Lemma 45.14.14. Assume given data (D0), (D1), and (D2') satisfying axioms (A1) – (A7). If $k''/k'/k$ are finite separable field extensions, then $H^0(\mathop{\mathrm{Spec}}(k')) \to H^0(\mathop{\mathrm{Spec}}(k''))$ is injective.
Proof. We may replace $k''$ by its normal closure over $k$ which is Galois over $k$, see Fields, Lemma 9.21.5. Then $k''$ is Galois over $k'$ as well, see Fields, Lemma 9.21.4. We deduce we have an isomorphism
\[ k' \otimes _ k k'' \longrightarrow \prod \nolimits _{\sigma \in \text{Gal}(k''/k')} k'',\quad \eta \otimes \zeta \longmapsto (\eta \sigma (\zeta ))_\sigma \]
This produces an isomorphism $\coprod _\sigma \mathop{\mathrm{Spec}}(k'') \to \mathop{\mathrm{Spec}}(k') \times \mathop{\mathrm{Spec}}(k'')$ which on cohomology produces the isomorphism
\[ H^*(\mathop{\mathrm{Spec}}(k')) \otimes _ F H^*(\mathop{\mathrm{Spec}}(k'')) \to \prod \nolimits _\sigma H^*(\mathop{\mathrm{Spec}}(k'')),\quad a' \otimes a'' \longmapsto (\pi ^*a' \cup \mathop{\mathrm{Spec}}(\sigma )^*a'')_\sigma \]
where $\pi : \mathop{\mathrm{Spec}}(k'') \to \mathop{\mathrm{Spec}}(k')$ is the morphism corresponding to the inclusion of $k'$ in $k''$. We conclude the lemma is true by taking $a'' = 1$. $\square$
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