I had some questions about the "Spectrum of a ring" section in the article about the Zariski topology. Assume for simplicity that $A=\mathbb C[x,y].$
Thus, $V(S)$ is "the same as" the maximal ideals containing $S$.
(Above $V(S)$ means the set of complex pairs $(a,b)$ such that $f(a,b)=0$ for all $f\in S$.) I understand it in the following sense: $$(a,b)\in V(S)\iff\forall f\in S, f(a,b)=0\iff \forall f\in S, f\in \ker(ev_{a,b})\iff S\subset (x-a,y-b)$$
But what exactly does "the same as" mean? I feel it's somehow related to $V'(S):=\{m\in MaxSpec(A): S\subset m\}$, but I can't formalize this.
Next, given $a\in A$, the article says that $a$ can be regarded as a function $Spec(A)\to Frac(A/I)$. Why do we need to consider $Frac(A/I)$ -- why not just $A/I$? What bad would happen if we considered $a$ as a map $Spec(A)\to A/I$?
More generally, $V(I)$ for any ideal $I$ is the common set on which all the "functions" in $ I$ vanish, which is formally similar to the classical definition.
Common set of what? Do they mean the set $\{P\in Spec(A):\forall f\in I, f(P)=0\}$? Or what is meant here? The set $\{(a,b)\in \mathbb C^2: \forall f\in I, f(a,b)=0\}$ is also a "common set on which all the "functions" in $ I$ vanish", but if that's what is meant, then it's not clear how it is "formally similar to the classical definition" -- it is the classical definition.
In fact, they agree in the sense that when $A$ is the ring of polynomials over some algebraically closed field $k$, the maximal ideals of $A$ are (as discussed in the previous paragraph) identified with $n$-tuples of elements of $k$, their residue fields are just $k$, and the "evaluation" maps are actually evaluation of polynomials at the corresponding $n$-tuples.
How to unpack this? And to begin with, what is the formal statement? Is it that there is a bijection between $\{(a,b)\in \mathbb C^2: \forall f\in I, f(a,b)=0\}$ and $\{P\in Spec\mathbb C[x,y]: I\subset P\}$? I guess this boils down to my very first question.