# Basic algebraic geometry questions

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.

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