The Stacks project

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. $\square$

Comments (3)

Comment #789 by Anfang Zhou on

Typo.In the statement of this lemma, it should be "Given any section ".

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 be a topological space, let be a sheaf of rings over and let be sheaves of -algebras. Suppose is an ideal sheaf with . Then is naturally a -module. Let be a morphism of -algebras ( becomes an -module via ). Denote to the set of -algebra homomorphisms that lift to . Then is a -torsor via

Proof. The map is well-defined: It is clear that is -linear, that it preserves the multiplicative unit and that as a map of sheaves of abelian groups, it's a lifting of to . It is left to show that is multiplicative. For this, we need first to prove the following: Let , and let be local sections respectively of and of , over the same open subset . Then we claim that , i.e., that . It suffices to verify the equality on germs at , and we leave this as an exercise to the reader (use the natural -module structure on ). Now, one has Thus, and the map from the statement constitutes a group action of on . It is clear that the action is free. To show transitivity, let . We have to show that . On the one hand, it is clear that is -linear. On the other hand, using the same facts as in the last computation, we get which shows Leibniz's rule for .

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