Affine schemes

We define affine schemes as a representable functor.

DEFINITION 2.2   Let $R$ be a ring. Then we denote by $\operatorname{Spec}(R)$ the affine scheme with coordinate ring $R$.

For any affine scheme $\operatorname{Spec}(R)$ and for any ring $S$, we define the $S$-valued point of $\operatorname{Spec}(R)$ by

$$\operatorname{Spec}(R)(S)=\operatorname{Hom}_{\operatorname{ring}}(R,S)$$

LEMMA 2.3   Let $k$ be a ring. Let $\{f_1,f_2,\dots,f_m\}$ be a set of equations in $n$-variables $X_1,X_2,\dots,X_n$ over $k$. Let us put

$$A=k[X_1,X_2,\dots,X_n]/(f_1,f_2,\dots,f_m).$$

Then we have a natural identification

$$V(f_1,f_2,\dots,f_m)(K) =\operatorname{Spec}(A)(K)$$

for any algebra $K$ over $k$.

COROLLARY 2.4   We employ the assumption as the Lemma. Then:

1. When the “target algebra” $K$ is given, the set of solutions $V(f_1,f_2,\dots,f_m)(K)$ depends only on the affine coordinate ring $A$.
2. For any element $P\in \operatorname{Spec}(A)(K)$, the “evaluation map”
   $$A \ni f \mapsto \operatorname{eval}_P(f)\in K$$
   is defined in an obvious way. Thus every element of $A$ may be regarded as a $K$-valued function on $\operatorname{Spec}(A)(K)$.