Definition 75.5.1. Let $i : Z \to X$ be an immersion. The *conormal sheaf $\mathcal{C}_{Z/X}$ of $Z$ in $X$* or the *conormal sheaf of $i$* is the quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{I}/\mathcal{I}^2$ described above.

## 75.5 Conormal sheaf of an immersion

Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the corresponding quasi-coherent sheaf of ideals, see Morphisms of Spaces, Lemma 66.13.1. Consider the short exact sequence

of quasi-coherent sheaves on $X$. Since the sheaf $\mathcal{I}/\mathcal{I}^2$ is annihilated by $\mathcal{I}$ it corresponds to a sheaf on $Z$ by Morphisms of Spaces, Lemma 66.14.1. This quasi-coherent $\mathcal{O}_ Z$-module is the *conormal sheaf of $Z$ in $X$* and is often denoted $\mathcal{I}/\mathcal{I}^2$ by the abuse of notation mentioned in Morphisms of Spaces, Section 66.14.

In case $i : Z \to X$ is a (locally closed) immersion we define the conormal sheaf of $i$ as the conormal sheaf of the closed immersion $i : Z \to X \setminus \partial Z$, see Morphisms of Spaces, Remark 66.12.4. It is often denoted $\mathcal{I}/\mathcal{I}^2$ where $\mathcal{I}$ is the ideal sheaf of the closed immersion $i : Z \to X \setminus \partial Z$.

In [IV Definition 16.1.2, EGA] this sheaf is denoted $\mathcal{N}_{Z/X}$. We will not follow this convention since we would like to reserve the notation $\mathcal{N}_{Z/X}$ for the *normal sheaf of the immersion*. It is defined as

provided the conormal sheaf is of finite presentation (otherwise the normal sheaf may not even be quasi-coherent). We will come back to the normal sheaf later (insert future reference here).

Lemma 75.5.2. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion. Let $\varphi : U \to X$ be an étale morphism where $U$ is a scheme. Set $Z_ U = U \times _ X Z$ which is a locally closed subscheme of $U$. Then

canonically and functorially in $U$.

**Proof.**
Let $T \subset X$ be a closed subspace such that $i$ defines a closed immersion into $X \setminus T$. Let $\mathcal{I}$ be the quasi-coherent sheaf of ideals on $X \setminus T$ defining $Z$. Then the lemma just states that $\mathcal{I}|_{U \setminus \varphi ^{-1}(T)}$ is the sheaf of ideals of the immersion $Z_ U \to U \setminus \varphi ^{-1}(T)$. This is clear from the construction of $\mathcal{I}$ in Morphisms of Spaces, Lemma 66.13.1.
$\square$

Lemma 75.5.3. Let $S$ be a scheme. Let

be a commutative diagram of algebraic spaces over $S$. Assume $i$, $i'$ immersions. There is a canonical map of $\mathcal{O}_ Z$-modules

**Proof.**
First find open subspaces $U' \subset X'$ and $U \subset X$ such that $g(U) \subset U'$ and such that $i(Z) \subset U$ and $i(Z') \subset U'$ are closed (proof existence omitted). Replacing $X$ by $U$ and $X'$ by $U'$ we may assume that $i$ and $i'$ are closed immersions. Let $\mathcal{I}' \subset \mathcal{O}_{X'}$ and $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaves of ideals associated to $i'$ and $i$, see Morphisms of Spaces, Lemma 66.13.1. Consider the composition

Since $g(i(Z)) \subset Z'$ we conclude this composition is zero (see statement on factorizations in Morphisms of Spaces, Lemma 66.13.1). Thus we obtain a commutative diagram

The lower row is exact since $g^{-1}$ is an exact functor. By exactness we also see that $(g^{-1}\mathcal{I}')^2 = g^{-1}((\mathcal{I}')^2)$. Hence the diagram induces a map $g^{-1}(\mathcal{I}'/(\mathcal{I}')^2) \to \mathcal{I}/\mathcal{I}^2$. Pulling back (using $i^{-1}$ for example) to $Z$ we obtain $i^{-1}g^{-1}(\mathcal{I}'/(\mathcal{I}')^2) \to \mathcal{C}_{Z/X}$. Since $i^{-1}g^{-1} = f^{-1}(i')^{-1}$ this gives a map $f^{-1}\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}$, which induces the desired map. $\square$

Lemma 75.5.4. Let $S$ be a scheme. The conormal sheaf of Definition 75.5.1, and its functoriality of Lemma 75.5.3 satisfy the following properties:

If $Z \to X$ is an immersion of schemes over $S$, then the conormal sheaf agrees with the one from Morphisms, Definition 29.31.1.

If in Lemma 75.5.3 all the spaces are schemes, then the map $f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}$ is the same as the one constructed in Morphisms, Lemma 29.31.3.

Given a commutative diagram

\[ \xymatrix{ Z \ar[r]_ i \ar[d]_ f & X \ar[d]^ g \\ Z' \ar[r]^{i'} \ar[d]_{f'} & X' \ar[d]^{g'} \\ Z'' \ar[r]^{i''} & X'' } \]then the map $(f' \circ f)^*\mathcal{C}_{Z''/X''} \to \mathcal{C}_{Z/X}$ is the same as the composition of $f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}$ with the pullback by $f$ of $(f')^*\mathcal{C}_{Z''/X''} \to \mathcal{C}_{Z'/X'}$

**Proof.**
Omitted. Note that Part (1) is a special case of Lemma 75.5.2.
$\square$

Lemma 75.5.5. Let $S$ be a scheme. Let

be a fibre product diagram of algebraic spaces over $S$. Assume $i$, $i'$ immersions. Then the canonical map $f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}$ of Lemma 75.5.3 is surjective. If $g$ is flat, then it is an isomorphism.

**Proof.**
Choose a commutative diagram

where $U$, $U'$ are schemes and the horizontal arrows are surjective and étale, see Spaces, Lemma 64.11.6. Then using Lemmas 75.5.2 and 75.5.4 we see that the question reduces to the case of a morphism of schemes. In the schemes case this is Morphisms, Lemma 29.31.4. $\square$

Lemma 75.5.6. Let $S$ be a scheme. Let $Z \to Y \to X$ be immersions of algebraic spaces. Then there is a canonical exact sequence

where the maps come from Lemma 75.5.3 and $i : Z \to Y$ is the first morphism.

**Proof.**
Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. Via Lemmas 75.5.2 and 75.5.4 the exactness of the sequence translates immediately into the exactness of the corresponding sequence for the immersions of schemes $Z \times _ X U \to Y \times _ X U \to U$. Hence the lemma follows from Morphisms, Lemma 29.31.5.
$\square$

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