Lemma 15.33.3. Given a commutative diagram of fields

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

are finite dimensional and

Lemma 15.33.3. Given a commutative diagram of fields

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

with $k \subset k'$ and $K \subset K'$ 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

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

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 15.33.1 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. $\square$

Your email address will not be published.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.

## Comments (0)