The Stacks project

Lemma 15.33.1 (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 10.152.11 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 10.132.2 and 10.132.13. The transcendence degree of $K$ over $k$ is the dimension of $A$ (by Algebra, Lemma 10.115.1) which is $n - c$ and we win. $\square$


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