17.28 Modules of differentials
In this section we briefly explain how to define the module of relative differentials for a morphism of ringed spaces. We suggest the reader take a look at the corresponding section in the chapter on commutative algebra (Algebra, Section 10.131).
Definition 17.28.1. Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $\mathcal{F}$ be an $\mathcal{O}_2$-module. An $\mathcal{O}_1$-derivation or more precisely a $\varphi $-derivation into $\mathcal{F}$ is a map $D : \mathcal{O}_2 \to \mathcal{F}$ which is additive, annihilates the image of $\mathcal{O}_1 \to \mathcal{O}_2$, and satisfies the Leibniz rule
\[ D(ab) = aD(b) + D(a)b \]
for all $a, b$ local sections of $\mathcal{O}_2$ (wherever they are both defined). We denote $\text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F})$ the set of $\varphi $-derivations into $\mathcal{F}$.
This is the sheaf theoretic analogue of Algebra, Definition 10.131.1. Given a derivation $D : \mathcal{O}_2 \to \mathcal{F}$ as in the definition the map on global sections
\[ D : \Gamma (X, \mathcal{O}_2) \longrightarrow \Gamma (X, \mathcal{F}) \]
is a $\Gamma (X, \mathcal{O}_1)$-derivation as in the algebra definition. Note that if $\alpha : \mathcal{F} \to \mathcal{G}$ is a map of $\mathcal{O}_2$-modules, then there is an induced map
\[ \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F}) \longrightarrow \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{G}) \]
given by the rule $D \mapsto \alpha \circ D$. In other words we obtain a functor.
Lemma 17.28.2. Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. The functor
\[ \textit{Mod}(\mathcal{O}_2) \longrightarrow \textit{Ab}, \quad \mathcal{F} \longmapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F}) \]
is representable.
Proof.
This is proved in exactly the same way as the analogous statement in algebra. During this proof, for any sheaf of sets $\mathcal{F}$ on $X$, let us denote $\mathcal{O}_2[\mathcal{F}]$ the sheafification of the presheaf $U \mapsto \mathcal{O}_2(U)[\mathcal{F}(U)]$ where this denotes the free $\mathcal{O}_2(U)$-module on the set $\mathcal{F}(U)$. For $s \in \mathcal{F}(U)$ we denote $[s]$ the corresponding section of $\mathcal{O}_2[\mathcal{F}]$ over $U$. If $\mathcal{F}$ is a sheaf of $\mathcal{O}_2$-modules, then there is a canonical map
\[ c : \mathcal{O}_2[\mathcal{F}] \longrightarrow \mathcal{F} \]
which on the presheaf level is given by the rule $\sum f_ s[s] \mapsto \sum f_ s s$. We will employ the short hand $[s] \mapsto s$ to describe this map and similarly for other maps below. Consider the map of $\mathcal{O}_2$-modules
17.28.2.1
\begin{equation} \label{modules-equation-define-module-differentials} \begin{matrix} \mathcal{O}_2[\mathcal{O}_2 \times \mathcal{O}_2] \oplus \mathcal{O}_2[\mathcal{O}_2 \times \mathcal{O}_2] \oplus \mathcal{O}_2[\mathcal{O}_1]
& \longrightarrow
& \mathcal{O}_2[\mathcal{O}_2]
\\ [(a, b)] \oplus [(f, g)] \oplus [h]
& \longmapsto
& [a + b] - [a] - [b] +
\\ & & [fg] - g[f] - f[g] +
\\ & & [\varphi (h)]
\end{matrix} \end{equation}
with short hand notation as above. Set $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$ equal to the cokernel of this map. Then it is clear that there exists a map of sheaves of sets
\[ \text{d} : \mathcal{O}_2 \longrightarrow \Omega _{\mathcal{O}_2/\mathcal{O}_1} \]
mapping a local section $f$ to the image of $[f]$ in $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$. By construction $\text{d}$ is a $\mathcal{O}_1$-derivation. Next, let $\mathcal{F}$ be a sheaf of $\mathcal{O}_2$-modules and let $D : \mathcal{O}_2 \to \mathcal{F}$ be a $\mathcal{O}_1$-derivation. Then we can consider the $\mathcal{O}_2$-linear map $\mathcal{O}_2[\mathcal{O}_2] \to \mathcal{F}$ which sends $[g]$ to $D(g)$. It follows from the definition of a derivation that this map annihilates sections in the image of the map (17.28.2.1) and hence defines a map
\[ \alpha _ D : \Omega _{\mathcal{O}_2/\mathcal{O}_1} \longrightarrow \mathcal{F} \]
Since it is clear that $D = \alpha _ D \circ \text{d}$ the lemma is proved.
$\square$
Definition 17.28.3. Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. The module of differentials of $\varphi $ is the object representing the functor $\mathcal{F} \mapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F})$ which exists by Lemma 17.28.2. It is denoted $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$, and the universal $\varphi $-derivation is denoted $\text{d} : \mathcal{O}_2 \to \Omega _{\mathcal{O}_2/\mathcal{O}_1}$.
Note that $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$ is the cokernel of the map (17.28.2.1) of $\mathcal{O}_2$-modules. Moreover the map $\text{d}$ is described by the rule that $\text{d}f$ is the image of the local section $[f]$.
Lemma 17.28.4. Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Then $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$ is the sheaf associated to the presheaf $U \mapsto \Omega _{\mathcal{O}_2(U)/\mathcal{O}_1(U)}$.
Proof.
Consider the map (17.28.2.1). There is a similar map of presheaves whose value on the open $U$ is
\[ \mathcal{O}_2(U)[\mathcal{O}_2(U) \times \mathcal{O}_2(U)] \oplus \mathcal{O}_2(U)[\mathcal{O}_2(U) \times \mathcal{O}_2(U)] \oplus \mathcal{O}_2(U)[\mathcal{O}_1(U)] \longrightarrow \mathcal{O}_2(U)[\mathcal{O}_2(U)] \]
The cokernel of this map has value $\Omega _{\mathcal{O}_2(U)/\mathcal{O}_1(U)}$ over $U$ by the construction of the module of differentials in Algebra, Definition 10.131.2. On the other hand, the sheaves in (17.28.2.1) are the sheafifications of the presheaves above. Thus the result follows as sheafification is exact.
$\square$
Lemma 17.28.5. Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. For $U \subset X$ open there is a canonical isomorphism
\[ \Omega _{\mathcal{O}_2/\mathcal{O}_1}|_ U = \Omega _{(\mathcal{O}_2|_ U)/(\mathcal{O}_1|_ U)} \]
compatible with universal derivations.
Proof.
Holds because $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$ is the cokernel of the map (17.28.2.1).
$\square$
Lemma 17.28.6. Let $f : Y \to X$ be a continuous map of topological spaces. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Then there is a canonical identification $f^{-1}\Omega _{\mathcal{O}_2/\mathcal{O}_1} = \Omega _{f^{-1}\mathcal{O}_2/f^{-1}\mathcal{O}_1}$ compatible with universal derivations.
Proof.
This holds because the sheaf $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$ is the cokernel of the map (17.28.2.1) and a similar statement holds for $\Omega _{f^{-1}\mathcal{O}_2/f^{-1}\mathcal{O}_1}$, because the functor $f^{-1}$ is exact, and because $f^{-1}(\mathcal{O}_2[\mathcal{O}_2]) = f^{-1}\mathcal{O}_2[f^{-1}\mathcal{O}_2]$, $f^{-1}(\mathcal{O}_2[\mathcal{O}_2 \times \mathcal{O}_2]) = f^{-1}\mathcal{O}_2[f^{-1}\mathcal{O}_2 \times f^{-1}\mathcal{O}_2]$, and $f^{-1}(\mathcal{O}_2[\mathcal{O}_1]) = f^{-1}\mathcal{O}_2[f^{-1}\mathcal{O}_1]$.
$\square$
Lemma 17.28.7. Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Let $x \in X$. Then we have $\Omega _{\mathcal{O}_2/\mathcal{O}_1, x} = \Omega _{\mathcal{O}_{2, x}/\mathcal{O}_{1, x}}$.
Proof.
This is a special case of Lemma 17.28.6 for the inclusion map $\{ x\} \to X$. An alternative proof is to use Lemma 17.28.4, Sheaves, Lemma 6.17.2, and Algebra, Lemma 10.131.5.
$\square$
Lemma 17.28.8. Let $X$ be a topological space. Let
\[ \xymatrix{ \mathcal{O}_2 \ar[r]_\varphi & \mathcal{O}_2' \\ \mathcal{O}_1 \ar[r] \ar[u] & \mathcal{O}'_1 \ar[u] } \]
be a commutative diagram of sheaves of rings on $X$. The map $\mathcal{O}_2 \to \mathcal{O}'_2$ composed with the map $\text{d} : \mathcal{O}'_2 \to \Omega _{\mathcal{O}'_2/\mathcal{O}'_1}$ is a $\mathcal{O}_1$-derivation. Hence we obtain a canonical map of $\mathcal{O}_2$-modules $\Omega _{\mathcal{O}_2/\mathcal{O}_1} \to \Omega _{\mathcal{O}'_2/\mathcal{O}'_1}$. It is uniquely characterized by the property that $\text{d}(f) \mapsto \text{d}(\varphi (f))$ for any local section $f$ of $\mathcal{O}_2$. In this way $\Omega _{-/-}$ becomes a functor on the category of arrows of sheaves of rings.
Proof.
This lemma proves itself.
$\square$
Lemma 17.28.9. In Lemma 17.28.8 suppose that $\mathcal{O}_2 \to \mathcal{O}'_2$ is surjective with kernel $\mathcal{I} \subset \mathcal{O}_2$ and assume that $\mathcal{O}_1 = \mathcal{O}'_1$. Then there is a canonical exact sequence of $\mathcal{O}'_2$-modules
\[ \mathcal{I}/\mathcal{I}^2 \longrightarrow \Omega _{\mathcal{O}_2/\mathcal{O}_1} \otimes _{\mathcal{O}_2} \mathcal{O}'_2 \longrightarrow \Omega _{\mathcal{O}'_2/\mathcal{O}_1} \longrightarrow 0 \]
The leftmost map is characterized by the rule that a local section $f$ of $\mathcal{I}$ maps to $\text{d}f \otimes 1$.
Proof.
For a local section $f$ of $\mathcal{I}$ denote $\overline{f}$ the image of $f$ in $\mathcal{I}/\mathcal{I}^2$. To show that the map $\overline{f} \mapsto \text{d}f \otimes 1$ is well defined we just have to check that $\text{d} f_1f_2 \otimes 1 = 0$ if $f_1, f_2$ are local sections of $\mathcal{I}$. And this is clear from the Leibniz rule $\text{d} f_1f_2 \otimes 1 = (f_1 \text{d}f_2 + f_2 \text{d} f_1 )\otimes 1 = \text{d}f_2 \otimes f_1 + \text{d}f_1 \otimes f_2 = 0$. A similar computation show this map is $\mathcal{O}'_2 = \mathcal{O}_2/\mathcal{I}$-linear. The map on the right is the one from Lemma 17.28.8. To see that the sequence is exact, we can check on stalks (Lemma 17.3.1). By Lemma 17.28.7 this follows from Algebra, Lemma 10.131.9.
$\square$
Definition 17.28.10. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (S, \mathcal{O}_ S)$ be a morphism of ringed spaces.
Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. An $S$-derivation into $\mathcal{F}$ is a $f^{-1}\mathcal{O}_ S$-derivation, or more precisely a $f^\sharp $-derivation in the sense of Definition 17.28.1. We denote $\text{Der}_ S(\mathcal{O}_ X, \mathcal{F})$ the set of $S$-derivations into $\mathcal{F}$.
The sheaf of differentials $\Omega _{X/S}$ of $X$ over $S$ is the module of differentials $\Omega _{\mathcal{O}_ X/f^{-1}\mathcal{O}_ S}$ endowed with its universal $S$-derivation $\text{d}_{X/S} : \mathcal{O}_ X \to \Omega _{X/S}$.
Here is a particular situation where derivations come up naturally.
Lemma 17.28.11. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (S, \mathcal{O}_ S)$ be a morphism of ringed spaces. Consider a short exact sequence
\[ 0 \to \mathcal{I} \to \mathcal{A} \to \mathcal{O}_ X \to 0 \]
Here $\mathcal{A}$ is a sheaf of $f^{-1}\mathcal{O}_ S$-algebras, $\pi : \mathcal{A} \to \mathcal{O}_ X$ is a surjection of sheaves of $f^{-1}\mathcal{O}_ S$-algebras, and $\mathcal{I} = \mathop{\mathrm{Ker}}(\pi )$ is its kernel. Assume $\mathcal{I}$ an ideal sheaf with square zero in $\mathcal{A}$. So $\mathcal{I}$ has a natural structure of an $\mathcal{O}_ X$-module. A section $s : \mathcal{O}_ X \to \mathcal{A}$ of $\pi $ is a $f^{-1}\mathcal{O}_ S$-algebra map such that $\pi \circ s = \text{id}$. Given any section $s : \mathcal{O}_ X \to \mathcal{A}$ of $\pi $ and any $S$-derivation $D : \mathcal{O}_ X \to \mathcal{I}$ the map
\[ s + D : \mathcal{O}_ X \to \mathcal{A} \]
is a section of $\pi $ and every section $s'$ is of the form $s + D$ for a unique $S$-derivation $D$.
Proof.
Recall that the $\mathcal{O}_ X$-module structure on $\mathcal{I}$ is given by $h \tau = \tilde h \tau $ (multiplication in $\mathcal{A}$) where $h$ is a local section of $\mathcal{O}_ X$, and $\tilde h$ is a local lift of $h$ to a local section of $\mathcal{A}$, and $\tau $ is a local section of $\mathcal{I}$. In particular, given $s$, we may use $\tilde h = s(h)$. To verify that $s + D$ is a homomorphism of sheaves of rings we compute
\begin{eqnarray*} (s + D)(ab) & = & s(ab) + D(ab) \\ & = & s(a)s(b) + aD(b) + D(a)b \\ & = & s(a) s(b) + s(a)D(b) + D(a)s(b) \\ & = & (s(a) + D(a))(s(b) + D(b)) \end{eqnarray*}
by the Leibniz rule. In the same manner one shows $s + D$ is a $f^{-1}\mathcal{O}_ S$-algebra map because $D$ is an $S$-derivation. Conversely, given $s'$ we set $D = s' - s$. Details omitted.
$\square$
Lemma 17.28.12. Let
\[ \xymatrix{ X' \ar[d]_{h'} \ar[r]_ f & X \ar[d]^ h \\ S' \ar[r]^ g & S } \]
be a commutative diagram of ringed spaces.
The canonical map $\mathcal{O}_ X \to f_*\mathcal{O}_{X'}$ composed with $f_*\text{d}_{X'/S'} : f_*\mathcal{O}_{X'} \to f_*\Omega _{X'/S'}$ is a $S$-derivation and we obtain a canonical map of $\mathcal{O}_ X$-modules $\Omega _{X/S} \to f_*\Omega _{X'/S'}$.
The commutative diagram
\[ \xymatrix{ f^{-1}\mathcal{O}_ X \ar[r] & \mathcal{O}_{X'} \\ f^{-1}h^{-1}\mathcal{O}_ S \ar[u] \ar[r] & (h')^{-1}\mathcal{O}_{S'} \ar[u] } \]
induces by Lemmas 17.28.6 and 17.28.8 a canonical map $f^{-1}\Omega _{X/S} \to \Omega _{X'/S'}$.
These two maps correspond (via adjointness of $f_*$ and $f^*$ and via $f^*\Omega _{X/S} = f^{-1}\Omega _{X/S} \otimes _{f^{-1}\mathcal{O}_ X} \mathcal{O}_{X'}$ and Sheaves, Lemma 6.20.2) to the same $\mathcal{O}_{X'}$-module homomorphism
\[ c_ f : f^*\Omega _{X/S} \longrightarrow \Omega _{X'/S'} \]
which is uniquely characterized by the property that $f^*\text{d}_{X/S}(a)$ maps to $\text{d}_{X'/S'}(f^*a)$ for any local section $a$ of $\mathcal{O}_ X$.
Proof.
Omitted.
$\square$
Lemma 17.28.13. Let
\[ \xymatrix{ X'' \ar[d] \ar[r]_ g & X' \ar[d] \ar[r]_ f & X \ar[d] \\ S'' \ar[r] & S' \ar[r] & S } \]
be a commutative diagram of ringed spaces. With notation as in Lemma 17.28.12 we have
\[ c_{f \circ g} = c_ g \circ g^* c_ f \]
as maps $(f \circ g)^*\Omega _{X/S} \to \Omega _{X''/S''}$.
Proof.
Omitted.
$\square$
Lemma 17.28.14. Let $f : X \to Y$, $g : Y \to S$ be morphisms of ringed spaces Then there is a canonical exact sequence
\[ f^*\Omega _{Y/S} \to \Omega _{X/S} \to \Omega _{X/Y} \to 0 \]
where the maps come from applications of Lemma 17.28.12.
Proof.
By taking induced maps in stalks at $x \in X$ and using Lemma 17.28.7 we obtain a sequence
\[ \mathcal{O}_{X, x} \otimes _{\mathcal{O}_{Y, f(x)}} \Omega _{\mathcal{O}_{Y, f(x)}/\mathcal{O}_{S, g(f(x))}} \to \Omega _{\mathcal{O}_{X, x}/\mathcal{O}_{S, g(f(x))}} \to \Omega _{\mathcal{O}_{X, x}/\mathcal{O}_{Y, f(x)}} \to 0 \]
It suffices to see that the maps of the sequence are the same as the ones in Algebra, Lemma 10.131.7. This is because the characterization of the maps in Lemma 17.28.12 and because via the isomorphism of Sheaves, Lemma 6.26.4 we have $(f^*s)_ x = s_ x \otimes 1$, for any local section $s$ of a sheaf of $\mathcal{O}_ Y$-modules.
$\square$
Comments (2)
Comment #3221 by Herman Rohrbach on
Comment #3323 by Johan on