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Math Help - Affine Variety - Single Points and maximal ideals

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    Super Member Bernhard's Avatar
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    Affine Variety - Single Points and maximal ideals

    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  V is called irreducible if it cannot be written as  V = V_1 \cup V_2 where  V_1  and  V_2  are proper algebraic sets in  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  V is irreducible if and only if  \mathcal{I}(V) is a prime ideal.

    Corollary 18. The affine algebraic set  V is a variety if and only if its coordinate ring  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  \mathbb{A}^n are affine varieties since their corresponding ideals in  k[A^n] are maximal ideals."

    I do not follow this reasoning.

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

    imply that single points in  A^n are affine varieties.

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

    I would appreciate some help.

    Peter
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    Last edited by Bernhard; November 7th 2013 at 09:32 PM.
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