The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04BP. Beware of the difference between the letter 'O' and the digit '0'.