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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