Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Example 2

  • #1
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I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 2 on page 660 reads as follows: (see attachment)

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(2) Over any field k, the ideal of functions vanishing at [itex] (a_1, a_2, ... ... ... a_n) \in \mathbb{A}^n [/itex] is a maximal ideal since it is the kernel of the surjective ring homomorphism from [itex] k[x_1, x_2, ... ... x_n] [/itex] to the field k given by evaluation at [itex] (a_1, a_2, ... ... ... a_n) [/itex].

It follows that [itex] \mathcal{I}((a_1, a_2, ... ... ... a_n)) = (x - a_1, x - a_2, ... ... ... , x - a_n) [/itex]

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I can see that [itex] (x - a_1, x - a_2, ... ... ... , x - a_n) [/itex] gives zeros for each polynomial in [itex] k[ \mathbb{A}^n ] [/itex] - indeed, to take a specific example involving [itex] \mathbb{R} [x,y] [/itex] we have for, let us say, a particular polynomial [itex] g \in \mathbb{R} [x,y] [/itex] where g is as follows:

[itex] g(x,y) = 6(x - a_1)^3 + 11(x - a_1)^2(y - a_2) + 12(y - a_2)^2 [/itex]

so in this case, clearly [itex] g(a_1, a_2) = 0 [/itex] ... ... ... and, of course, other polynomials in [itex] \mathbb{R} [x,y] [/itex] similarly.

BUT ... ... I cannot understand D&Fs reference to maximal ideals. Why is it necessary to reason about maximal ideals.

Since I am obviously missing something, can someone please help by explaining what is going on in this example.

Another issue I have is why do D&F write [itex] \mathcal{I}((a_1, a_2, ... ... ... a_n)) [/itex] with 'double' parentheses and not just [itex] \mathcal{I}(a_1, a_2, ... ... ... a_n) [/itex]?

Would appreciate some help.

Peter

Note - see attachment for definition of [itex] \mathcal{I}(A) [/itex]
 

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  • #2
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The ideal ##I=\mathcal{I}((a_1,...,a_n))## is the set of all polynomials vanish on the point ##\mathbf{a} = (a_1,....,a_n)##.

The reasoning is as follows. Note that the polynomials ##X_i-a_i## certainly vanish on this point, as is easily checked. So ##X_i-a_i\in I## certainly. Thus the ideal generated by these polynomials is in ##I## too. So ##(X_1-a_1,...,X_n-a_n)\subseteq I##. But perhaps ##I## contains more! Here is our information about maximality comes in, because if ##I## contains more, it must be the entire ring ##k[X_1,...,X_n]##. But this can not be since the constant polynomial ##1## does not vanish on the set.
 
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  • #3
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Excellent!!! Now see the link to maximal ideals!

But will reflect on this further so I am sure I have understood all angles of the theory

Thank you so much for this critical help ... I can now progress with more confidence!

Peter
 

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