The Stacks project

Lemma 15.34.1 (Cartier equality). Let $K/k$ be a finitely generated field extension. Then $\Omega _{K/k}$ and $H_1(L_{K/k})$ are finite dimensional and $\text{trdeg}_ k(K) = \dim _ K \Omega _{K/k} - \dim _ K H_1(L_{K/k})$.

Proof. We can find a global complete intersection $A = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ over $k$ such that $K$ is isomorphic to the fraction field of $A$, see Algebra, Lemma 10.158.11 and its proof. In this case we see that $\mathop{N\! L}\nolimits _{K/k}$ is homotopy equivalent to the complex

\[ \bigoplus \nolimits _{j = 1, \ldots , c} K \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} K\text{d}x_ i \]

by Algebra, Lemmas 10.134.2 and 10.134.13. The transcendence degree of $K$ over $k$ is the dimension of $A$ (by Algebra, Lemma 10.116.1) which is $n - c$ and we win. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 15.34: Cartier's equality and geometric regularity

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Your email address will not be published.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 07E1. Beware of the difference between the letter 'O' and the digit '0'.