The Stacks project

Lemma 15.34.3. Given a commutative diagram of fields

\[ \xymatrix{ K \ar[r] & K' \\ k \ar[u] \ar[r] & k' \ar[u] } \]

with $k'/k$ and $K'/K$ finitely generated field extensions the kernel and cokernel of the maps

\[ \alpha : \Omega _{K/k} \otimes _ K K' \to \Omega _{K'/k'} \quad \text{and}\quad \beta : H_1(L_{K/k}) \otimes _ K K' \to H_1(L_{K'/k'}) \]

are finite dimensional and

\[ \dim \mathop{\mathrm{Ker}}(\alpha ) - \dim \mathop{\mathrm{Coker}}(\alpha ) -\dim \mathop{\mathrm{Ker}}(\beta ) + \dim \mathop{\mathrm{Coker}}(\beta ) = \text{trdeg}_ k(k') - \text{trdeg}_ K(K') \]

Proof. The Jacobi-Zariski sequences for $k \subset k' \subset K'$ and $k \subset K \subset K'$ are

\[ 0 \to H_1(L_{k'/k}) \otimes K' \to H_1(L_{K'/k}) \to H_1(L_{K'/k'}) \to \Omega _{k'/k} \otimes K' \to \Omega _{K'/k} \to \Omega _{K'/k'} \to 0 \]


\[ 0 \to H_1(L_{K/k}) \otimes K' \to H_1(L_{K'/k}) \to H_1(L_{K'/K}) \to \Omega _{K/k} \otimes K' \to \Omega _{K'/k} \to \Omega _{K'/K} \to 0 \]

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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