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 $X \rightarrow X_0$.