Bijection between solutions of polynomials equations and spectrum of the quotient ring.

Question

I would like to see the connection between the set of solutions of a system of polynomial equations and a spectrum of the quotient polynomial ring.

Given $f_1, f_2,\cdots, f_m \in k[x_1,x_2,\cdots, x_n]$ where $k$ is a field. Let $L$ be an extension field of $k$, denote by $X(L)=\{a\in L^n\;|\;f_i(a)=0,\; i = 1,\cdots,m\}$ the set of solutions in $L$. Consider $$\bigsqcup_{L\supset k} X(L)/\sim$$ where $a\sim b$ iff $a\in L_1^n$ and $b\in L_2^n$ and there exists field $L$ and embeddings of $L_1$ and $L_2$ into $L$ such that $a$ and $b$ map to the same element in $L^n$.

This set should be in natural bijective correspondence with $\mathrm{Spec} (A_X)$ where $A_X= k[x_1,x_2,\dots ,x_n]/(f_1,f_2,\dots , f_m)$.

To see this bijection, we map $a\mapsto \ker(e_a)$, where $e_a:A_x\rightarrow L$ is the evaluation homomorphism, and $L$ is an extension field that contains $a$. I failed to see why this is a bijection.

Answer 1

This is magnificently explained in full generality, with plenty of motivation, in the Introduction to the 1971 Springer edition of Dieudonné-Grothendieck's EGA I.
Strangely, this point of view is not used in the book itself, nor even mentioned in the original 1960 I.H.E.S. edition, nor alluded to in the subsequent volumes nor, come to think of it, explained in detail in any other book on scheme theory. Great pity, that!

Answer 2

Define $X = \mathrm{Spec} (A_X)$.

In modern language of algebraic geometry (due to Grothendieck), the points you are considering are called geometric points. They correspond by definition to morphisms $\mathrm{Spec}(L) \to X$, where $L$ is an extension field of $k$.

Now if you have a prime ideal $P$ of $A_X$, consider the composition $A \to A_X/P \to Quot(A_X/P)$. Defining $L := Quot(A_X/P)$ this provides you a morphism $\mathrm{Spec}(L) \to X$, i.e. an $L$-point of $X$.

If conversely you have a field extension $L$ of $k$ and a point $a = (a_1, \dots, a_n) \in X(L)$, consider the homomorphism of $k$-algebras $e_a : A_X \to L$ that is defined by "evaluating at a", i.e. by sending $x_i$ to $a_i$ for $i=1,\dots,n$. The kernel of this morphism is a prime ideal of $A_X$, i.e. an element of $\mathrm{Spec}(A_X)$.

If I am not mistaken, these maps should provide a bijection between certain sets. I am not sure if the ones you mentioned in your question are the right ones. You can find details in EGA I 3.4.5