I am self-studying the following notes: Introduction to Commutative algebra and algebraic geometry.
There is a sudden run of statements from end of page 6 / beginning of page 7 which leave me confused. I have numbered the statements for clarity:
My questions:
is $(2)$ a contraction as per Definition 1.17 ? what is the domain and target of the map $(2)$ anyway ?
how does this map $(2)$ induce $(3)$ ?
Does map $(4)$ refer to the Nullstellentstaz (weak form) as stated in statement $(0)$ / theorem 1.16 ?
how would $(4)$ induce map $(5)$ ?
Many thanks for any guidance.
I appreciate that the notation in the book isn't super clear.
$(2)$ is a map between $\operatorname{Max}(k[x]/I)$, the set of maximal ideals of the quotient ring $k[x] / I$, and $\{ m \in \operatorname{Max}(k[x]) : I \subset m \}$, the set of maximal ideals of $k[x]$ that contain $I$.
The mapping sends each maximal ideal $m \in \operatorname{Max}(k[x] / I)$ to $\sigma^{-1}(m) \in \operatorname{Max}(k[x])$, where $\sigma : k[x] \to k[x] / I$ is the natural quotient homomorphism. In this sense, the mapping is a contraction.
(And as noted in the second half of Remark 1.18, $\sigma^{-1}(m)$ is a maximal ideal in $k[x]$ when $m$ is a maximal ideal in $k[x]/I$, since $\sigma$ is surjective.)
Note that $\{ m \in \operatorname{Max}(k[x]) : I \subset m \}$ is in correspondence with $V(I)$, as pointed out earlier in your book quote: you associate each $m = (x_1 - a_1, \dots, x_n - a_n)$ with the point $(a_1, \dots, a_n) \in k^n$ via the Nullstellensatz.
Putting everything together, we have a correspondence between $\operatorname{Max}(k[x] / I)$ and $V(I)$, as claimed.