# not understand functor of points definition

This is definition of scheme functor in Mumford, Oda, Algebraic Geometry II Chpt 1.

Let $$F$$ be a covariant functor from Ring category to set category s.t. for any ring $$R$$, $$Spec(R)=\cup_iD(f_i)$$, $$F(R)$$ is the equalizer of $$\prod_iF(R_{f_i})\to\prod_{ij}F(R_{f_if_j})$$. Then $$F$$ is called sheaf in zariski topology.

Let covariant functor $$F:Ring\to Set$$ with $$x\in F(R)$$ called open set if (a) $$Hom(Spec(-), Spec(R))\xrightarrow{x_\star} F$$ where $$x_\star$$ is pushforward and both $$Hom(Spec(-), Spec(R)), F$$ are functors from ring category to sets and (b) For all ring $$S,\eta\in F(S)$$, then $$\eta_\star^{-1}[x_\star(Hom(Spec(-),Spec(R)))]$$ is a subfunctor of $$Hom(Spec(-),Spec(S))$$ and it is of the form $$Hom^{I}(Spec(A),Spec(S))=\{f\in Hom(Spec(A), Spec(S))|f^\star(I)A=A\}$$ for some ideal $$I\subset S$$.

$$F$$ is a scheme functor if $$F$$ is sheaf in zariski topology and for all field $$k$$ there exists open sets $$x_i\in F(R_i)$$ s.t. $$F(k)=\cup_i (x_i)_\star(Hom(Spec(k), Spec(R_i))$$.

$$\textbf{Q:}$$ Why is $$F(k)=\cup_i (x_i)_\star(Hom(Spec(k), Spec(R_i))$$ required to check $$F(R)=Hom(Spec(R), X)$$ for scheme $$X$$?(i.e. $$F\to Hom(-, X)$$ is equivalence of functors.) This seems to assemble pointwise information only. Furthermore $$Spec(R_i)$$ is basically forming covering of $$X$$ and the only part of information missing here is gluing information which I think the glueing part is identified by checking at $$k-$$points overlapped part.

• It's hard to understand what, exactly, your question is. It should be clear that the condition is required: the $k$-points of a scheme $X$ are certainly a union of the $k$-points of some set of representables $\{R_i\}$. Are you having trouble with why we're using fields at all? Why we're using all fields? What do you mean by "pointwise information?" – Kevin Carlson Feb 11 at 2:31
• @KevinCarlson If I demand all rings $R$ instead field $k$, then it seems much plausible as you account all $R$ valued points. However accounting residue field points only seems a very restrictive sense of information. Why such a logic jump(from all rings to field) is natural? – user45765 Feb 11 at 2:42
• @KevinCarlson "Pointwise information" means that $k-$valued points without scheme structure to me. – user45765 Feb 11 at 2:43