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.
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 $k$.