The Stacks project

Lemma 99.15.8. Let $k$ be a field and let $x = (X \to \mathop{\mathrm{Spec}}(k))$ be an object of $\mathcal{X} = \mathcal{C}\! \mathit{urves}$ over $\mathop{\mathrm{Spec}}(k)$.

  1. If $k$ is of finite type over $\mathbf{Z}$, then the vector spaces $T\mathcal{F}_{\mathcal{X}, k, x}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x})$ (see Artin's Axioms, Section 98.8) are finite dimensional, and

  2. in general the vector spaces $T_ x(k)$ and $\text{Inf}_ x(k)$ (see Artin's Axioms, Section 98.21) are finite dimensional.

Proof. This is immediate from the fully faithful embedding (99.15.1.1) and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.8). $\square$

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Your email address will not be published.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 0D57. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 99.15.8 as pdf