Definition 76.5.1. 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.
76.5 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 67.13.1. Consider the short exact sequence
\[ 0 \to \mathcal{I}^2 \to \mathcal{I} \to \mathcal{I}/\mathcal{I}^2 \to 0 \]
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 67.14.1. 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 67.14.
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 67.12.4. 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
\[ \mathcal{N}_{Z/X} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Z}(\mathcal{C}_{Z/X}, \mathcal{O}_ Z) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Z}(\mathcal{I}/\mathcal{I}^2, \mathcal{O}_ Z) \]
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 76.5.2. 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 $U$.
\[ \mathcal{C}_{Z/X}|_{Z_ U} = \mathcal{C}_{Z_ U/U} \]
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 67.13.1. $\square$
Lemma 76.5.3. 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
\[ \xymatrix{ Z \ar[r]_ i \ar[d]_ f & X \ar[d]^ g \\ Z' \ar[r]^{i'} & X' } \]
\[ f^*\mathcal{C}_{Z'/X'} \longrightarrow \mathcal{C}_{Z/X} \]
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 67.13.1. Consider the composition
\[ g^{-1}\mathcal{I}' \to g^{-1}\mathcal{O}_{X'} \xrightarrow {g^\sharp } \mathcal{O}_ X \to \mathcal{O}_ X/\mathcal{I} = i_*\mathcal{O}_ Z \]
Since $g(i(Z)) \subset Z'$ we conclude this composition is zero (see statement on factorizations in Morphisms of Spaces, Lemma 67.13.1). Thus we obtain a commutative diagram
\[ \xymatrix{ 0 \ar[r] & \mathcal{I} \ar[r] & \mathcal{O}_ X \ar[r] & i_*\mathcal{O}_ Z \ar[r] & 0 \\ 0 \ar[r] & g^{-1}\mathcal{I}' \ar[r] \ar[u] & g^{-1}\mathcal{O}_{X'} \ar[r] \ar[u] & g^{-1}i'_*\mathcal{O}_{Z'} \ar[r] \ar[u] & 0 } \]
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 76.5.4. Let $S$ be a scheme. The conormal sheaf of Definition 76.5.1, and its functoriality of Lemma 76.5.3 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 76.5.3 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
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 $(f')^*\mathcal{C}_{Z''/X''} \to \mathcal{C}_{Z'/X'}$
\[ \xymatrix{ Z \ar[r]_ i \ar[d]_ f & X \ar[d]^ g \\ Z' \ar[r]^{i'} \ar[d]_{f'} & X' \ar[d]^{g'} \\ Z'' \ar[r]^{i''} & X'' } \]
Proof. Omitted. Note that Part (1) is a special case of Lemma 76.5.2. $\square$
Lemma 76.5.5. 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 76.5.3 is surjective. If $g$ is flat, then it is an isomorphism.
\[ \xymatrix{ Z \ar[r]_ i \ar[d]_ f & X \ar[d]^ g \\ Z' \ar[r]^{i'} & X' } \]
Proof. Choose a commutative diagram
\[ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ U' \ar[r] & X' } \]
where $U$, $U'$ are schemes and the horizontal arrows are surjective and étale, see Spaces, Lemma 65.11.6. Then using Lemmas 76.5.2 and 76.5.4 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 76.5.6. 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 76.5.3 and $i : Z \to Y$ is the first morphism.
\[ i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 \]
Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. Via Lemmas 76.5.2 and 76.5.4 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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