What are $F$-structures all about? 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?
 A: 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:


*

*Yes.  See comments

*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.  

*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 $A \to F$.

*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.
