We suggest the reader take a look at the corresponding section in the chapter on commutative algebra (Algebra, Section 10.131) and the corresponding section in the chapter on sheaves of modules (Modules, Section 17.28).
Definition 29.32.1. Let $f : X \to S$ be a morphism of schemes. The sheaf of differentials $\Omega _{X/S}$ of $X$ over $S$ is the sheaf of differentials of $f$ viewed as a morphism of ringed spaces (Modules, Definition 17.28.10) equipped with its universal $S$-derivation
\[ \text{d}_{X/S} : \mathcal{O}_ X \longrightarrow \Omega _{X/S}. \]
It turns out that $\Omega _{X/S}$ is a quasi-coherent $\mathcal{O}_ X$-module for example as it is isomorphic to the conormal sheaf of the diagonal morphism $\Delta : X \to X \times _ S X$ (Lemma 29.32.7). We have defined the module of differentials of $X$ over $S$ using a universal property, namely as the receptacle of the universal derivation. If you have any other construction of the sheaf of relative differentials which satisfies this universal property then, by the Yoneda lemma, it will be canonically isomorphic to the one defined above. For convenience we restate the universal property here.
Lemma 29.32.2. Let $f : X \to S$ be a morphism of schemes. The map is an isomorphism of functors $\textit{Mod}(\mathcal{O}_ X) \to \textit{Sets}$.
\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/S}, \mathcal{F}) \longrightarrow \text{Der}_ S(\mathcal{O}_ X, \mathcal{F}),\quad \alpha \longmapsto \alpha \circ \text{d}_{X/S} \]
Proof. This is just a restatement of the definition. $\square$
Lemma 29.32.3. Let $f : X \to S$ be a morphism of schemes. Let $U \subset X$, $V \subset S$ be open subschemes such that $f(U) \subset V$. Then there is a unique isomorphism $\Omega _{X/S}|_ U = \Omega _{U/V}$ of $\mathcal{O}_ U$-modules such that $\text{d}_{X/S}|_ U = \text{d}_{U/V}$.
Proof. This is a special case of Modules, Lemma 17.28.5 if we use the canonical identification $f^{-1}\mathcal{O}_ S|_ U = (f|_ U)^{-1}\mathcal{O}_ V$. $\square$
From now on we will use these canonical identifications and simply write $\Omega _{U/S}$ or $\Omega _{U/V}$ for the restriction of $\Omega _{X/S}$ to $U$.
Lemma 29.32.4. Let $R \to A$ be a ring map. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules on $X = \mathop{\mathrm{Spec}}(A)$. Set $S = \mathop{\mathrm{Spec}}(R)$. The rule which associates to an $S$-derivation on $\mathcal{F}$ its action on global sections defines a bijection between the set of $S$-derivations of $\mathcal{F}$ and the set of $R$-derivations on $M = \Gamma (X, \mathcal{F})$.
Proof. Let $D : A \to M$ be an $R$-derivation. We have to show there exists a unique $S$-derivation on $\mathcal{F}$ which gives rise to $D$ on global sections. Let $U = D(f) \subset X$ be a standard affine open. Any element of $\Gamma (U, \mathcal{O}_ X)$ is of the form $a/f^ n$ for some $a \in A$ and $n \geq 0$. By the Leibniz rule we have
in $\Gamma (U, \mathcal{F})$. Since $f$ acts invertibly on $\Gamma (U, \mathcal{F})$ this completely determines the value of $D(a/f^ n) \in \Gamma (U, \mathcal{F})$. This proves uniqueness. Existence follows by simply defining
and proving this has all the desired properties (on the basis of standard opens of $X$). Details omitted. $\square$
Lemma 29.32.5. Let $f : X \to S$ be a morphism of schemes. For any pair of affine opens $\mathop{\mathrm{Spec}}(A) = U \subset X$, $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $f(U) \subset V$ there is a unique isomorphism compatible with $\text{d}_{X/S}$ and $\text{d} : A \to \Omega _{A/R}$.
\[ \Gamma (U, \Omega _{X/S}) = \Omega _{A/R}. \]
Proof. By Lemma 29.32.3 we may replace $X$ and $S$ by $U$ and $V$. Thus we may assume $X = \mathop{\mathrm{Spec}}(A)$ and $S = \mathop{\mathrm{Spec}}(R)$ and we have to show the lemma with $U = X$ and $V = S$. Consider the $A$-module $M = \Gamma (X, \Omega _{X/S})$ together with the $R$-derivation $\text{d}_{X/S} : A \to M$. Let $N$ be another $A$-module and denote $\widetilde{N}$ the quasi-coherent $\mathcal{O}_ X$-module associated to $N$, see Schemes, Section 26.7. Precomposing by $\text{d}_{X/S} : A \to M$ we get an arrow
Using Lemmas 29.32.2 and 29.32.4 we get identifications
Taking global sections determines an arrow $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/S}, \widetilde{N}) \to \mathop{\mathrm{Hom}}\nolimits _ R(M, N)$. Combining this arrow and the identifications above we get an arrow
Checking what happens on global sections, we find that $\alpha $ and $\beta $ are each others inverse. Hence we see that $\text{d}_{X/S} : A \to M$ satisfies the same universal property as $\text{d} : A \to \Omega _{A/R}$, see Algebra, Lemma 10.131.3. Thus the Yoneda lemma (Categories, Lemma 4.3.5) implies there is a unique isomorphism of $A$-modules $M \cong \Omega _{A/R}$ compatible with derivations. $\square$
Remark 29.32.6. The lemma above gives a second way of constructing the module of differentials. Namely, let $f : X \to S$ be a morphism of schemes. Consider the collection of all affine opens $U \subset X$ which map into an affine open of $S$. These form a basis for the topology on $X$. Thus it suffices to define $\Gamma (U, \Omega _{X/S})$ for such $U$. We simply set $\Gamma (U, \Omega _{X/S}) = \Omega _{A/R}$ if $A$, $R$ are as in Lemma 29.32.5 above. This works, but it takes somewhat more algebraic preliminaries to construct the restriction mappings and to verify the sheaf condition with this ansatz.
The following lemma gives yet another way to define the sheaf of differentials and it in particular shows that $\Omega _{X/S}$ is quasi-coherent if $X$ and $S$ are schemes.
Lemma 29.32.7. Let $f : X \to S$ be a morphism of schemes. There is a canonical isomorphism between $\Omega _{X/S}$ and the conormal sheaf of the diagonal morphism $\Delta _{X/S} : X \longrightarrow X \times _ S X$.
Proof. We first establish the existence of a couple of “global” sheaves and global maps of sheaves, and further down we describe the constructions over some affine opens.
Recall that $\Delta = \Delta _{X/S} : X \to X \times _ S X$ is an immersion, see Schemes, Lemma 26.21.2. Let $\mathcal{J}$ be the ideal sheaf of the immersion which lives over some open subscheme $W$ of $X \times _ S X$ such that $\Delta (X) \subset W$ is closed. Let us take the one that was found in the proof of Schemes, Lemma 26.21.2. Note that the sheaf of rings $\mathcal{O}_ W/\mathcal{J}^2$ is supported on $\Delta (X)$. Moreover it sits in a short exact sequence of sheaves
Using $\Delta ^{-1}$ we can think of this as a surjection of sheaves of $f^{-1}\mathcal{O}_ S$-algebras with kernel the conormal sheaf of $\Delta $ (see Definition 29.31.1 and Lemma 29.31.2).
This places us in the situation of Modules, Lemma 17.28.11. The projection morphisms $p_ i : X \times _ S X \to X$, $i = 1, 2$ induce maps of sheaves of rings $(p_ i)^\sharp : (p_ i)^{-1}\mathcal{O}_ X \to \mathcal{O}_{X \times _ S X}$. We may restrict to $W$ and quotient by $\mathcal{J}^2$ to get $(p_ i)^{-1}\mathcal{O}_ X \to \mathcal{O}_ W/\mathcal{J}^2$. Since $\Delta ^{-1}p_ i^{-1}\mathcal{O}_ X = \mathcal{O}_ X$ we get maps
Both $s_1$ and $s_2$ are sections to the map $\Delta ^{-1}(\mathcal{O}_ W/\mathcal{J}^2) \to \mathcal{O}_ X$, as in Modules, Lemma 17.28.11. Thus we get an $S$-derivation $\text{d} = s_2 - s_1 : \mathcal{O}_ X \to \mathcal{C}_{X/X \times _ S X}$. By the universal property of the module of differentials we find a unique $\mathcal{O}_ X$-linear map
To see the map is an isomorphism, let us work this out over suitable affine opens. We can cover $X$ by affine opens $\mathop{\mathrm{Spec}}(A) = U \subset X$ whose image is contained in an affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$. According to the proof of Schemes, Lemma 26.21.2 $U \times _ V U \subset X \times _ S X$ is an affine open contained in the open $W$ mentioned above. Also $U \times _ V U = \mathop{\mathrm{Spec}}(A \otimes _ R A)$. The sheaf $\mathcal{J}$ corresponds to the ideal $J = \mathop{\mathrm{Ker}}(A \otimes _ R A \to A)$. The short exact sequence to the short exact sequence of $A \otimes _ R A$-modules
The sections $s_ i$ correspond to the ring maps
By Lemma 29.31.2 we have $\Gamma (U, \mathcal{C}_{X/X \times _ S X}) = J/J^2$ and by Lemma 29.32.5 we have $\Gamma (U, \Omega _{X/S}) = \Omega _{A/R}$. The map above is the map $a \text{d}b \mapsto a \otimes b - ab \otimes 1$ which is shown to be an isomorphism in Algebra, Lemma 10.131.13. $\square$
Lemma 29.32.8. Let be a commutative diagram of schemes. The canonical map $\mathcal{O}_ X \to f_*\mathcal{O}_{X'}$ composed with the map $f_*\text{d}_{X'/S'} : f_*\mathcal{O}_{X'} \to f_*\Omega _{X'/S'}$ is a $S$-derivation. Hence we obtain a canonical map of $\mathcal{O}_ X$-modules $\Omega _{X/S} \to f_*\Omega _{X'/S'}$, and by adjointness of $f_*$ and $f^*$ a canonical $\mathcal{O}_{X'}$-module homomorphism It is uniquely characterized by the property that $f^*\text{d}_{X/S}(h)$ maps to $\text{d}_{X'/S'}(f^* h)$ for any local section $h$ of $\mathcal{O}_ X$.
\[ \xymatrix{ X' \ar[d] \ar[r]_ f & X \ar[d] \\ S' \ar[r] & S } \]
\[ c_ f : f^*\Omega _{X/S} \longrightarrow \Omega _{X'/S'}. \]
Proof. This is a special case of Modules, Lemma 17.28.12. In the case of schemes we can also use the functoriality of the conormal sheaves (see Lemma 29.31.3) and Lemma 29.32.7 to define $c_ f$. Or we can use the characterization in the last line of the lemma to glue maps defined on affine patches (see Algebra, Equation (10.131.4.1)). $\square$
Lemma 29.32.9. Let $f : X \to Y$, $g : Y \to S$ be morphisms of schemes. Then there is a canonical exact sequence where the maps come from applications of Lemma 29.32.8.
\[ f^*\Omega _{Y/S} \to \Omega _{X/S} \to \Omega _{X/Y} \to 0 \]
Proof. This is the sheafified version of Algebra, Lemma 10.131.7. Alternatively, there is a general version for morphisms of ringed spaces, see Modules, Lemma 17.28.14. $\square$
Lemma 29.32.10. Let $X \to S$ be a morphism of schemes. Let $g : S' \to S$ be a morphism of schemes. Let $X' = X_{S'}$ be the base change of $X$. Denote $g' : X' \to X$ the projection. Then the map of Lemma 29.32.8 is an isomorphism.
\[ (g')^*\Omega _{X/S} \to \Omega _{X'/S'} \]
Proof. This is the sheafified version of Algebra, Lemma 10.131.12. $\square$
Lemma 29.32.11. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Let $p : X \times _ S Y \to X$ and $q : X \times _ S Y \to Y$ be the projection morphisms. The maps from Lemma 29.32.8 give an isomorphism.
\[ p^*\Omega _{X/S} \oplus q^*\Omega _{Y/S} \longrightarrow \Omega _{X \times _ S Y/S} \]
Proof. By Lemma 29.32.10 the composition $p^*\Omega _{X/S} \to \Omega _{X \times _ S Y/S} \to \Omega _{X \times _ S Y/Y}$ is an isomorphism, and similarly for $q$. Moreover, the cokernel of $p^*\Omega _{X/S} \to \Omega _{X \times _ S Y/S}$ is $\Omega _{X \times _ S Y/X}$ by Lemma 29.32.9. The result follows. $\square$
Lemma 29.32.12. Let $f : X \to S$ be a morphism of schemes. If $f$ is locally of finite type, then $\Omega _{X/S}$ is a finite type $\mathcal{O}_ X$-module.
Proof. Immediate from Algebra, Lemma 10.131.16, Lemma 29.32.5, Lemma 29.15.2, and Properties, Lemma 28.16.1. $\square$
Lemma 29.32.13. Let $f : X \to S$ be a morphism of schemes. If $f$ is locally of finite presentation, then $\Omega _{X/S}$ is an $\mathcal{O}_ X$-module of finite presentation.
Proof. Immediate from Algebra, Lemma 10.131.15, Lemma 29.32.5, Lemma 29.21.2, and Properties, Lemma 28.16.2. $\square$
Lemma 29.32.14. If $X \to S$ is an immersion, or more generally a monomorphism, then $\Omega _{X/S}$ is zero.
Proof. This is true because $\Delta _{X/S}$ is an isomorphism in this case and hence has trivial conormal sheaf. Hence $\Omega _{X/S} = 0$ by Lemma 29.32.7. The algebraic version is Algebra, Lemma 10.131.4. $\square$
Lemma 29.32.15. Let $i : Z \to X$ be an immersion of schemes over $S$. There is a canonical exact sequence where the first arrow is induced by $\text{d}_{X/S}$ and the second arrow comes from Lemma 29.32.8.
\[ \mathcal{C}_{Z/X} \to i^*\Omega _{X/S} \to \Omega _{Z/S} \to 0 \]
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. $\square$
Lemma 29.32.16. Let $i : Z \to X$ be an immersion of schemes over $S$, and assume $i$ (locally) has a left inverse. Then the canonical sequence of Lemma 29.32.15 is (locally) split exact. In particular, if $s : S \to X$ is a section of the structure morphism $X \to S$ then the map $\mathcal{C}_{S/X} \to s^*\Omega _{X/S}$ induced by $\text{d}_{X/S}$ is an isomorphism.
\[ 0 \to \mathcal{C}_{Z/X} \to i^*\Omega _{X/S} \to \Omega _{Z/S} \to 0 \]
Proof. Follows from Algebra, Lemma 10.131.10. Clarification: if $g : X \to Z$ is a left inverse of $i$, then $i^*c_ g$ is a right inverse of the map $i^*\Omega _{X/S} \to \Omega _{Z/S}$. Also, if $s$ is a section, then it is an immersion $s : Z = S \to X$ over $S$ (see Schemes, Lemma 26.21.11) and in that case $\Omega _{Z/S} = 0$. $\square$
Remark 29.32.17. Let $X \to S$ be a morphism of schemes. According to Lemma 29.32.11 we have On the other hand, the diagonal morphism $\Delta : X \to X \times _ S X$ is an immersion, which locally has a left inverse. Hence by Lemma 29.32.16 we obtain a canonical short exact sequence Note that the right arrow is $(1, 1)$ which is indeed a split surjection. On the other hand, by Lemma 29.32.7 we have an identification $\Omega _{X/S} = \mathcal{C}_{X/X \times _ S X}$. Because we chose $\text{d}_{X/S}(f) = s_2(f) - s_1(f)$ in this identification it turns out that the left arrow is the map $(-1, 1)$1.
\[ \Omega _{X \times _ S X/S} = \text{pr}_1^*\Omega _{X/S} \oplus \text{pr}_2^*\Omega _{X/S} \]
\[ 0 \to \mathcal{C}_{X/X \times _ S X} \to \Omega _{X/S} \oplus \Omega _{X/S} \to \Omega _{X/S} \to 0 \]
Lemma 29.32.18. Let be a commutative diagram of schemes where $i$ and $j$ are immersions. Then there is a canonical exact sequence where the first arrow comes from Lemma 29.31.3 and the second from Lemma 29.32.15.
\[ \xymatrix{ Z \ar[r]_ i \ar[rd]_ j & X \ar[d] \\ & Y } \]
\[ \mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega _{X/Y} \to 0 \]
Proof. The algebraic version of this is Algebra, Lemma 10.134.7. $\square$
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