AG1. Algebraic Sets, Zariski Topology and Affine Varieties - Ryan Lok-Wing Pang

Algebraic Sets, Zariski Topology and Affine Varieties

Ryan, Lok-Wing Pang

Department of MathematicsHong Kong University of Science and Technology

Introduction to Algebraic Geometry, 2015

Ryan, Lok-Wing Pang (HKUST) Algebraic Sets, Zariski Topology and Affine VarietiesIntroduction to Algebraic Geometry, 2015 1

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Outline

1 Algebraic Sets

2 Zariski Topology

3 Affine Varieties


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Affine Spaces and Algebraic Sets

Let k be a field and k be a fixed algebraic closure of k . We define theaffine n-space over k to be An

k = {(a1, · · · , an)|ai ∈ kn}.

Defintion (Algebraic Set)

An algebraic set X ⊆ Ank is the set of common zeros of a collection of

polynomials f1, · · · , fm ∈ k[x1, · · · , xn]. We denote this set byV (f1, · · · , fm) = {(a1 · · · , an) ∈ An

k |fi (a1, · · · , an) = 0 ∀i = 1, · · · ,m}.

More generally, for an ideal I E k[x1, · · · , xn], we defineV (I ) = {(a1 · · · , an) ∈ An

k |fi (a1, · · · , an) = 0 ∀fi ∈ I}.It is easy to see that if I = (f1, · · · , fm), then V (I ) = V (f1, · · · , fm).Conversely, since k[x1, · · · , xn] is a Noetherian ring (by Hilbert BasisTheorem), every ideal I is finitely generated, and hence every V (I )can be written as V (f1, · · · , fm) for some finite collection of fi .


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k = {(a1, · · · , an)|ai ∈ kn}.




k |fi (a1, · · · , an) = 0 ∀i = 1, · · · ,m}.




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k = {(a1, · · · , an)|ai ∈ kn}.




k |fi (a1, · · · , an) = 0 ∀i = 1, · · · ,m}.


k |fi (a1, · · · , an) = 0 ∀fi ∈ I}.

It is easy to see that if I = (f1, · · · , fm), then V (I ) = V (f1, · · · , fm).Conversely, since k[x1, · · · , xn] is a Noetherian ring (by Hilbert BasisTheorem), every ideal I is finitely generated, and hence every V (I )can be written as V (f1, · · · , fm) for some finite collection of fi .


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k = {(a1, · · · , an)|ai ∈ kn}.




k |fi (a1, · · · , an) = 0 ∀i = 1, · · · ,m}.




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Examples

Algebraic sets can be extremely hard to be determined.

The algebraic set V : xn + yn = 1 is defined over Q. Andrew Wilesproved the Fermat’s Last Theorem in 1995 which stated that for alln ≥ 3,

V (Q) =

{{(1, 0), (0, 1)} if n is odd,{(±1, 0), (0,±1)} if n is even.

In fact, by Hilbert’s tenth problem (which is now solved), given afinite system of polynomial equations with coefficients in Q, theredoes NOT exist a finite algorithm to determine the system has asolution in Q or not.


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Examples



V (Q) =




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Examples



V (Q) =




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Zariski Topology

We then define the Zariski Topology on Ank . First, we need a

lemma:

Lemma

Let I , J, Iα be ideals in k[x1, · · · , xn], then we have(i) V (I ) ∪ V (J) = V (I ∩ J),(ii)

⋂α V (Iα) = V (

∑α Iα),

(iii) V (I ) ⊃ V (J) for√I ⊂√J, where

√I = {f ∈ k[x1, · · · , xn]|f m ∈ I}

for some m ∈ Z+ is called the radical of I .

Remark. If I =√I , then we call I an radical ideal.

Remark 2. Hence algebraic set behaves like closed set.


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Zariski Topology


lemma:

Lemma


⋂α V (Iα) = V (

∑α Iα),


√I = {f ∈ k[x1, · · · , xn]|f m ∈ I}





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Zariski Topology


lemma:

Lemma


⋂α V (Iα) = V (

∑α Iα),


√I = {f ∈ k[x1, · · · , xn]|f m ∈ I}





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Zariski Topology


lemma:

Lemma


⋂α V (Iα) = V (

∑α Iα),


√I = {f ∈ k[x1, · · · , xn]|f m ∈ I}





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Proof of lemma

Proof.

First observe that V (I ) ⊃ V (J) for I ⊂ J, henceV (I ∩ J) ⊃ V (I ),V (J) and V (I ) ∪ V (J) ⊂ V (I ∩ J).

Conversely, let P ∈ V (I ∩ J). If P 6∈ V (I ), then there exists apolynomial f ∈ I such that f (P) 6= 0.

Now, for an arbitrary element g ∈ J, we have h = fg ∈ I ∩ J. Henceh(P) = f (P)g(P) = 0 and g(P) = 0, which implies P ∈ V (J).Therefore V (I ∩ J) ⊂ V (I ) ∪ V (J), completing the proof of (i).

For (ii), since Iα ⊂∑

α Iα, we have V (Iα) ⊃ V (∑

α Iα) and hence⋂α V (Iα) ⊃ V (

∑α Iα) .

For each α, we can write Iα in terms of generators by Noetherianproperty: Iα = (fα1 , · · · , fαkα

). Then for P ∈⋂

α V (Iα), we havefαj (P) = 0 for all j = 1, · · · , kα. On the other hand, the set {fαk

}generates the ideal

∑α Iα and hence P ∈ V (

∑α Iα).

(iii) is left as an exercise.


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Proof of lemma

Proof.







∑α Iα) .






∑α Iα).



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Proof of lemma

Proof.







∑α Iα) .






∑α Iα).



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Proof of lemma

Proof.







∑α Iα) .






∑α Iα).



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Proof of lemma

Proof.







∑α Iα) .






∑α Iα).



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Proof of lemma

Proof.







∑α Iα) .






∑α Iα).



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Zariski Topology

Defintion (Zariski Topology)

We define the Zariski topology on Ank by taking the open subsets to be the

complements of the algebraic sets. By the previous lemma and the factthat ∅ = V (1),An

k = V (0), this is indeed a topology.

Let us consider the Zariski Topology on the Affine line A1k for an

algebraically closed field k. Since k[x ] is a PID, every algebraic set isthe set of zeros of a single polynomial. Since k is algebraically closed,f (x) = an(x − α1) · · · (x − αn) for f (x) ∈ k[x ], an, αi ∈ k . HenceV (f ) = {α1, · · · , αn}. Hence all algebraic sets not equal to A1

k arefinite.

Notice that this topology is not Hausdorff.


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Zariski Topology







k arefinite.



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Zariski Topology







k arefinite.



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Affine Varieties

An algebraic set V in Ank is said to be irreducible if V = V1 ∪ V2

implies V1 = ∅ or V2 = ∅, where V1 and V2 are algebraic sets in Ank .

Defintion (Affine Varieties and Quasi-Affine Varieties)

An affine varieties is an irreducible closed subset of Ank with the induced

topology (i.e. an affine variety is an irreducible algebraic set V in Ank). An

open subset of an affine variety is called a quasi-affine variety.

Next, we show that there exists a bijective correspondence betweenradical ideals and affine varieties. Recall that we have constructed avariety (a set) V (I ) from an ideal I E k[x1, · · · , xn]. Conversely, wecan construct an ideal from a set.

Defintion (Ideal of X )

Let X ⊂ Ank be any subset, then the ideal of X is

I (X ) = {f ∈ k[x1, · · · , xn]|f (P) = 0 ∀P ∈ X}.

One can easily check that this is indeed an ideal.


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Affine Varieties










I (X ) = {f ∈ k[x1, · · · , xn]|f (P) = 0 ∀P ∈ X}.



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Affine Varieties










I (X ) = {f ∈ k[x1, · · · , xn]|f (P) = 0 ∀P ∈ X}.



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Affine Varieties










I (X ) = {f ∈ k[x1, · · · , xn]|f (P) = 0 ∀P ∈ X}.



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Affine Varieties










I (X ) = {f ∈ k[x1, · · · , xn]|f (P) = 0 ∀P ∈ X}.

One can easily check that this is indeed an ideal.Ryan, Lok-Wing Pang (HKUST) Algebraic Sets, Zariski Topology and Affine Varieties

Introduction to Algebraic Geometry, 2015 8/ 19

Hilbert’s Nullstellensatz

Theorem (Hilbert’s Nullstellensatz)

Let k be an algebraically closed field. Then for any idealJ E k[x1, · · · , xn], we have

I (V (J)) =√J.

Corollary (Ideal-Variety Correspondence)

Let k be an algebraically closed field. Then there exists a bijectionbetween algebraic sets in An

k and radical ideals in k[x1, · · · , xn]:{V ⊆ An

k} ←→ {I E k[x1, · · · , xn]}V 7−→ I (V )

V (J)←− [ JFurthermore, an algebraic set is an affine variety iff its ideal is a prime(hence radical) ideal. i.e. There exists a bijection between affine vareitiesV in An

k and Spec k[x1, · · · , xn]:{V ⊆ An

k} ←→ Spec k[x1, · · · , xn]


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Hilbert’s Nullstellensatz

Theorem (Hilbert’s Nullstellensatz)

Let k be an algebraically closed field. Then for any idealJ E k[x1, · · · , xn], we have

I (V (J)) =√J.

Corollary (Ideal-Variety Correspondence)

Let k be an algebraically closed field. Then there exists a bijectionbetween algebraic sets in An

k and radical ideals in k[x1, · · · , xn]:{V ⊆ An

k} ←→ {I E k[x1, · · · , xn]}V 7−→ I (V )

V (J)←− [ JFurthermore, an algebraic set is an affine variety iff its ideal is a prime(hence radical) ideal. i.e. There exists a bijection between affine vareitiesV in An

k and Spec k[x1, · · · , xn]:{V ⊆ An

k} ←→ Spec k[x1, · · · , xn]


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Proof

Proof.

The first part is immediate from the Hilbert’s Nullstellensatz.

For the last part, if V is irreducible, we show that I (V ) is prime. Iffg ∈ I (V ), then V ⊆ V (fg) = V (f ) ∪ V (g). HenceV = (V (f )∩V )∪ (V (g)∩V ), both being closed subsets of V . SinceV is irreducible, we have either V = V (f ) ∩ V or V = V (g) ∩ V . i.e.V ⊆ V (f ) or V ⊆ V (g). Hence either f ∈ I (V ) or g ∈ I (V ).

Conversely, if V is reducible, let V = V1 ∪ V2 be a non-trivialdecomposition. Then we have V1,V2 ⊂ V and henceI (V ) ⊂ I (V1), I (V2). Consider f ∈ I (V1)\I (V ) and g ∈ I (V2)\I (V ),then fg ∈ I (V ) since it vanishes on V = V1 ∪ V2. Hence I (V ) is nota prime ideal.


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Proof

Proof.





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Proof

Proof.





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Coordinate Ring of an Affine Variety

Defintion (Coordinate Ring)

Let V ⊆ Ank be an algebraic set, the coordinate ring of V is the quotient

ring k[V ] = k[x1 · · · , xn]/I (V ).

Immediately we have the following characterization of affine variety:

Theorem

Let V ⊆ Ank be an algebraic set, then the following are equivalent:

1) V is an affine variety.2) V is an irreducible topological space.3) I (V ) is a prime ideal in k[x1, · · · , xn].4) k[V ] is an integral domain.

Proof.

Only the last part is new, which follows from the fact that a quotient ringR/I is an integral domain iff I is prime.


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ring k[V ] = k[x1 · · · , xn]/I (V ).


Theorem



Proof.



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ring k[V ] = k[x1 · · · , xn]/I (V ).


Theorem



Proof.



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ring k[V ] = k[x1 · · · , xn]/I (V ).


Theorem



Proof.



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k[Ank ] = k[x1, · · · , xn].

k[pt] = k .

Food for thought: Prove that A2k\{(0, 0)} is not an affine variety, but

its coordinate ring is the same as k[x , y ].

Hence the coordinate ring is not enough to determine the variety.


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k[Ank ] = k[x1, · · · , xn].

k[pt] = k .





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k[Ank ] = k[x1, · · · , xn].

k[pt] = k .





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k[Ank ] = k[x1, · · · , xn].

k[pt] = k .





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Regular and Rational Maps

Defintion (Regular Functions)

Let X ⊆ Ank be an affine variety. A function f : X → k is regular at a

point P ∈ X if there exists open neighborhood U containing P andpolynomials g , h ∈ k[x1, · · · , xn] such that h(Q) 6= 0 ∀Q ∈ U and f = g/hon U. f is regular on X if it is regular at every point of X .

Observe that the set of regular functions on X form a ring, denotedby O(X ).

Theorem

Let X ⊆ Ank be an affine variety, then k[X ] ∼= O(X ).

Two polynomials f and g define the same function on X iffg − f ∈ I (X ). Hence the elements of k[X ] or O(X ) can be identifiedwith the polynomials from X to k.


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Theorem




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Theorem




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Theorem




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We are now ready to define the category of affine varieties.

Defintion (Regular Maps)

A map f : X → Y between two varieties X ⊆ Ank ,Y ⊆ Am

k is regular ifthere exists m regular functions f1, · · · , fm on X such thatf (x) = (f1(x), · · · , fm(x)) for all x ∈ X .

It is routine to check that composition of two morphisms is a morphism,hence we have a category.

A regular function on X is exactly the same thing as a regular mapX → A1

k .


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k .


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k .


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k .


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Rational Maps

Let X be an affine variety, define k(X ) = Frac k[X ] to be the field ofrational functions of X .

Defintion (Rational Maps)

Let X ,Y be varieties. A rational map ϕ : X → Y ⊆ Amk is an m-tuple of

rational functions ϕ1, · · · , ϕm ∈ k(X ) such that for all points x ∈ X atwhich the ϕi are regular, ϕ(x) = (ϕ1(x), · · · , ϕm(x)) ∈ Y

Defintion (Birational Equivalence)

A rational map ϕ : X → Y is a biratioanl if there exists a rational mapψ : Y → X such that ψ ◦ ϕ = idX and ϕ ◦ ψ = idY as rational maps.


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Rational Maps








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Rational Maps








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Examples

Example (Veronese Maps)

Fix N, d ,M =(N+d

d

)− 1, define the Vernoese map νd as follows:

PN −→ PM

[x0 : · · · : xN ] 7−→ [xd0 : xd−10 x1 : · · · : xdN ]

is a regular map.For example if N = 2, d = 2,M = 6− 1 = 5, then[x0 : x1 : x2] 7→ [x2

0 : x0x1 : x0x2 : x1x2 : x22 ].

Example (Frobenius Maps)

Let k = Fp for a prime p, Fi (x1, · · · , xn) ∈ Fp[x1, · · · , xn], i = 1, 2, · · · ,mand X = V (F1, · · · ,Fm). Define Frobp : An

k → Ank by

Frobp(α1, · · · , αn) = (αp1 , · · · , α

pn). This is obviously a regular map. It

also maps X to itself since if Fi (α) = 0, then from the properties of fieldsof characteristic p, we have Fi (α

p1 , · · · , α

pn) = (Fi (α1, · · · , αn))p = 0. The

map Frobp : X → X is called the Frobenius map.


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Examples

Example (Veronese Maps)

Fix N, d ,M =(N+d

d

)− 1, define the Vernoese map νd as follows:

PN −→ PM

[x0 : · · · : xN ] 7−→ [xd0 : xd−10 x1 : · · · : xdN ]

is a regular map.For example if N = 2, d = 2,M = 6− 1 = 5, then[x0 : x1 : x2] 7→ [x2

0 : x0x1 : x0x2 : x1x2 : x22 ].

Example (Frobenius Maps)

Let k = Fp for a prime p, Fi (x1, · · · , xn) ∈ Fp[x1, · · · , xn], i = 1, 2, · · · ,mand X = V (F1, · · · ,Fm). Define Frobp : An

k → Ank by

Frobp(α1, · · · , αn) = (αp1 , · · · , α

pn). This is obviously a regular map. It

also maps X to itself since if Fi (α) = 0, then from the properties of fieldsof characteristic p, we have Fi (α

p1 , · · · , α

pn) = (Fi (α1, · · · , αn))p = 0. The

map Frobp : X → X is called the Frobenius map.


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Contravariant Equivalence of Categories

We end by stating the following very strong theorem:

Theorem

Let X and Y be affine varieties. Then there is a bijective correspondence:

Hom (X ,Y )∼−→ Hom (k[Y ] ∼= O(Y ),O(X )),

where the left Hom means morphisms (regular maps) of varieties, and theright Hom means k-algebras homomorphisms.

The theorem extends to a contravariant functor from the category ofaffine varieties to the category of k-algebras.


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Theorem


Hom (X ,Y )∼−→ Hom (k[Y ] ∼= O(Y ),O(X )),




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Theorem


Hom (X ,Y )∼−→ Hom (k[Y ] ∼= O(Y ),O(X )),




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References

[1] M. Atiyah, Introduction to Commutative Algebra, Westview Press,1994.

[2] D. Eisenbud, Commutative Algebra: with a View Toward AlgebraicGeometry. Springer-Verlag, 1990.

[3] I. Shafarevich - Basic Algebraic Geometry 1, Springer-Verlag, 2007.

[4] K. Ueno - Algebraic Geometry 1: From Algebraic Varieties to Schemes,American Mathematical Society, 1999.


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Thanks

Thanks!


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Documents

AG1. Algebraic Sets, Zariski Topology and Affine Varieties - Ryan Lok-Wing Pang