18.33 Modules of differentials
In this section we briefly explain how to define the module of relative differentials for a morphism of ringed topoi. We suggest the reader take a look at the corresponding section in the chapter on commutative algebra (Algebra, Section 10.131).
Definition 18.33.1. Let $\mathcal{C}$ be a site. 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. A $\mathcal{O}_1$-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 18.33.1. Given a derivation $D : \mathcal{O}_2 \to \mathcal{F}$ as in the definition the map on global sections
\[ D : \Gamma (\mathcal{O}_2) \longrightarrow \Gamma (\mathcal{F}) \]
clearly is a $\Gamma (\mathcal{O}_1)$-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 18.33.2. Let $\mathcal{C}$ be a site. 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 $\mathcal{C}$, 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
18.33.2.1
\begin{equation} \label{sites-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 (18.33.2.1) 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 18.33.3. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. The module of differentials of the ring map $\varphi $ is the object representing the functor $\mathcal{F} \mapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F})$ which exists by Lemma 18.33.2. It is denoted $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$, and the universal $\varphi $-derivation is denoted $\text{d} : \mathcal{O}_2 \to \Omega _{\mathcal{O}_2/\mathcal{O}_1}$.
Since this module and the derivation form the universal object representing a functor, this notion is clearly intrinsic (i.e., does not depend on the choice of the site underlying the ringed topos, see Section 18.18). Note that $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$ is the cokernel of the map (18.33.2.1) 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 18.33.4. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of presheaves of rings. 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 (18.33.2.1). There is a similar map of presheaves whose value on $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ 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 (18.33.2.1) are the sheafifications of the presheaves above. Thus the result follows as sheafification is exact.
$\square$
Lemma 18.33.5. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $\mathcal{C}$. 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 (18.33.2.1) 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 18.33.6. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. For any object $U$ of $\mathcal{C}$ 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.
This is a special case of Lemma 18.33.5.
$\square$
Lemma 18.33.7. Let $\mathcal{C}$ be a site. 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 $\mathcal{C}$. 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 18.33.8. In Lemma 18.33.7 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 $\text{d} f_1f_2 \otimes 1 = (f_1 \text{d}f_2 + f_2 \text{d} f_1 )\otimes 1 = \text{d}f_2 \otimes f_1 + \text{d}f_2 \otimes f_1 = 0$. 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 18.33.7.
To see that the sequence is exact, we argue as follows. Let $\mathcal{O}''_2 \subset \mathcal{O}'_2$ be the presheaf of $\mathcal{O}_1$-algebras whose value on $U$ is the image of $\mathcal{O}_2(U) \to \mathcal{O}'_2(U)$. By Algebra, Lemma 10.131.9 the sequences
\[ \mathcal{I}(U)/\mathcal{I}(U)^2 \longrightarrow \Omega _{\mathcal{O}_2(U)/\mathcal{O}_1(U)} \otimes _{\mathcal{O}_2(U)} \mathcal{O}''_2(U) \longrightarrow \Omega _{\mathcal{O}''_2(U)/\mathcal{O}_1(U)} \longrightarrow 0 \]
are exact for all objects $U$ of $\mathcal{C}$. Since sheafification is exact this gives an exact sequence of sheaves of $(\mathcal{O}'_2)^\# $-modules. By Lemma 18.33.4 and the fact that $(\mathcal{O}''_2)^\# = \mathcal{O}'_2$ we conclude.
$\square$
Here is a particular situation where derivations come up naturally.
Lemma 18.33.9. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Consider a short exact sequence
\[ 0 \to \mathcal{F} \to \mathcal{A} \to \mathcal{O}_2 \to 0 \]
Here $\mathcal{A}$ is a sheaf of $\mathcal{O}_1$-algebras, $\pi : \mathcal{A} \to \mathcal{O}_2$ is a surjection of sheaves of $\mathcal{O}_1$-algebras, and $\mathcal{F} = \mathop{\mathrm{Ker}}(\pi )$ is its kernel. Assume $\mathcal{F}$ an ideal sheaf with square zero in $\mathcal{A}$. So $\mathcal{F}$ has a natural structure of an $\mathcal{O}_2$-module. A section $s : \mathcal{O}_2 \to \mathcal{A}$ of $\pi $ is a $\mathcal{O}_1$-algebra map such that $\pi \circ s = \text{id}$. Given any section $s : \mathcal{O}_2 \to \mathcal{F}$ of $\pi $ and any $\varphi $-derivation $D : \mathcal{O}_1 \to \mathcal{F}$ the map
\[ s + D : \mathcal{O}_1 \to \mathcal{A} \]
is a section of $\pi $ and every section $s'$ is of the form $s + D$ for a unique $\varphi $-derivation $D$.
Proof.
Recall that the $\mathcal{O}_2$-module structure on $\mathcal{F}$ is given by $h \tau = \tilde h \tau $ (multiplication in $\mathcal{A}$) where $h$ is a local section of $\mathcal{O}_2$, and $\tilde h$ is a local lift of $h$ to a local section of $\mathcal{A}$, and $\tau $ is a local section of $\mathcal{F}$. 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 $\mathcal{O}_1$-algebra map because $D$ is an $\mathcal{O}_1$-derivation. Conversely, given $s'$ we set $D = s' - s$. Details omitted.
$\square$
Definition 18.33.10. Let $X = (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ and $Y = (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be ringed topoi. Let $(f, f^\sharp ) : X \to Y$ be a morphism of ringed topoi. In this situation
for a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules a $Y$-derivation $D : \mathcal{O} \to \mathcal{F}$ is just a $f^\sharp $-derivation, and
the sheaf of differentials $\Omega _{X/Y}$ of $X$ over $Y$ is the module of differentials of $f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$, see Definition 18.33.3.
Thus $\Omega _{X/Y}$ comes equipped with a universal $Y$-derivation $\text{d}_{X/Y} : \mathcal{O} \longrightarrow \Omega _{X/Y}$. We sometimes write $\Omega _{X/Y} = \Omega _ f$.
Recall that $f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$ so that this definition makes sense.
Lemma 18.33.11. Let $X = (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ X), \mathcal{O}_ X)$, $Y = (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ Y), \mathcal{O}_ Y)$, $X' = (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X'}), \mathcal{O}_{X'})$, and $Y' = (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{Y'}), \mathcal{O}_{Y'})$ be ringed topoi. Let
\[ \xymatrix{ X' \ar[d] \ar[r]_ f & X \ar[d] \\ Y' \ar[r] & Y } \]
be a commutative diagram of morphisms of ringed topoi. 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 clear except for the last assertion. Let us explain the meaning of this. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ X)$ and let $t \in \mathcal{O}_ X(U)$. This is what it means for $t$ to be a local section of $\mathcal{O}_ X$. Now, we may think of $t$ as a map of sheaves of sets $t : h_ U^\# \to \mathcal{O}_ X$. Then $f^{-1}t : f^{-1}h_ U^\# \to f^{-1}\mathcal{O}_ X$. By $f^*t$ we mean the composition
\[ \xymatrix{ f^{-1}h_ U^\# \ar[rr]^{f^{-1}t} \ar@/^4ex/[rrrr]^{f^*t} & & f^{-1}\mathcal{O}_ X \ar[rr]^{f^\sharp } & & \mathcal{O}_{X'} } \]
Note that $\text{d}_{X/Y}(t) \in \Omega _{X/Y}(U)$. Hence we may think of $\text{d}_{X/Y}(t)$ as a map $\text{d}_{X/Y}(t) : h_ U^\# \to \Omega _{X/Y}$. Then $f^{-1}\text{d}_{X/Y}(t) : f^{-1}h_ U^\# \to f^{-1}\Omega _{X/Y}$. By $f^*\text{d}_{X/Y}(t)$ we mean the composition
\[ \xymatrix{ f^{-1}h_ U^\# \ar[rr]^{f^{-1}\text{d}_{X/Y}(t)} \ar@/^4ex/[rrrr]^{f^*\text{d}_{X/Y}(t)} & & f^{-1}\Omega _{X/Y} \ar[rr]^{1 \otimes \text{id}} & & f^*\Omega _{X/Y} } \]
OK, and now the statement of the lemma means that we have
\[ c_ f \circ f^*t = f^*\text{d}_{X/Y}(t) \]
as maps from $f^{-1}h_ U^\# $ to $\Omega _{X'/Y'}$. We omit the verification that this property holds for $c_ f$ as defined in the lemma. (Hint: The first map $c'_ f : \Omega _{X/Y} \to f_*\Omega _{X'/Y'}$ satisfies $c'_ f(\text{d}_{X/Y}(t)) = f_*\text{d}_{X'/Y'}(f^\sharp (t))$ as sections of $f_*\Omega _{X'/Y'}$ over $U$, and you have to turn this into the equality above by using adjunction.) The reason that this uniquely characterizes $c_ f$ is that the images of $f^*\text{d}_{X/Y}(t)$ generate the $\mathcal{O}_{X'}$-module $f^*\Omega _{X/Y}$ simply because the local sections $\text{d}_{X/Y}(t)$ generate the $\mathcal{O}_ X$-module $\Omega _{X/Y}$.
$\square$
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