Affine Varieties - Single Points and maximal ideals

[Math Amateur]

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 $$V = V_1 \cup V_2$$ where $$V_1$$ and $$V_2$$ 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 $$\mathcal{I}(V)$$ 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 $$\mathbb{A}^n$$ 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

[This has also been posted on MHF]

 

[jvicens]

Hello Peter,

Thank you for bringing this forum post to my attention. I am a scientist and I would be happy to explain the reasoning behind the statement in Example 1 on page 681 of Dummit and Foote.

First, let's review the definitions and propositions mentioned in the post. An affine algebraic set is a set of points in affine space that satisfy a set of polynomial equations. An affine variety is an irreducible affine algebraic set, meaning it cannot be written as the union of two proper algebraic sets.

Proposition 17 states that an affine algebraic set V is irreducible if and only if its corresponding ideal \mathcal{I}(V) is a prime ideal. This proposition is important because it links the geometric concept of irreducibility to the algebraic concept of prime ideals.

Corollary 18 states that an affine algebraic set V is a variety if and only if its coordinate ring k[V] is an integral domain. This means that the coordinate ring of V, which is the set of polynomials that vanish on V, has no zero divisors. This is important because it tells us that the coordinate ring of a variety is a nice algebraic structure.

Now, let's look at Example 1 on page 681. In this example, D&F are considering single points in affine space \mathbb{A}^n. These points are defined by a single polynomial equation, so their corresponding ideal in k[A^n] is generated by a single polynomial. Since k[A^n] is a polynomial ring, this ideal is maximal. This means that it is not contained in any other proper ideal of k[A^n].

Now, recall that a prime ideal is an ideal that is not contained in any other proper ideal except itself. Therefore, since the ideal corresponding to a single point in \mathbb{A}^n is maximal, it must also be prime. By Proposition 17, this means that the single point is an irreducible affine algebraic set. And by Corollary 18, since the coordinate ring of this single point is an integral domain, the single point is also an affine variety.

I hope this explanation helps to clarify the reasoning behind the statement in Example 1. Let me know if you have any further questions.