The definition of an affine scheme using a functor

I lately am reading this pdf. So I found the definition of an affine scheme.

Let $$\textbf{Ring}$$ denote the category of commutative rings with identity. For $$R\in \textbf{Ring}$$, an affine scheme is a functor following form:

$$\begin{array}{rrcl} \operatorname{Spec}(R): & \textbf{Ring} & \to & \textbf{Set} \\ & A & \mapsto & Hom_{\textbf{Ring}}(R,A) \end{array}$$

I like this definition because of very simple, but I can't understand this definition is the same as usual definition.

That is, a affine scheme is a locally ringed space $$(X, \mathcal{O}_X)$$ isomorphic to the spectrum (as a set of prime ideal) $$(\operatorname{Spec}(R), \mathcal{O}_{\operatorname{Spec}(R)})$$ of a commutative ring R.

Why are these the same? I'd appreciate if you could answer this question.

• Thank you for your answer! I did not know "Functor of points".i will learn. – undertate Mar 7 '20 at 10:57

These are not "the same", but they do give equivalent categories.

Precisely, let us write $$\hat{R}$$ for the functor $$\textbf{Ring}\to \textbf{Set}$$ you call $$\operatorname{Spec}(R)$$ (to avoid using the same symbol for two different things), and $$\textbf{LocRingSpc}$$ for the category of locally ringed spaces.

Then you can define two functors $$\begin{array}{rrcl} F: &\textbf{Ring}^{op} & \to & \textbf{Set}^{\textbf{Ring}} \\ & R & \mapsto & \hat{R} \end{array}$$ and $$\begin{array}{rrcl} G: &\textbf{Ring}^{op} & \to & \textbf{LocRingSpc} \\ & R & \mapsto & (\operatorname{Spec}(R), \mathcal{O}_{\operatorname{Spec}(R)}). \end{array}$$

Then it turns out that these two functors are fully faithful (by Yoneda lemma for $$F$$ and a standard basic theorem of algebraic geometry for $$G$$), so the (essential) images of $$F$$ and $$G$$ are equivalent categories, which both could be called "the category of affine schemes", although it is more natural to use that term for the image of $$G$$. You could call the image of $$F$$ "functorial affine schemes" and the image of $$G$$ "topological affine schemes" if you want. Of course they are also just equivalent to $$\textbf{Ring}^{op}$$.

You can go directly from one to the other by restricting $$\begin{array}{rcl} \textbf{LocRingSpc} & \to & \textbf{Set}^{\textbf{Ring}} \\ (X,\mathcal{O}_X) & \mapsto & (R\mapsto Hom_{\textbf{LocRingSpc}}(\operatorname{Spec}(R),X)). \end{array}$$ to the subcategory of "topological affine schemes".

The point is that it is arguably more natural to extend affine schemes to general schemes by using the locally ringed space definition than by the functorial definition. But in any case the above functor will still give an equivalence between "topological schemes" and "functorial schemes" to pursue with this (arbitrary) terminology.

• Thank you for your courteous reply! I understood that there is equivalence of categories between the subcategory of "topological affine schemes" and the category of "functorial affine schemes". Is there any problem by this "difference" in an enlargement of scheme theory? For example, Flat morphism need $\mathcal{O}_X$ in definition. It seems difficult to define flat morphism in the theory of "functorial affine schemes". – undertate Mar 7 '20 at 11:37