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,