Ideal of functions disappearing at (a_1, a_2, .... .... , a_n)

[Math Amateur]

I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ...

At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ...

I need someone to help me to fully understand the reasoning/analysis behind the statements in Example (2) on Page 660 of D&F ...

On page 660 (in Section 15.1) of D&F we find the following text and examples (I am specifically focused on Example (2)):
https://www.physicsforums.com/attachments/4742

I need help with understanding the analysis/proof/thinking behind the statement of Example 2.

Could someone please explain the basis for the statement of Example 2 regarding the ideal \(\displaystyle I\), say, of functions vanishing on \(\displaystyle (a_1, a_2, \ ... \ ... \ , a_n) \in \mathbb{A}^n\) ... ...

Specifically, how (in what way) is \(\displaystyle I\) the kernel of a surjective ring homomorphism from \(\displaystyle k[ x_1, x_2, \ ... \ ... \ , x_n]\) to the field \(\displaystyle k\) ... what exactly is the homomorphism?

Further, why does I being the kernel of such a ring homomorphism imply that \(\displaystyle I\) is a maximal ideal? Can someone unpack this statement explaining the implication step by step ...

Hope someone can help ...

Peter

 

[mathbalarka]

(fair warning : Dummit-Foote doesn't have much algebraic geometry. I'd recommend Eisenbud's book or Reid's book for learning commutative algebra with a flavor of algebraic geometry)

The surjective ring homomorphism $\psi : k[x_1, \cdots, x_n] \to k$ here is taking a polynomial $f(x_1, x_2, \cdots, x_n)$ and evaluating it at the point $(a_1, \cdots, a_n)$ to obtain the scalar $f(a_1, \cdots, a_n) \in k$. That's it.

Ideal of functions vanishing at $(a_1, \cdots, a_n) \in \Bbb A^n$ is precisely the collection of polynomials $f \in k[x_1, \cdots, x_n]$ such that $f(a_1, \cdots, a_n) = 0$. That said, it's clear that this ideal is kernel of the evaluation morphism $\psi$, and thus is a maximal ideal. (Recall that an ideal is maximal iff quotient ring is a field)

 

[Math Amateur]

(fair warning : Dummit-Foote doesn't have much algebraic geometry. I'd recommend Eisenbud's book or Reid's book for learning commutative algebra with a flavor of algebraic geometry)

The surjective ring homomorphism $\psi : k[x_1, \cdots, x_n] \to k$ here is taking a polynomial $f(x_1, x_2, \cdots, x_n)$ and evaluating it at the point $(a_1, \cdots, a_n)$ to obtain the scalar $f(a_1, \cdots, a_n) \in k$. That's it.

Ideal of functions vanishing at $(a_1, \cdots, a_n) \in \Bbb A^n$ is precisely the collection of polynomials $f \in k[x_1, \cdots, x_n]$ such that $f(a_1, \cdots, a_n) = 0$. That said, it's clear that this ideal is kernel of the evaluation morphism $\psi$, and thus is a maximal ideal. (Recall that an ideal is maximal iff quotient ring is a field)

Hi Mathbalarka,

Just reflecting on what you have said ...

I do have Eisenbud's book and it looks really interesting ... but I think I need something more elementary ... just to start with anyway ...

Peter

 

[mathbalarka]

I do agree that Eisenbud is pretty advanced. However, if you know enough ring theory (which I think you do), you can start off with Atiyah-MacDonald and Reid's Undergraduate Commutative Algebra. Both are undergrad books, and complement each another quite well in the sense that A-M's theory is dry, but exercises are good and Reid's exercises are not-so-good, but the theory is excellent. It provides a lot of geometric intuition for a few things.

 

[Math Amateur]

I do agree that Eisenbud is pretty advanced. However, if you know enough ring theory (which I think you do), you can start off with Atiyah-MacDonald and Reid's Undergraduate Commutative Algebra. Both are undergrad books, and complement each another quite well in the sense that A-M's theory is dry, but exercises are good and Reid's exercises are not-so-good, but the theory is excellent. It provides a lot of geometric intuition for a few things.

Hi Mathbalarka,

Am now referencing Reid's book on Commutative Algebra and also his book on Algebraic Geometry ... thanks for the lead to Miles Reid's books ... ...

I will try to finish the Chapter in Dummit and Foote ... because I really like D&Fs exposition of mathematics ... then work a bit with Reid's books ... and then try Eisenbud again ... ...

By the way ... thanks again on the excellent advice regarding helpful books ... ...

Peter