The Stacks project

74.7 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 10.130), 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 74.7.1. 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 74.7.5.

Definition 74.7.2. 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 64.21.3. The universal $Y$-derivation will be denoted $\text{d}_{X/Y} : \mathcal{O}_ X \to \Omega _{X/Y}$.

By Lemma 74.7.1 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 74.7.3. 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 64.18.10 and Equation (64.18.10.1) 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 74.7.4. 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 74.7.3 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 64.29.6. $\square$

Remark 74.7.5. Now that we know that $\Omega _{X/Y}$ is quasi-coherent we can attempt to construct it in another manner. For example we can use the result of Properties of Spaces, Section 64.32 to construct the sheaf of differentials by glueing. For example if $Y$ is a scheme and if $U \to X$ is a surjective étale morphism from a scheme towards $X$, then we see that $\Omega _{U/Y}$ is a quasi-coherent $\mathcal{O}_ U$-module, and since $s, t : R \to U$ are étale we get an isomorphism

\[ \alpha : s^*\Omega _{U/Y} \to \Omega _{R/Y} \to t^*\Omega _{U/Y} \]

by using Morphisms, Lemma 29.33.16. You check that this satisfies the cocycle condition and you're done. If $Y$ is not a scheme, then you define $\Omega _{U/Y}$ as the cokernel of the map $(U \to Y)^*\Omega _{Y/S} \to \Omega _{U/S}$, and proceed as before. This two step process is a little bit ugly. Another possibility is to glue the sheaves $\Omega _{U/V}$ for any diagram as in Lemma 74.7.3 but this is not very elegant either. Both approaches will work however, and will give a slightly more elementary construction of the sheaf of differentials.

Lemma 74.7.6. 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 74.7.7. 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 74.7.8. 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 74.7.6.

Proof. Follows from the schemes version, see Morphisms, Lemma 29.32.9, of this result via étale localization, see Lemma 74.7.3. $\square$

Lemma 74.7.9. 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 74.7.3. $\square$

Lemma 74.7.10. 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 74.7.6.

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 74.7.3 and 74.5.2. 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 65.14.1 this corresponds to a map $\mathcal{C}_{Z/X} \to i^*\Omega _{X/S}$ as desired. $\square$

Lemma 74.7.11. 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 74.7.10 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 74.7.3 and 74.5.2. $\square$

Lemma 74.7.12. 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 74.7.6 is an isomorphism.

Proof. Follows from the schemes version, see Morphisms, Lemma 29.32.10 and étale localization, see Lemma 74.7.3. $\square$

Lemma 74.7.13. 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 74.7.6

\[ p^*\Omega _{X/B} \oplus q^*\Omega _{Y/B} \longrightarrow \Omega _{X \times _ B Y/B} \]

give an isomorphism.

Proof. Follows from the schemes version, see Morphisms, Lemma 29.32.11 and étale localization, see Lemma 74.7.3. $\square$

Lemma 74.7.14. 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 74.7.3. $\square$

Lemma 74.7.15. 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 74.7.3. $\square$

Lemma 74.7.16. 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 74.7.3. Hence this follows from the case of schemes, see Morphisms, Lemma 29.33.12. $\square$


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