and

3. the map (75.6.1.2) is an isomorphism.

Proof. Follows from the case of schemes (Divisors, Lemma 31.21.5) via étale localization (use Definition 75.44.2 and Lemma 75.6.2). $\square$

Lemma 75.44.7. Let $S$ be a scheme. Let $Z \to Y \to X$ be immersions of algebraic spaces over $S$. Assume that $Z \to Y$ is $H_1$-regular. Then the canonical sequence of Lemma 75.5.6

$0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0$

is exact and (étale) locally split.

Proof. Since $\mathcal{C}_{Z/Y}$ is finite locally free (see Lemma 75.44.6 and Lemma 75.44.3) it suffices to prove that the sequence is exact. It suffices to show that the first map is injective as the sequence is already right exact in general. After étale localization on $X$ this reduces to the case of schemes, see Divisors, Lemma 31.21.6. $\square$

A composition of quasi-regular immersions may not be quasi-regular, see Algebra, Remark 10.69.8. The other types of regular immersions are preserved under composition.

Lemma 75.44.8. Let $S$ be a scheme. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of algebraic spaces over $S$.

1. If $i$ and $j$ are Koszul-regular immersions, so is $j \circ i$.

2. If $i$ and $j$ are $H_1$-regular immersions, so is $j \circ i$.

3. If $i$ is an $H_1$-regular immersion and $j$ is a quasi-regular immersion, then $j \circ i$ is a quasi-regular immersion.

Proof. Immediate from the case of schemes, see Divisors, Lemma 31.21.7. $\square$

Lemma 75.44.9. Let $S$ be a scheme. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of algebraic spaces over $S$. Assume that the sequence

$0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0$

of Lemma 75.5.6 is exact and locally split.

1. If $j \circ i$ is a quasi-regular immersion, so is $i$.

2. If $j \circ i$ is a $H_1$-regular immersion, so is $i$.

3. If both $j$ and $j \circ i$ are Koszul-regular immersions, so is $i$.

Proof. Immediate from the case of schemes, see Divisors, Lemma 31.21.8. $\square$

Lemma 75.44.10. Let $S$ be a scheme. Let $i : Z \to Y$ and $j : Y \to X$ be immersions of algebraic spaces over $S$. Assume $X$ is locally Noetherian. The following are equivalent

1. $i$ and $j$ are Koszul regular immersions,

2. $i$ and $j \circ i$ are Koszul regular immersions,

3. $j \circ i$ is a Koszul regular immersion and the conormal sequence

$0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0$

is exact and locally split.

Proof. Immediate from the case of schemes, see Divisors, Lemma 31.21.9. $\square$

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