Lang's Introduction to Algebraic Geometry conventions

Question

As I mention in my question regarding Lang's definition of a generic point of the variety, the conventions he follows in his book Introduction to Algebraic Geometry seem weird to me. He considers a universal domain $\Omega$ which is algebraically closed and shares much of the properties of that of complex numbers. In particular, $\Omega$ has infinite degree of transcende over its prime field. So far so good.

The problems begin when he considers other fields $K,L,\dots$ such that their corresponding prime fields are exactly that of $\Omega$ and the latter is a field extension of infinite degree of transcende of any of them. Then, all the algebraic objects will be defined over $K,L,\dots$. These are stated at the beginning of Section 1 in Chapter II.

For example, a variety over $K$, as defined on page 25, is the set

$$ V=\{(x)\in \Omega^n \mid f(x)=0 \quad \forall f\in \mathfrak p\}, $$ where $\mathfrak p$ is a prime ideal of $K[X_1,\dots,X_n]$, and the topology is the Zariski topology given by $K[X_1,\dots,X_n]$ (that is why Theorem 4 on page 25 remains true, even if $\mathfrak p$ is no a prime ideal of $\Omega[X_1,\dots,X_n]$, e.g. $K=\mathbb Q$, $\Omega=\mathbb C$, $n=1$ and $\mathfrak p$ generated by the polynomial $f(X)=X^2+1$).

Notice that if $\Omega=\mathbb C$ as usual in most practical scenarios, his assumptions forbid considering the field of real numbers.

Why does he make such assumptions? And how much of his theory remain true when the subfields of $\Omega$ are finite algebraic extensions?

Answer 1

I have not read Lang but I get the impression it is the same framework as Weil's Foundations of algebraic geometry. Weil explains, regarding the universal domain:

What is required of the “universal domain” is merely that there be room enough in it for all our fields and all our constructions. If, as could easily have been done (at the cost of some inconvenience), we had systematically avoided the use of algebraically closed fields and of algebraic closures, and if we had set an a priori limitation on the dimension of all varieties, and on the degree of transcendency of all fields over the prime field, then in view of the fact that only a finite number of fields, of points and of varieties have been considered in this book, it would have been possible to use, as the “universal domain”, a finitely generated extension of the prime field, viz. one of a sufficiently large degree of transcendency, and containing enough algebraic elements over the purely trascendental extension of the prime field of that degree; one would thus have carried out, so far as algebraic geometry is concerned, the strict Kroneckerian programme, as formulated in the famous Grundzüge.

So in some sense the requirement that all fields considered be subfields of the universal domain such that the universal domain be of infinite transcendence degree over the subfield is just a matter of convenience... but I would say it is not possible to accommodate in Weil's framework varieties defined over subfields such that the universal domain is algebraic over the subfield. The reason is simple: generic points (in the sense of Weil) of varieties of (strictly) positive dimension defined over a field $k$ necessarily have coordinates in a transcendental extension of $k$. So, in Weil's framework, the only varieties defined over $\mathbb{R}$ in the universal domain $\mathbb{C}$ are 0-dimensional, i.e. points.

I have the impression that the transcendence degree condition is not something that bothers algebraic number theorists, though. After all, $\mathbb{C}$ is of infinite transcendence degree over $\bar{\mathbb{Q}}$.