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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