Affine Varieties - Single Points and maximal ideals

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In summary, in Dummit and Foote Chapter 15, Section 15.3, the definition of an affine variety is given as a nonempty affine algebraic set that cannot be written as the union of two proper algebraic sets. They also prove that an affine algebraic set is irreducible if and only if its corresponding ideal is a prime ideal, and that an affine algebraic set is a variety if and only if its coordinate ring is an integral domain. In Example 1 on page 681, they use this knowledge to show that single points in A^n are affine varieties because their corresponding ideals in k[A^n] are maximal ideals, which are always prime. This is seen through Proposition 17 and Corollary
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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 [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.

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Dummit and Foote then prove the following results:

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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.

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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.

Peter
 

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Maximal ideals are always prime.
 
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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.

Peter
 

1. What is an affine variety?

An affine variety is a subset of n-dimensional space defined by a set of polynomial equations. It is the common zero locus of a finite number of polynomials in n variables.

2. What is a single point in an affine variety?

A single point in an affine variety is a specific solution to the polynomial equations defining the variety. It is a set of values for the variables that satisfy all of the equations simultaneously.

3. How do you find single points in an affine variety?

To find single points in an affine variety, you must first determine the defining polynomial equations. Then, you can solve the equations simultaneously to find the values of the variables that satisfy all of them. These values will be the coordinates of the single point in the affine variety.

4. What is a maximal ideal in an affine variety?

A maximal ideal is an ideal in the coordinate ring of an affine variety that is not properly contained in any other ideal. In other words, it is an ideal that cannot be extended any further. Maximal ideals are important in algebraic geometry because they correspond to single points in the affine variety.

5. How do maximal ideals relate to single points in an affine variety?

In an affine variety, every single point corresponds to a unique maximal ideal in the coordinate ring. Similarly, every maximal ideal corresponds to a single point in the affine variety. This correspondence is known as the Nullstellensatz theorem and is a fundamental concept in algebraic geometry.

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