The Stacks project

Lemma 89.4.5. Let $S$ be an object of $\widehat{\mathcal{C}}_\Lambda $. Then $\dim _ k \text{Der}_\Lambda (S, k) < \infty $.

Proof. Let $x_1, \ldots , x_ n \in \mathfrak m_ S$ map to a $k$-basis for the relative cotangent space $\mathfrak m_ S/(\mathfrak m_\Lambda S + \mathfrak m_ S^2)$. Choose $y_1, \ldots , y_ m \in S$ whose images in $k$ generate $k$ over $k'$. We claim that $\dim _ k \text{Der}_\Lambda (S, k) \leq n + m$. To see this it suffices to prove that if $D(x_ i) = 0$ and $D(y_ j) = 0$, then $D = 0$. Let $a \in S$. We can find a polynomial $P = \sum \lambda _ J y^ J$ with $\lambda _ J \in \Lambda $ whose image in $k$ is the same as the image of $a$ in $k$. Then we see that $D(a - P) = D(a) - D(P) = D(a)$ by our assumption that $D(y_ j) = 0$ for all $j$. Thus we may assume $a \in \mathfrak m_ S$. Write $a = \sum a_ i x_ i$ with $a_ i \in S$. By the Leibniz rule

\[ D(a) = \sum x_ iD(a_ i) + \sum a_ iD(x_ i) = \sum x_ iD(a_ i) \]

as we assumed $D(x_ i) = 0$. We have $\sum x_ iD(a_ i) = 0$ as multiplication by $x_ i$ is zero on $k$. $\square$


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Your email address will not be published.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 06SC. Beware of the difference between the letter 'O' and the digit '0'.