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. $\square$

Comment #789 by Anfang Zhou on

Typo.In the statement of this lemma, it should be "Given any section $s : \mathcal{O}_X \to \mathcal{A}$".

Comment #8565 by on

Instead of this lemma, one could prove the following more general result and obtain Lemma 17.28.11 as a Corollary:

Lemma. Let $X$ be a topological space, let $\mathcal{O}$ be a sheaf of rings over $X$ and let $\mathcal{A},\mathcal{B}$ be sheaves of $\mathcal{O}$-algebras. Suppose $\mathcal{I}\subset\mathcal{B}$ is an ideal sheaf with $\mathcal{I}^2=0$. Then $\mathcal{I}$ is naturally a $\mathcal{B}/\mathcal{I}$-module. Let $\varphi:\mathcal{A}\to\mathcal{B}/\mathcal{I}$ be a morphism of $\mathcal{O}$-algebras ($\mathcal{I}$ becomes an $\mathcal{A}$-module via $\varphi$). Denote $\operatorname{Lift}_\mathcal{O}(\varphi,\mathcal{B})$ to the set of $\mathcal{O}$-algebra homomorphisms $\mathcal{A}\to\mathcal{B}$ that lift $\varphi$ to $\mathcal{B}$. Then $\operatorname{Lift}_\mathcal{O}(\varphi,\mathcal{B})$ is a $\operatorname{Der}_\mathcal{O}(\mathcal{A},\mathcal{I})$-torsor via

Proof. The map is well-defined: It is clear that $D+\psi$ is $\mathcal{O}$-linear, that it preserves the multiplicative unit and that as a map of sheaves of abelian groups, it's a lifting of $\varphi$ to $\mathcal{B}$. It is left to show that $D+\psi$ is multiplicative. For this, we need first to prove the following: Let $\psi\in\operatorname{Lift}_\mathcal{O}(\varphi,\mathcal{B})$, and let $h,\tau$ be local sections respectively of $\mathcal{A}$ and of $\mathcal{I}$, over the same open subset $U\subset X$. Then we claim that $h\tau=\psi(h)\tau$, i.e., that $\varphi(h)\tau=\psi(h)\tau$. It suffices to verify the equality on germs at $x\in U$, and we leave this as an exercise to the reader (use the natural $\mathcal{B}_x/\mathcal{I}_x$-module structure on $\mathcal{I}_x$). Now, one has Thus, $D+\psi\in \operatorname{Lift}_\mathcal{O}(\varphi,\mathcal{B})$ and the map from the statement constitutes a group action of $\operatorname{Der}_\mathcal{O}(\mathcal{A},\mathcal{I})$ on $\operatorname{Lift}_\mathcal{O}(\varphi,\mathcal{B})$. It is clear that the action is free. To show transitivity, let $\psi,\psi'\in\operatorname{Lift}_\mathcal{O}(\varphi,\mathcal{B})$. We have to show that $D=\psi'-\psi\in\operatorname{Der}_\mathcal{O}(\mathcal{A},\mathcal{I})$. On the one hand, it is clear that $D$ is $\mathcal{O}$-linear. On the other hand, using the same facts as in the last computation, we get which shows Leibniz's rule for $D$. $\square$

Comment #9144 by on

Yes, I think this is fine except I would use perhaps "pseudo-torsor" in stead of "torsor". You can also sheafify (so look at the sheaf of local sections), etc. Going to leave as is for now.

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