The Stacks project

Lemma 76.7.1. Let $f : X \to Y$ be a morphism of schemes. Let $f_{small} : X_{\acute{e}tale}\to Y_{\acute{e}tale}$ be the associated morphism of small étale sites, see Descent, Remark 35.8.4. Then there is a canonical isomorphism

\[ (\Omega _{X/Y})^ a = \Omega _{X_{\acute{e}tale}/Y_{\acute{e}tale}} \]

compatible with universal derivations. Here the first module is the sheaf on $X_{\acute{e}tale}$ associated to the quasi-coherent $\mathcal{O}_ X$-module $\Omega _{X/Y}$, see Morphisms, Definition 29.32.1, and the second module is the one from Modules on Sites, Definition 18.33.3.

Proof. Let $h : U \to X$ be an étale morphism. In this case the natural map $h^*\Omega _{X/Y} \to \Omega _{U/Y}$ is an isomorphism, see More on Morphisms, Lemma 37.9.9. This means that there is a natural $\mathcal{O}_{Y_{\acute{e}tale}}$-derivation

\[ \text{d}^ a : \mathcal{O}_{X_{\acute{e}tale}} \longrightarrow (\Omega _{X/Y})^ a \]

since we have just seen that the value of $(\Omega _{X/Y})^ a$ on any object $U$ of $X_{\acute{e}tale}$ is canonically identified with $\Gamma (U, \Omega _{U/Y})$. By the universal property of $\text{d}_{X/Y} : \mathcal{O}_{X_{\acute{e}tale}} \to \Omega _{X_{\acute{e}tale}/Y_{\acute{e}tale}}$ there is a unique $\mathcal{O}_{X_{\acute{e}tale}}$-linear map $c : \Omega _{X_{\acute{e}tale}/Y_{\acute{e}tale}} \to (\Omega _{X/Y})^ a$ such that $\text{d}^ a = c \circ \text{d}_{X/Y}$.

Conversely, suppose that $\mathcal{F}$ is an $\mathcal{O}_{X_{\acute{e}tale}}$-module and $D : \mathcal{O}_{X_{\acute{e}tale}} \to \mathcal{F}$ is a $\mathcal{O}_{Y_{\acute{e}tale}}$-derivation. Then we can simply restrict $D$ to the small Zariski site $X_{Zar}$ of $X$. Since sheaves on $X_{Zar}$ agree with sheaves on $X$, see Descent, Remark 35.8.3, we see that $D|_{X_{Zar}} : \mathcal{O}_ X \to \mathcal{F}|_{X_{Zar}}$ is just a “usual” $Y$-derivation. Hence we obtain a map $\psi : \Omega _{X/Y} \longrightarrow \mathcal{F}|_{X_{Zar}}$ such that $D|_{X_{Zar}} = \psi \circ \text{d}$. In particular, if we apply this with $\mathcal{F} = \Omega _{X_{\acute{e}tale}/Y_{\acute{e}tale}}$ we obtain a map

\[ c' : \Omega _{X/Y} \longrightarrow \Omega _{X_{\acute{e}tale}/Y_{\acute{e}tale}}|_{X_{Zar}} \]

Consider the morphism of ringed sites $\text{id}_{small, {\acute{e}tale}, Zar} : X_{\acute{e}tale}\to X_{Zar}$ discussed in Descent, Remark 35.8.4 and Lemma 35.8.5. Since the restriction functor $\mathcal{F} \mapsto \mathcal{F}|_{X_{Zar}}$ is equal to $\text{id}_{small, {\acute{e}tale}, Zar, *}$, since $\text{id}_{small, {\acute{e}tale}, Zar}^*$ is left adjoint to $\text{id}_{small, {\acute{e}tale}, Zar, *}$ and since $(\Omega _{X/Y})^ a = \text{id}_{small, {\acute{e}tale}, Zar}^*\Omega _{X/Y}$ we see that $c'$ is adjoint to a map

\[ c'' : (\Omega _{X/Y})^ a \longrightarrow \Omega _{X_{\acute{e}tale}/Y_{\acute{e}tale}}. \]

We claim that $c''$ and $c'$ are mutually inverse. This claim finishes the proof of the lemma. To see this it is enough to show that $c''(\text{d}(f)) = \text{d}_{X/Y}(f)$ and $c(\text{d}_{X/Y}(f)) = \text{d}(f)$ if $f$ is a local section of $\mathcal{O}_ X$ over an open of $X$. We omit the verification. $\square$

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