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Affine Varieties - Single Points and maximal ideals

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
In Dummit and Foote Chapter 15, Section 15.3: Radicals and Affine Varieties on page 679 we find the following definition of affine variety: (see attachment)

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Definition. A nonempty affine algebraic set \(\displaystyle V \) is called irreducible if it cannot be written as [TEX] V = V_1 \cup V_2 [/TEX] where [TEX] V_1 [/TEX] and [TEX] V_2 [/TEX] are proper algebraic sets in \(\displaystyle V \).

An irreducible affine algebraic set is called an affine variety.

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Dummit and Foote then prove the following results:

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Proposition 17. The affine algebraic set \(\displaystyle V \) is irreducible if and only if [TEX] \mathcal{I}(V) [/TEX] is a prime ideal.

Corollary 18. The affine algebraic set \(\displaystyle V \) is a variety if and only if its coordinate ring \(\displaystyle k[V] \) is an integral domain.

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Then in Example 1 on page 681 (see attachment) D&F write:

"Single points in [TEX] \mathbb{A}^n [/TEX] are affine varieties since their corresponding ideals in [TEX] k[A^n] [/TEX] are maximal ideals."

I do not follow this reasoning.

Can someone please explain why the fact that ideals in [TEX] k[A^n] [/TEX] that correspond to single points are maximal

imply that single points in [TEX] A^n [/TEX] are affine varieties.

Presumably Proposition 17 and Corollary 18 are involved but I cannot see the link.

I would appreciate some help.

Peter

[This has also been posted on MHF]
 
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