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