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Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Properties of the map I

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, the set [TEX] \mathcal{I} (A) [/TEX] is defined in the following text on page 660: (see attachment)

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"While the ideal whose locus determines a particular algebraic set V is not unique, there is a unique largest ideal that determines V, given by the set of all polynomials that vanish on V.

In general, for any subset A of [TEX] \mathbb{A}^n [/TEX] define

[TEX] \mathcal{I}(A) = \{ f \in k[x_1, x_2, ... \ ... \ , x_n ] \ | \ f ( a_1, a_2, ... \ ... \ , a_n) = 0 [/TEX] for all [TEX] ( a_1, a_2, ... \ ... \ , a_n) \in A \} [/TEX]"

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Then at the top of page 661 D&F write: (see attachment)

"The following properties of the map [TEX] \mathcal{I} [/TEX] are very easy exercises ...

Among these easy exercises is [TEX] \mathcal{I}( \mathbb{A}^n ) = 0 [/TEX]

Despite this being an easy exercise, I cannot see exactly why [TEX] \mathcal{I}( \mathbb{A}^n ) = 0 [/TEX] :(

Can someone please help?

Peter

[This has also been posted on MHF]