A standard variant of the Lefschetz principle I am studying this paper, and I am a little confused about the terminology "a standard variant of the Lefschetz principle".
Assume I have a smooth and proper morphism of schemes $X\longrightarrow\operatorname{Spec}(k)$, where $k$ is a field. Assume that $X$ is connected. The author writes:

A standard variant of the ''Lefschetz principle'' guarantees the
existence of the following very noncanonical data: a finitely
generated $\mathbb{Z}$-Algebra $A$, a ring homomorphism $A\longrightarrow k$, a smooth proper $A$-scheme $\mathfrak{X}$, and an isomorphism $\mathfrak{X}_{k}\cong X$.

What exactly is he talking about here? Is $\mathfrak{X}$ some sort of a Néron model, and how do I construct it (or prove its existence)? Why does (as is claimed by Eric Wofsey's answer) Chow's Lemma help me here?
Thanks in advance.
 A: This is just saying that defining the scheme $X$ involves only finitely many elements of $k$.  Cover $X$ with finitely many affine $k$-schemes.  Each of these affine $k$-schemes are actually defined over some finitely generated subring of $k$, since the corresponding $k$-algebras have a presentation that involves only finitely many specific elements of $k$.  To glue these affine $k$-schemes together, you just have to cover their pairwise intersections with finitely many affine open subschemes, which again is witnessed by finitely many elements of $k$.  To witness that these $k$-algebras are smooth over $k$ uses finitely many more elements of $k$.  To witness that the scheme you get by gluing them together is proper again needs only finitely many elements of $k$ (using Chow's lemma).
Taken together, this gives a finitely generated subring $A\subseteq k$ which contains all the elements of $k$ needed to construct the scheme $X$ and prove it is smooth and proper over $k$.  So, you can carry out this construction and obtain a smooth and proper scheme $\mathfrak{X}$ over $A$ which becomes $X$ after base-changing to $k$.
