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Is the following proposition true? If yes, I would like to know the proof, preferably without referring to books or papers.

Proposition Let $k$ be a field. Let $P$ be the prime subfield of $k$. Let $X$ be a sceheme of finite type over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then there exist a subfield $k_0$ of $k$ which is finitely generated over $P$ and a scheme $X_0$ of finite type over $k_0$ and a coherent $\mathcal{O}_{X_0}$-module $\mathcal{F_0}$ such that $X = X_0 \times_{k_0} k$ and $\mathcal{F}$ is the pull back of $\mathcal{F}_0$ by the projection $X \rightarrow X_0$.

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It is true by the theory of limits of schemes of finite presentation: EGA IV §8. –  Andrea Dec 6 '12 at 9:19
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1 Answer

Since $X$ is of finite type over $k$, it is covered by finitely many affines. To give a sheaf $\mathcal F$ on $X$ is to give a sheaf on these finite affines and some glueing data.

Thus, to answer your question, we may and do assume $X$ is affine, say $X=$ Spec $A$ with $A=k[x_1,\ldots,x_n]/I$. The ideal $I$ is finitely generated. Thus, there are $f_i \in k[x_1,\ldots,x_n]$ $(i=1,\ldots,m)$ such that $I=(f_1,\ldots,f_m)$. Let $k_0$ be the field generated by the coefficients of $f_1,\ldots,f_m$. This is a finitely generated field over the prime field of $k$. Let $A_0 = k_0[x_1,\ldots,x_n]/(f_1,\ldots,f_m)$, where you view $f_1,\ldots,f_m$ as elements of $k_0[x_1,\ldots,x_n]$. Clearly, $X_0 = $ Spec $A_0$ fulfills the sought property.

Let $M$ be a finitely generated $A$-module, i.e., a coherent sheaf on $X$. Then, $M$ descends to an $A_0$-module by a similar argument as before (replacing $k_0$ maybe by a slightly bigger extension.)

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