# Scheme of finite type over a field $K$ v.s. $K$-scheme

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,

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I wouldn't call $\operatorname{Spec} K$ ridiculous. The map of sheaves carries real information, as you yourself say! – Dylan Moreland Nov 13 '11 at 21:41
I'll try to write more later, but I think Borel often works over an algebraically closed field in that book. In that case, one does have functoriality for $\operatorname{maxSpec}$ applied to f.g. $K$-algebras. See Chapter 3 of Milne's notes for a clean presentation of this. – Dylan Moreland Nov 13 '11 at 21:53
Thanks ! Yeah i'm looking there. He also does at some point the switch to only considering max ideals. I'll try to understand why, can't see it for now – Bogdan Nov 13 '11 at 22:03
I guess the point is that it feels simpler. The points of your space correspond to the points you would plot if you could plot a variety over an algebraically closed field. And moreover you get a category anti-equivalent to (maybe I need more adjectives here) f.g. $k$-algebras (so you could say that the maxSpec with its sheaf knows everything about the ring) and this is what you wanted. – Dylan Moreland Nov 13 '11 at 22:09
Haha yeah actually i've just seen this equivalence of categories. Too lazy to work all the adjectives out for the moment though, thanks for the help :) – Bogdan Nov 14 '11 at 21:46

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!

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Okay that's really great for the functoriality, thanks a lot for the explanation, and for all ! – Bogdan Nov 14 '11 at 21:44