Lemma 29.32.15. Let $i : Z \to X$ be an immersion of schemes over $S$. There is a canonical exact sequence

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

where the first arrow is induced by $\text{d}_{X/S}$ and the second arrow comes from Lemma 29.32.8.

Proof. This is the sheafified version of Algebra, Lemma 10.131.9. However we should make sure we can define the first arrow globally. Hence we explain the meaning of “induced by $\text{d}_{X/S}$” here. Namely, we may assume that $i$ is a closed immersion by shrinking $X$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the sheaf of ideals corresponding to $Z \subset X$. Then $\text{d}_{X/S} : \mathcal{I} \to \Omega _{X/S}$ maps the subsheaf $\mathcal{I}^2 \subset \mathcal{I}$ to $\mathcal{I}\Omega _{X/S}$. Hence it induces a map $\mathcal{I}/\mathcal{I}^2 \to \Omega _{X/S}/\mathcal{I}\Omega _{X/S}$ which is $\mathcal{O}_ X/\mathcal{I}$-linear. By Lemma 29.4.1 this corresponds to a map $\mathcal{C}_{Z/X} \to i^*\Omega _{X/S}$ as desired. $\square$

Comment #8579 by on

Suggested alternative proof:

We can assume that $i$ is a closed immersion by shrinking $X$ if necessary. Let $\mathcal{I}\subset\mathcal{O}_X$ be the ideal sheaf associated to $Z$. Consider the sequence Modules, Lemma 17.28.9, by setting $\mathcal{O}_2\to\mathcal{O}_2'$ equal to $\mathcal{O}_X\to\mathcal{O}_X/\mathcal{I}$ and $\mathcal{O}_1=f^{-1}\mathcal{O}_S$, where $f:X\to S$ is the structure morphism. Since $i^*$ is right exact, we can pull back along $i$ and get an exact sequence where in the second term we have used Modules, Lemma 17.16.4. Hence, the second term is $i^*\Omega_{X/S}$. On the other hand, by Modules, Lemma 17.28.6, the third term equals where the last step is done using formula $\mathcal{F}\otimes_{\mathcal{O}_Y}\mathcal{O}_Y/\mathcal{J}\cong\mathcal{F}/\mathcal{J}\mathcal{F}$, for $(Y,\mathcal{O}_Y)$ a ringed space, $\mathcal{J}\subset\mathcal{O}_Y$ an ideal sheaf and $\mathcal{F}$ an $\mathcal{O}_Y$-module.

Comment #8580 by on

Okay, I forgot to justify the last claim from the statement in the alternative proof: one sees that the resulting pulled-back map $i^*\Omega_{X/S}\to\Omega_{Z/S}$ equals $c_i$ (Lemma 29.32.8) by applying Modules, Lemma 17.28.13, taking $S=S'=S''$ and $X''\to X'\to X$ to be the morphisms of ringed spaces $(Z,\mathcal{O}_Z)\to (X,\mathcal{O}_X/\mathcal{I})\to(X,\mathcal{O}_X)$.

I'm not sure if any of this is any better the already existing proof. I just felt it was nice to connect Lemma 29.32 to the similar-looking Modules, Lemma 17.28.9.

Last thing: maybe one could make explicit in the statement that the maps are $\mathcal{O}_Z$-linear.

There are also:

• 2 comment(s) on Section 29.32: Sheaf of differentials of a morphism

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