The Stacks project

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:

  1. 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.

  2. 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.

  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. $\square$

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