What is the purpose of introducing the notion of points of value?

Question

The following is a quotation from Siegfried Bosch「Algebraic Geometry and Commutative Algebra」(6.7 The Affine n-Space) .

For any $R$-algebra $R'$ let $\mathbb{A}^n_S(R')$ be the set of all $S$-morphisms $\mathop{Spec} R' \to \mathbb{A}^n_S$; this is the set of so-called $R'$-valued points of $\mathbb{A}^n_R$. Since the set of $R$-homomorphisms $R[t_1, \cdots , t_n] \to R'$ corresponds bijectively to the set $(R')^n$ of all $n$-tuples with entries in $R'$, via the map $\varphi \mapsto (\varphi(t_1), \cdots , \varphi(t_n))$, we see that the set of $R'$-valued points of $\mathbb{A}^n_R$ is given by $$ \mathbb{A}^n_R (R')=(R')^n. $$

My question is ;

In this case, $(R')^n$ and $\mathbb{A}^n_R(R')$ coincide each other. Why the author introduce the notion of $R'$-valued points instead of $(R')^n$? When is the notion of point with value useful?

Thanks.

Answer 1

Of course I can't read Bosch's mind but his notation might be preparation for a more general situation, namely:
Let $I\subset R[T_1,\cdots,T_n]$ be an ideal and define for any $R$-algebra $R'$ the subset $V_I(R')\subset \mathbb A^n_{R}(R')=(R')^n$ by the requirement $$V_I(R')=\{(r'_1,\cdots,r'_n)\in (R')^n\vert (\forall f\in I) \; f(r'_1,\cdots,r'_n)=0 \}\subset \mathbb A^n_R(R') $$ Bosch has thus introduced a functor $$ \mathbb A^n_R :\operatorname {\mathcal {Alg}}_R\to \operatorname {\mathcal {Sets}}:R'\mapsto (R')^n$$ and the above is a subfunctor $$ V_I :\operatorname {\mathcal {Alg}}_R\to \operatorname {\mathcal {Sets}}:R'\mapsto V_I(R')$$ for which we write $V_I \subset \mathbb A^n_R$.
A fundamental point is that $V_I$ is representable by the $R-$algebra $A:=R[T_1,\cdots,T_n]/I$, which means that we have functorial bijections $$Hom_{R-Alg}(A,R')\to V_I(R'):\varphi\mapsto (\overline {T_1},\cdots, \overline {T_n}) $$ This opens the way to an even more general notion of points $ X(T)=Hom_{Schemes}(T,X)$ of an arbitrary, non-affine, scheme X with values in an other arbitrary scheme $T$ .
All this is admirably explained in much detail in these notes (in French) by Ducros .

Answer 2

From Siegfried Bosch「Algebraic Geometry and Commutative Algebra」(7.2 Fiber Products)

First note that the universal property of fiber products just says that these products respect the formation of $T$-valued points, namely, $$(X \times_S Y)(T) = X(T) \times Y(T) $$ for $S$-schemes $T$, recall that $X(T) = \mathrm{Hom}_S(T,X)$ is called the set of $T$-valued points of an $S$-scheme $X$. However, such a nice behavior cannot be expected from its underlying topological space.