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 $\text{trdeg}_ k(K) = \dim _ K \Omega _{K/k} - \dim _ K H_1(L_{K/k})$.
15.34 Cartier's equality and geometric regularity
A reference for this section and the next is [Section 39, MatCA]. In order to comfortably read this section the reader should be familiar with the naive cotangent complex and its properties, see Algebra, Section 10.134.
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. $\square$
Lemma 15.34.2. Let $M/L/K$ be field extensions. Then the Jacobi-Zariski sequence is exact.
\[ 0 \to H_1(L_{L/K}) \otimes _ L M \to H_1(L_{M/K}) \to H_1(L_{M/L}) \to \Omega _{L/K} \otimes _ L M \to \Omega _{M/K} \to \Omega _{M/L} \to 0 \]
Proof. Combine Lemma 15.33.7 with Algebra, Lemma 10.158.11. $\square$
Lemma 15.34.3. Given a commutative diagram of fields with $k'/k$ and $K'/K$ finitely generated field extensions the kernel and cokernel of the maps are finite dimensional and
\[ \xymatrix{ K \ar[r] & K' \\ k \ar[u] \ar[r] & k' \ar[u] } \]
\[ \alpha : \Omega _{K/k} \otimes _ K K' \to \Omega _{K'/k'} \quad \text{and}\quad \beta : H_1(L_{K/k}) \otimes _ K K' \to H_1(L_{K'/k'}) \]
\[ \dim \mathop{\mathrm{Ker}}(\alpha ) - \dim \mathop{\mathrm{Coker}}(\alpha ) -\dim \mathop{\mathrm{Ker}}(\beta ) + \dim \mathop{\mathrm{Coker}}(\beta ) = \text{trdeg}_ k(k') - \text{trdeg}_ K(K') \]
Proof. The Jacobi-Zariski sequences for $k \subset k' \subset K'$ and $k \subset K \subset K'$ are
\[ 0 \to H_1(L_{k'/k}) \otimes K' \to H_1(L_{K'/k}) \to H_1(L_{K'/k'}) \to \Omega _{k'/k} \otimes K' \to \Omega _{K'/k} \to \Omega _{K'/k'} \to 0 \]
and
\[ 0 \to H_1(L_{K/k}) \otimes K' \to H_1(L_{K'/k}) \to H_1(L_{K'/K}) \to \Omega _{K/k} \otimes K' \to \Omega _{K'/k} \to \Omega _{K'/K} \to 0 \]
By Lemma 15.34.1 the vector spaces $\Omega _{k'/k}$, $\Omega _{K'/K}$, $H_1(L_{K'/K})$, and $H_1(L_{k'/k})$ are finite dimensional and the alternating sum of their dimensions is $\text{trdeg}_ k(k') - \text{trdeg}_ K(K')$. The lemma follows. $\square$
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