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Affine Varieties - Single Points and maximal ideals

  1. Nov 8, 2013 #1
    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)


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

    An irreducible affine algebraic set is called an affine variety.


    Dummit and Foote then prove the following results:


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

    Corollary 18. The affine algebraic set [itex] V [/itex] is a variety if and only if its coordinate ring [itex] k[V] [/itex] is an integral domain.


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

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

    I do not follow this reasoning.

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

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

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

    I would appreciate some help.


    Attached Files:

    Last edited: Nov 8, 2013
  2. jcsd
  3. Nov 8, 2013 #2
    Maximal ideals are always prime.
  4. Nov 8, 2013 #3
    Thanks R136a1

    Was looking for that relationship in D&F - just found it in D&F ch 7 page 256 ...

    Corollary 14: Assume R is commutative. Every maximal ideal of R is a prime ideal

    Mind you, it was your post got me looking again :-)

    Thanks again.

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