What are $F$-structures all about?

Question

This is a pretty open ended question. I'm reading Springer's book on algebraic groups and am very confused about these "$F$-structures." If $k$ is an algebraically closed field, and $A$ is an affine $k$-algebra, we can put a topology on the set $S$ of maximal ideals of $A$, or equivalently the set of $k$-algebra homomorphisms $A \rightarrow k$, and make this topological space into a ringed space. This ringed space will be isomorphic to the ringed space of regular functions of a closed subset of $k^n$ in the Zariski topology, for some $n \geq 0$.

Now let $F$ be a subfield of $k$. A finitely generated $F$-subalgebra $A_0$ is said to be defined over $S$ if the obvious $k$-vector space homomorphism $k \otimes_F A_0 \rightarrow A$ is an isomorphism. Then the $F$-rational points is the set of $F$-algebra homomorphisms $A_0 \rightarrow F$.

1 . What is this homomorphism $k \otimes_F A_0 \rightarrow A$ normally? Is it always injective? That is, can I identify the tensor product $k \otimes_F A_0$ with the span of $A_0$ in $A$ as a $k$-vector space?

2 . What is the significance of the $F$-rational points? Is there a topology we can place on it, or a ringed space structure?

3 . To what extent can we discuss subfields $F$ of $k$ and their $F$-structures, or resulting topological spaces, while avoiding noncanonical or arbitrary measures? For example, I prefer to associate to an affine $k$-algebra the ringed space of maximal ideals of $A$, because I can define this topology without reference to a set of generators $x_1, ... , x_n$ of $A$, i.e. the results clearly don't depend on the choice of a closed subset $V$ of the Zariski topology in $k^n$ such that $A \cong k[X_1, ... , X_n]/I(V)$.

4 . Why are $F$-structures important?

Answer 1

Let's take a simple but representative example.

Suppose you want to consider the circle

$$S^1 = \{(x, y) \in \mathbb{R}^2 \; | \; x^2 + y^2 = 1\}$$

Now algebraic geometry works best over algebraically closed fields, and $\mathbb{R}$ isn't algebraically closed. But what we can do is consider the algebraic variety

$$C := V(x^2 + y^2 - 1) \subset \mathbb{A}^2_\mathbb{C}$$

or, equivalently, its coordinate $\mathbb{C}$-algebra

$$\mathbb{C}[C] = \mathbb{C}[x,y]/(x^2 + y^2 - 1)$$

Now, unlike an arbitrary $\mathbb{C}$-algebra, $\mathbb{C}[C]$ has a special property: we obtained it from an $\mathbb{R}$-algebra by extending scalars. That is,

$$\mathbb{C}[C] = \mathbb{R}[x,y]/(x^2 + y^2 - 1) \otimes_\mathbb{R} \mathbb{C}$$

This wouldn't be true for, say, the $\mathbb{C}$-algebra $A = \mathbb{C}[x,y]/(x+iy)$ -- there's no $\mathbb{R}$-subalgebra $A_0 < A$ which generates $A$ as a $\mathbb{C}$-algebra.

The $\mathbb{R}$-rational points of $\mathbb{C}$ are simply the points $S^1$ described above. Since $C$ is defined over $\mathbb{R}$, we can think of them as the solutions over $\mathbb{R}$ of a system of equations (in this case, the system is just the single equation $x^2 + y^2 = 1$) in $\mathbb{R}[x,y]$.

If we considered something that was not defined over $\mathbb{R}$, then we could still have $\mathbb{R}$-rational points -- for instance, the algebra $A$ corresponds to a complex line with a single $\mathbb{R}$-rational point at $(0,0)$, but it doesn't have a satisfying description as the solution to a corresponding system of polynomials over $\mathbb{R}$.

Of course in practice $F$ is more likely a finite field. This lets us consider things like ${\rm GL}_n(F)$, which are finite sets but where the corresponding variety over the algebraic closure ${\rm GL}_n(F)$ is infinite.

To answer your questions:

1. Yes. See comments
2. You want to think of them as a subset of the variety over the algebraic closure, which of course has the structure of a locally ringed space.
3. Nothing here depends on a choice of generators. If you like, the $F$-rational points are the maximal ideals which are the kernel of some $F$-algebra homomorphism $A \to F$.
4. See above.

This is all probably substantially easier to understand in scheme-theoretic terms. See e.g. Mumford's description of ${\rm Spec} \mathbb{R}[x]$ for instance.