76.7 Sheaf of differentials of a morphism
We suggest the reader take a look at the corresponding section in the chapter on commutative algebra (Algebra, Section 10.131), the corresponding section in the chapter on morphism of schemes (Morphisms, Section 29.32) as well as Modules on Sites, Section 18.33. We first show that the notion of sheaf of differentials for a morphism of schemes agrees with the corresponding morphism of small étale (ringed) sites.
To clearly state the following lemma we temporarily go back to denoting $\mathcal{F}^ a$ the sheaf of $\mathcal{O}_{X_{\acute{e}tale}}$-modules associated to a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ on the scheme $X$, see Descent, Definition 35.8.2.
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.
$\square$
This clears the way for the following definition. For an alternative, see Remark 76.7.5.
Definition 76.7.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The sheaf of differentials $\Omega _{X/Y}$ of $X$ over $Y$ is sheaf of differentials (Modules on Sites, Definition 18.33.10) for the morphism of ringed topoi
\[ (f_{small}, f^\sharp ) : (X_{\acute{e}tale}, \mathcal{O}_ X) \to (Y_{\acute{e}tale}, \mathcal{O}_ Y) \]
of Properties of Spaces, Lemma 66.21.3. The universal $Y$-derivation will be denoted $\text{d}_{X/Y} : \mathcal{O}_ X \to \Omega _{X/Y}$.
By Lemma 76.7.1 this does not conflict with the already existing notion in case $X$ and $Y$ are representable. From now on, if $X$ and $Y$ are representable, we no longer distinguish between the sheaf of differentials defined above and the one defined in Morphisms, Definition 29.32.1. We want to relate this to the usual modules of differentials for morphisms of schemes. Here is the key lemma.
Lemma 76.7.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Consider any commutative diagram
\[ \xymatrix{ U \ar[d]_ a \ar[r]_\psi & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]
where the vertical arrows are étale morphisms of algebraic spaces. Then
\[ \Omega _{X/Y}|_{U_{\acute{e}tale}} = \Omega _{U/V} \]
In particular, if $U$, $V$ are schemes, then this is equal to the usual sheaf of differentials of the morphism of schemes $U \to V$.
Proof.
By Properties of Spaces, Lemma 66.18.11 and Equation (66.18.11.1) we may think of the restriction of a sheaf on $X_{\acute{e}tale}$ to $U_{\acute{e}tale}$ as the pullback by $a_{small}$. Similarly for $b$. By Modules on Sites, Lemma 18.33.6 we have
\[ \Omega _{X/Y}|_{U_{\acute{e}tale}} = \Omega _{\mathcal{O}_{U_{\acute{e}tale}}/ a_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}}} \]
Since $a_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}} = \psi _{small}^{-1}b_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}} = \psi _{small}^{-1}\mathcal{O}_{V_{\acute{e}tale}}$ we see that the lemma holds.
$\square$
Lemma 76.7.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $\Omega _{X/Y}$ is a quasi-coherent $\mathcal{O}_ X$-module.
Proof.
Choose a diagram as in Lemma 76.7.3 with $a$ and $b$ surjective and $U$ and $V$ schemes. Then we see that $\Omega _{X/Y}|_ U = \Omega _{U/V}$ which is quasi-coherent (for example by Morphisms, Lemma 29.32.7). Hence we conclude that $\Omega _{X/Y}$ is quasi-coherent by Properties of Spaces, Lemma 66.29.6.
$\square$
Lemma 76.7.6. Let $S$ be a scheme. Let
\[ \xymatrix{ X' \ar[d] \ar[r]_ f & X \ar[d] \\ Y' \ar[r] & Y } \]
be a commutative diagram of algebraic spaces. The map $f^\sharp : \mathcal{O}_ X \to f_*\mathcal{O}_{X'}$ composed with the map $f_*\text{d}_{X'/Y'} : f_*\mathcal{O}_{X'} \to f_*\Omega _{X'/Y'}$ is a $Y$-derivation. Hence we obtain a canonical map of $\mathcal{O}_ X$-modules $\Omega _{X/Y} \to f_*\Omega _{X'/Y'}$, and by adjointness of $f_*$ and $f^*$ a canonical $\mathcal{O}_{X'}$-module homomorphism
\[ c_ f : f^*\Omega _{X/Y} \longrightarrow \Omega _{X'/Y'}. \]
It is uniquely characterized by the property that $f^*\text{d}_{X/Y}(t)$ mapsto $\text{d}_{X'/Y'}(f^* t)$ for any local section $t$ of $\mathcal{O}_ X$.
Proof.
This is a special case of Modules on Sites, Lemma 18.33.11.
$\square$
Lemma 76.7.7. Let $S$ be a scheme. Let
\[ \xymatrix{ X'' \ar[d] \ar[r]_ g & X' \ar[d] \ar[r]_ f & X \ar[d] \\ Y'' \ar[r] & Y' \ar[r] & Y } \]
be a commutative diagram of algebraic spaces over $S$. Then we have
\[ c_{f \circ g} = c_ g \circ g^* c_ f \]
as maps $(f \circ g)^*\Omega _{X/Y} \to \Omega _{X''/Y''}$.
Proof.
Omitted. Hint: Use the characterization of $c_ f, c_ g, c_{f \circ g}$ in terms of the effect these maps have on local sections.
$\square$
Lemma 76.7.8. Let $S$ be a scheme. Let $f : X \to Y$, $g : Y \to B$ be morphisms of algebraic spaces over $S$. Then there is a canonical exact sequence
\[ f^*\Omega _{Y/B} \to \Omega _{X/B} \to \Omega _{X/Y} \to 0 \]
where the maps come from applications of Lemma 76.7.6.
Proof.
Follows from the schemes version, see Morphisms, Lemma 29.32.9, of this result via étale localization, see Lemma 76.7.3.
$\square$
Lemma 76.7.9. Let $S$ be a scheme. If $X \to Y$ is an immersion of algebraic spaces over $S$ then $\Omega _{X/S}$ is zero.
Proof.
Follows from the schemes version, see Morphisms, Lemma 29.32.14, of this result via étale localization, see Lemma 76.7.3.
$\square$
Lemma 76.7.10. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $i : Z \to X$ be an immersion of algebraic spaces over $B$. There is a canonical exact sequence
\[ \mathcal{C}_{Z/X} \to i^*\Omega _{X/B} \to \Omega _{Z/B} \to 0 \]
where the first arrow is induced by $\text{d}_{X/B}$ and the second arrow comes from Lemma 76.7.6.
Proof.
This is the algebraic spaces version of Morphisms, Lemma 29.32.15 and will be a consequence of that lemma by étale localization, see Lemmas 76.7.3 and 76.5.2. However, we should make sure we can define the first arrow globally. Hence we explain the meaning of “induced by $\text{d}_{X/B}$” here. Namely, we may assume that $i$ is a closed immersion after replacing $X$ by an open subspace. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent 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 Morphisms of Spaces, Lemma 67.14.1 this corresponds to a map $\mathcal{C}_{Z/X} \to i^*\Omega _{X/S}$ as desired.
$\square$
Lemma 76.7.11. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $i : Z \to X$ be an immersion of algebraic spaces over $B$, and assume $i$ (étale locally) has a left inverse. Then the canonical sequence
\[ 0 \to \mathcal{C}_{Z/X} \to i^*\Omega _{X/B} \to \Omega _{Z/B} \to 0 \]
of Lemma 76.7.10 is (étale locally) split exact.
Proof.
Clarification: we claim that if $g : X \to Z$ is a left inverse of $i$ over $B$, then $i^*c_ g$ is a right inverse of the map $i^*\Omega _{X/B} \to \Omega _{Z/B}$. Having said this, the result follows from the corresponding result for morphisms of schemes by étale localization, see Lemmas 76.7.3 and 76.5.2.
$\square$
Lemma 76.7.12. Let $S$ be a scheme. Let $X \to Y$ be a morphism of algebraic spaces over $S$. Let $g : Y' \to Y$ be a morphism of algebraic spaces over $S$. Let $X' = X_{Y'}$ be the base change of $X$. Denote $g' : X' \to X$ the projection. Then the map
\[ (g')^*\Omega _{X/Y} \to \Omega _{X'/Y'} \]
of Lemma 76.7.6 is an isomorphism.
Proof.
Follows from the schemes version, see Morphisms, Lemma 29.32.10 and étale localization, see Lemma 76.7.3.
$\square$
Lemma 76.7.13. Let $S$ be a scheme. Let $f : X \to B$ and $g : Y \to B$ be morphisms of algebraic spaces over $S$ with the same target. Let $p : X \times _ B Y \to X$ and $q : X \times _ B Y \to Y$ be the projection morphisms. The maps from Lemma 76.7.6
\[ p^*\Omega _{X/B} \oplus q^*\Omega _{Y/B} \longrightarrow \Omega _{X \times _ B Y/B} \]
give an isomorphism.
Proof.
Follows from the schemes version, see Morphisms, Lemma 29.32.11 and étale localization, see Lemma 76.7.3.
$\square$
Lemma 76.7.14. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, then $\Omega _{X/Y}$ is a finite type $\mathcal{O}_ X$-module.
Proof.
Follows from the schemes version, see Morphisms, Lemma 29.32.12 and étale localization, see Lemma 76.7.3.
$\square$
Lemma 76.7.15. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite presentation, then $\Omega _{X/Y}$ is an $\mathcal{O}_ X$-module of finite presentation.
Proof.
Follows from the schemes version, see Morphisms, Lemma 29.32.13 and étale localization, see Lemma 76.7.3.
$\square$
Lemma 76.7.16. Let $S$ be a scheme. Let $f : X \to Y$ be a smooth morphism of algebraic spaces over $S$. Then the module of differentials $\Omega _{X/Y}$ is finite locally free.
Proof.
The statement is étale local on $X$ and $Y$ by Lemma 76.7.3. Hence this follows from the case of schemes, see Morphisms, Lemma 29.34.12.
$\square$
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