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Let $X$ be a proper $k$-scheme and $k \subset k'$ a field extension. Consider the fibre product \ base change $X' = X \otimes _k k'$.

Let $\mathcal{F} \in Coh(X)$ and $p: X' \to X$ the canonical projection (I think that it is a affine morphism (why ?).

Does and why $p_* (p^* (\mathcal{F})) = \mathcal{F} \otimes _k k'$ hold?

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    $\begingroup$ This is called the projection formula, see Hartshorne Residues and Duality, Proposition II.5.6 (though their is certainly an easier proof in your very simple case). $\endgroup$
    – abx
    Commented Dec 30, 2017 at 16:21
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    $\begingroup$ On a basis of affine opens, this is immediate from Hartshorne Algebraic Geometry, Proposition II.5.2(d,e). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Dec 30, 2017 at 17:27

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