I'm lost in some definitions about schemes. I have some trouble about two definitions of a scheme of finite type over $K$, for an alg.closed field $K$.
Version 1 (Hartshorn) : a scheme of finite type over $K$ is a scheme $X$ together with a morphism $X \to K$, where $X$ is a scheme (a locally ringed space $(X, \mathcal{O})$ with a cover of spectra of rings) and for me $K$ is the (ridiculous ?) scheme $\text{Spec} K = \{ (0) \}$ with the sheaf sending $\{(0)\}$ to $K$. So the morphism is a continuous function $f \colon X \to \{\ast\}$ irrelevant since only one possible, and a finite morphism of sheaves, that is reduced to a single ring hm $f^{\sharp} \colon K \to \mathcal{O}_X(X)$.
Version 2 : a scheme of finite type over $K$ is a scheme with a finite cover of spectra of finitely generated $K$-algebras.
For example, let's assume $X$ is affine, so $X = \text{Spec} R$ for some ring $R$, so in the first version it is the data of $\text{Spec} R$ with its topology and the sheaf associated, and we add a finite morphism $K \to \mathcal{O}_X(X) = R$. In the second version, it is a scheme of the form $\text{Spec}(A)$ for some finitely generated $K$-algebra $A$.
Oh, actually i think this makes the bridge between the two notions… Well… Sorry. I have however another question :
While studying algebraic groups in the Borel, he considers $K$-schemes, which are almost schemes of finite type over $K$, in the sense that the topological space is not the whole spectrum $\text{Spec} A$ but only $\text{max} A = \text{Spec}_K A$ of maximal ideals of the finitely generated $K$-algebra $A$. So clearly the topological space contains "less" points, what does it change ? Why does he do that ? There is a bijection between $\text{max} A$ and $\text{Hom}_K (A,K)$, but what does it bring along ? Because we lose the functoriality (inverse image of maximal ideal is not maximal) and the result is not a scheme anymore…
Sorry for this not linear question, I hope it's understandable, or I'll edit or delete..
Thanks for any hint or piece of information !!
Bogdan
P.S. Actually,
Best Answer
It seems you actually understand the situation very well!
Borel's definition is a hybrid between classical algebraic geometry and scheme theory.
It stems from the desire not to use the full machinery of schemes.
Technically Borel can get away with that approach because for a scheme $X$ of finite type over a field $K$, the subset of closed points $X_{cl} \subset X$ is very dense in $X$.
This means that the restriction map $Open(X) \to Open (X_{cl}): U \mapsto U\cap X_{cl}$ is bijective.
The reason for that is that a finitely generated algebra $A$ over $K$ is a Jacobson ring, meaning that every prime ideal in $A$ is the intersection of the maximal ideals which contain it.
And for Jacobson rings we actually have functoriality: given a morphism $A\to B$ between two Jacobson rings, the inverse image of a maximal ideal of $B$ is a maximal ideal of $A$.
But I feel that this ad hoc approach should be a temporary crutch. The sooner you handle full-fledged scheme theory, the better: I strongly encourage you to go on reading Hartshorne!