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. $\square$

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