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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:

enter image description here

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.

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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.

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  • $\begingroup$ Hi Kenny, I am Beñat Mencia from TCM-Cavendish. I lost your contact and I could only find you here. Please drop me an email at beinat.mencia@gmail.com to recover contact :) Then I will delete this comment. $\endgroup$
    – Mencia
    Jun 16 '20 at 15:55

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