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Chapter 1
Varieties
Outline:
- Affine varieties and examples.
- The basics of the algebra-geometry dictionary.
- Define Zariski topology (Zariski open and closed sets)
- Primary decomposition and decomposition into components. Combinatorial defini- tion of dimension.
- Regular functions. Complete the algebraic-geometric dictionary (equivalence of cat- egories). A whisper about schemes.
- Rational functions.
- Projective Varieties
- Do projection and elimination. Use it to define the dimension of a variety. Prove the weak Nullstellensatz.
- Appendix on Algebra.
1.1 Affine Varieties
The richness of algebraic geometry as a mathematical discipline comes from the interplay of algebra and geometry, as its basic objects are both geometrical and algebraic. The vivid intuition of geometry is expressed with precision via the language of algebra. Symbolic and numeric manipulation of algebraic objects give practical tools for applications. Let F be a field, which for us will be either the complex numbers C, the real numbers R, or the rational numbers Q. These different fields have their individual strengths and weaknesses. The complex numbers are algebraically closed; every univariate polnomial
2 CHAPTER 1. VARIETIES
has a complex root. Algebraic geometry works best when using an algebraically closed field, and most introductory texts restrict themselves to the complex numbers. However, quite often real number answers are needed, particularly in applications. Because of this, we will often consider real varieties and work over R. Symbolic computation provides many useful tools for algebraic geometry, but it requires a field such as Q, which can be represented on a computer. The set of all n-tuples (a 1 ,... , an) of numbers in F is called affine n-space and written An^ or An F when we want to indicate our field. We write An^ rather than Fn^ to emphasize that we are not doing linear algebra. Let x 1 ,... , xn be variables, which we regard as coordinate functions on An^ and write F[x 1 ,... , xn] for the ring of polynomials in the variables x 1 ,... , xn with coefficients in the field F. We may evaluate a polynomial f ∈ F[x 1 ,... , xn] at a point a ∈ An^ to get a number f (a) ∈ F, and so polynomials are also functions on An. We make the main definition of this book.
Definition. An affine variety is the set of common zeroes of a collection of polynomials. Given a set S ⊂ F[x 1 ,... , xn] of polynomials, the affine variety defined by S is the set
V(S) := {a ∈ An^ | f (a) = 0 for f ∈ S}.
This is a (affine) subvariety of An^ or simple a variety.
If X and Y are varieties with Y ⊂ X, then Y is a subvariety of X. The empty set ∅ = V(1) and affine space itself An^ = V(0) are varieties. Any linear or affine subspace L of An^ is a variety. Indeed, L has an equation Ax = b, where A is a matrix and b is a vector, and so L = V(Ax − b) is defined by the linear polynomials which form the rows of Ax − b. An important special case of this is when L = {a} is a point of An. Writing a = (a 1 ,... , an), then L is defined by the equations xi − ai = 0 for i = 1,... , n. Any finite subset Z ⊂ A^1 is a variety as Z = V(f ), where
f :=
z∈Z
(x − z)
is the monic polynomial with simple zeroes in Z. A non-constant polynomial p(x, y) in the variables x and y defines a plane curve V(p) ⊂ A^2. Here are the plane cubic curves V(p + 201 ), V(p), and V(p − 201 ), where p(x, y) := y^2 − x^3 − x^2.
4 CHAPTER 1. VARIETIES
x
y
z
x
y
z
Figure 1.2: Intersection of two quadrics.
right below.
The product V × W of two varieties V and W is again a variety. Suppose that V ⊂ An is defined by the polynomials f 1 ,... , fs ∈ F[x 1 ,... , xn] and the variety W ⊂ Am^ is defined by the polynomials g 1 ,... , gt ∈ F[y 1 ,... , ym]. Then X × Y ⊂ An^ × Am^ = An+m^ is defined by the polynomials f 1 ,... , fs, g 1 ,... , gt ∈ F[x 1 ,... , xn, y 1 ,... , ym].
The set Matn×n or Matn×n(F) of n × n matrices with entries in F is identified with the affine space An 2
. The special linear group is the set of matrices with determinant 1,
SLn := {M ∈ Matn×n | det M = 1} = V(det −1).
We will show that SLn is smooth, irreducible, and has dimension n^2 − 1. (We must first, of course, define these notions.)
We also point out some subsets of An^ which are not varieties. The set Z of integers is not a variety. The only polynomial vanishing at every integer is the zero polynomial, whose variety is all of A^1. The same is true for any other infinite subset of A^1 , for example, the infinite sequence { (^1) n | n = 1, 2 ,... } is not a subvariety of A^1.
Other subsets which are not varieties (for the same reasons) include the unit disc in
1.2. THE ALGEBRA-GEOMETRY DICTIONARY I: IDEAL-VARIETY CORRESPONDENCE 5
R^2 , {(x, y) ∈ R^2 | x^2 + y^2 ≤ 1 } or the complex numbers with positive real part.
x
y
unit disc
− 1 R^2
{z | Re(z) ≥ 0 }
−i
i
C
Sets like these last two which are defined by inequalities involving real polynomials are called semi-algebraic. We will study them later.
1.2 The algebra-geometry dictionary I: ideal-variety
correspondence
We defined varieties V(S) associated to sets S ⊂ F[x 1 ,... , xn] of polynomials,
V(S) = {a ∈ An^ | f (a) = 0 for all f ∈ S}.
We would like to invert this association. Given a subset Z of An, consider the collection of polynomials that vanish on Z,
I(Z) := {f ∈ F[x 1 ,... , xn] | f (z) = 0 for all z ∈ Z}.
The map I reverses inclusions so that Z ⊂ Y implies I(Z) ⊃ I(Y ). These two inclusion-reversing maps
{Subsets S of F[x 1 ,... , xn]}
−−→V
I
{Subsets Z of An} (1.1)
form the basis of the algebra-geometry dictionary of affine algebraic geometry. We will refine this correspondence to make it more precise. An ideal is a subset I ⊂ F[x 1 ,... , xn] which is closed under addition and under muntiplication by polynomials in F[x 1 ,... , xn]: If f, g ∈ I then f + g ∈ I and if we also have h ∈ F[x 1 ,... , xn], then hf ∈ I. The ideal 〈S〉 generated by a subset S of F[x 1 ,... , xn] is the smallest ideal containing S. This is the set of all expressions of the form h 1 f 1 + · · · + hmfm
where f 1 ,... , fm ∈ S and h 1 ,... , hm ∈ F[x 1 ,... , xn]. We work with ideals because of f , g, and h are polynomials and a ∈ An^ with f (a) = g(a) = 0, then (f + g)(a) = 0 and (hf )(a) = 0. Thus V(S) = V(〈S〉), and so we may restrict V to the ideals of F[x 1 ,... , xn]. In fact, we lose nothing if we restrict the left-hand-side of the correspondence (1.1) to the ideals of F[x 1 ,... , xn].
1.2. THE ALGEBRA-GEOMETRY DICTIONARY I: IDEAL-VARIETY CORRESPONDENCE 7
For example, when F = Q and n = 1, we have ∅ = V(1) = V(x^2 −2). The problem here is that the rational numbers are not algebraically closed and we need to work with a larger field (for example Q(
2)) to study V(x^2 − 2). When F = R and n = 1, ∅ 6 = V(x^2 − 2), but we have ∅ = V(1) = V(1 + x^2 ) = V(1 + x^4 ). While the problem here is again that the real numbers are not algebraically closed, we view this as a manifestation of positivity. The two polynomials 1 + x^2 and 1 + x^4 only take positive values. When working over R (as our interest in applications leads us to) we will sometimes take positivity of polynomials into account. The problem with the map V is more fundamental than these examples reveal and occurs even when F = C. When n = 1 we have { 0 } = V(x) = V(x^2 ), and when n = 2, we invite the reader to check that V(y − x^2 ) = V(y^2 − yx^2 , xy − x^3 ). Note that while x 6 ∈ 〈x^2 〉, we have x^2 ∈ 〈x^2 〉. Similarly, y − x^2 6 ∈ V(y^2 − yx^2 , xy − x^3 ), but
(y − x^2 )^2 = y^2 − yx^2 − x(xy − x^3 ) ∈ 〈y^2 − yx^2 , xy − x^3 〉.
In both cases, the lack of injectivity of the map V boils down to f and f m^ having the same set of zeroes, for any positive integer m. In particular, if f 1 ,... , fs are polynomials, then the two ideals
〈f 1 , f 2 ,... , fs〉 and 〈f 1 , f 22 , f 33 ,... , f (^) ss 〉
both define the same variety, and if f m^ ∈ I(Z), then f ∈ I(Z). We clarify this point with a definition. An ideal I ⊂ F[x 1 ,... , xn] is radical if whenever f m^ ∈ I for some m ≥ 1, then f ∈ I. The radical
I of an ideal I of F[x 1 ,... , xn] is defined to be √ I := {f ∈ F[x 1 ,... , xn] | f m^ ∈ I for some m ≥ 1 }.
This turns out to be an ideal. In fact it is the smallest radical ideal containing I. For example, we just showed that √ 〈y^2 − yx^2 , xy − x^3 〉 = 〈y − x^2 〉.
The reason for this definition is twofold: I(Z) is radical and also an ideal and its radical both define the same variety. We record these facts.
Lemma 1.2.4 For Z ⊂ An, I(Z) is a radical ideal. If I ⊂ F[x 1 ,... , xn] is an ideal, then V(I) = V(
I).
When F is algebraically closed, the precise nature of the correspondence (1.2) fol- lows from Hilbert’s Nullstellensatz (null=zeroes, stelle=places, satz=theorem), another of Hilbert’s foundational results in the 1890’s†^ that helped to lay the foundations of algebraic geometry and usher in twentieth century mathematics. We first state a weak form of the Nullstellensatz, which describes the ideals defining the empty set.
†Likely his 1890 paper, maybe the 1893 one.
8 CHAPTER 1. VARIETIES
Theorem 1.2.5 (Weak Nullstellensatz) If I is an ideal of C[x 1 ,... , xn] with V(I) = ∅, then I = C[x 1 ,... , xn].
Let a = (a 1 ,... , an) ∈ An, which is defined by the linear polynomials xi − ai. A polynomial f is equal to the constant f (a) modulo the ideal ma := 〈x 1 − a 1 ,... , xn − an〉 generated by these polynomials, thus the quotient ring F[x 1 ,... , xn]/ma is isomorphic to the field F and so ma is a maximal ideal. In the appendix we show that when F = C (or any other algebraically closed field), these are the only maximal ideals.
Theorem 1.2.6 The maximal ideals of C[x 1 ,... , xn] all have the form ma for some a ∈ An.
Proof of Weak Nullstellensatz. We prove the contrapositive, if I ( C[x 1 ,... , xn] is a proper ideal, then V(I) 6 = ∅. There is a maximal ideal ma with a ∈ An^ of C[x 1 ,... , xn] which contains I. But then {a} = V(ma) ⊂ V(I) ,
and so V(I) 6 = ∅. Thus if V(I) = ∅, we must have I = C[x 1 ,... , xn], which proves the weak Nullstellensatz.
We will later give a second proof that relies on geometric ideas. The Fundamental Theorem of Algebra states that any nonconstant polynomial f ∈ C[x] has a root (a solution to f (x) = 0). We recast the weak Nullstellensatz as the multivariate fundamental thoerem of algebra.
Theorem 1.2.7 (Multivariate Fundamental Theorem of Algebra) If the ideal gen- erated by polynomials f 1 ,... , fm ∈ C[x 1 ,... , xn] is not the whole ring C[x 1 ,... , xn], then the system of polynomial equations
f 1 (x) = f 2 (x) = · · · = fm(x) = 0
has a solution in An.
We now deduce the full Nullstellensatz, which we will use to complete the characteri- zation (1.2).
Theorem 1.2.8 (Nullstellensatz) If I ⊂ C[x 1 ,... , xn] is an ideal, then I(V(I)) =
I.
Proof. Since V(I) = V(
I), we have
I ⊂ I(V(I)). We show the other inclusion. Suppose that we have a polynomial f ∈ I(V(I)). Introduce a new variable t. Then the variety V(tf − 1 , I) ⊂ An+1^ defined by I and tf − 1 is empty. Indeed, if (a 1 ,... , an, b) were a point of this variety, then (a 1 ,... , an) would be a point of V(I). But then f (a 1 ,... , an) = 0, and so the polynomial tf − 1 evaluates to 1 at the point (a 1 ,... , an, b).
10 CHAPTER 1. VARIETIES
4. I(X ∪ Y ) =
I ∩ J =
I · J.
Example. It can happen that I · J 6 = I ∩ J. For example, if I = 〈xy − x^3 〉 and J = 〈y^2 − x^2 y〉, then I · J = 〈xy(y − x^2 )^2 〉, while I ∩ J = 〈xy(y − x^2 )〉.
This correspondence will be further refined in Section 1.5 to include maps between varieties. Because of this correspondence, each geometric concept has a corresponding algebraic concept, when F = C is algebraically closed. When F is not algebraically closed, this correspondence is not exact. In that case we will often use algebra to guide our geometric definitions.
Exercises for Section 1
- Show that no proper nonempty open subset S of Rn^ or Cn^ is a variety. Here, we mean open in the usual (Euclidean) topology on Rn^ and Cn. (Hint: Consider the Taylor expansion of any polynomial in I(S).)
- Verify the claim in the text that smallest ideal containing a set S ⊂ F[x 1 ,... , xn] of polynomials is the set of all expressions of the form
h 1 f 1 + · · · + hmfm
where f 1 ,... , fm ∈ S and h 1 ,... , hm ∈ F[x 1 ,... , xn].
- Prove that in A^2 , we have V(y − x^2 ) = V(y^3 − y^2 x^2 , x^2 y − x^4 ).
- Express the cubic space curve C with parametrization (t, t^2 , t^3 ) in each of the fol- lowing ways.
(a) The intersection of a quadric and a cubic hypersurface. (b) The intersection of two quadrics. (c) The intersection of three quadrics.
- Let I be an ideal of C[x 1 ,... , xn]. Show that √ I := {f ∈ F[x 1 ,... , xn] | f m^ ∈ I for some m ≥ 1 }
is an ideal, is radical, and is the smallest radical ideal containing I.
- If Y ( X are varieties, show that I(X) ( I(Y ).
- Suppose that I and J are radical ideals. Show that I ∩ J is also a radical ideal.
- Give radical ideals I and J for which I + J is not radical.
- Given ideals I and J show that {f · g | f ∈ I and g ∈ J} is an ideal.
1.3. GENERIC PROPERTIES OF VARIETIES 11
1.3 Generic properties of varieties
A useful feature in algebraic geometry is that many properties hold for almost all points of a variety or for almost all objects of a given type. For example, matrices are almost always invertible, univariate polynomials of degree d almost always have d distinct roots, and multivariate polynomials are almost always irreducible. We develop the terminology ‘generic’ and ‘Zariski open’ to describe this situation. A starting point is that intersections and unions of affine varieties behave well.
Theorem 1.3.1 The intersection of any collection of affine varieties is an affine variety. The union of any finite collection of affine varieties is an affine variety.
Proof. For the first statement, let {It | t ∈ T } be a collection of ideals in F[x 1 ,... , xn]. Then we have (^) ⋂
t∈T
V(It) = V
t∈T
It
Arguing by induction on the number of varieties, shows that it suffices to establish the second statement for the union of two varieties but that case is Lemma 1.2.10 (3).
Theorem 1.3.1 shows that affine varieties have the same properties as the closed sets of a topology on An. This was observed by Oscar Zariski.
Definition. We call an affine variety a Zariski closed set. The complement of a Zariski closed set is a Zariski open set. The Zariski topology on An^ is the topology whose closed sets are the affine varieties in An. The Zariski closure of a subset Z ⊂ An^ is the smallest variety containing Z, which is V(I(Z)), by Lemma 1.2.3. Any subvariety X of An^ inherits its Zariski topology from An, the closed subsets are simply the subvarieties of X. A subset Z ⊂ X of a variety X is Zariski dense in X if its closure is X.
We emphasize that the purpose of this terminology is to aid our discussion of varieties, and not because we will use notions from topology in any essential way. This Zariski topology is behaves quite differently from the usual Euclidean topology on Rn^ or Cn^ with which we are familiar. A topology on a space may be defined by giving a collection of basic open sets which generate the topology—any open set is a union or a finite intersection of basic open sets. In the Euclidean topology, the basic open sets are balls. The ball with radius ǫ > 0 centered at z ∈ Anis
B(z, ǫ) := {a ∈ An^ |
|ai − zi|^2 < ǫ}.
In the Zariski topology, the basic open sets are complements of hypersurfaces, called principal open sets.†^ Let f ∈ F[x 1 ,... , xn] and set
Uf := {a ∈ An^ | f (a) 6 = 0}. †Is this the terminology you want?
1.3. GENERIC PROPERTIES OF VARIETIES 13
The last statement of Theorem 1.3.2 leads to the important notions of genericity and generic sets and properties.
Definition. Let X be a variety. A subset Y ⊂ X is called generic if it contains a Zariski dense open subset of X. A property is generic if the set of points on which it holds is a generic set. Points of a generic set are called general points.
This notion of general depends on the context, and so care must be exercised in its use. For example, the general quadratic polynomial ax^2 + bx + c does not vanish when x = 0. (We just need to avoid quadratics with c = 0.) On the other hand, the general quadratic polynomial has two roots, as we need only avoid quadratics with b^2 − 4 ac = 0. The quadratic x^2 − 2 x + 1 is general in the first sense, but not in the second, while the quadratic x^2 +x is general in the second sense, but not in the first. Despite this ambiguity, we will see that this is a very useful concept. When F is R or C, generic sets are dense in the Euclidean topology, by Theorem 1.3.2(6). Thus generic properties hold almost everywhere, in the standard sense.
Example. The generic n × n matrix is invertible, as it is a nonempty principal open subset of Matn×n = An×n. It is the complement of the variety V(det) of singular matrices. Define the general linear group GLn to be the set of all invertible matrices,
GLn := {M ∈ Matn×n | det(M ) 6 = 0} = Udet.
Example. The general univariate polynomial of degree n has n distinct complex roots. Identify An^ with the set of univariate polynomials of degree n via
(a 1 ,... , an) ∈ An^7 −→ xn^ + a 1 xn−^1 + · · · + an− 1 x + an ∈ F[x]. (1.4)
The classical discriminant ∆ ∈ F[a 1 ,... , an] is a polynomial of degree 2n − 1 which vanishes precisely when the polynomial (1.4) has a repeated factor. This identifies the set of polynomials with n distinct complex roots as the set U∆. The discriminant of a quadric x^2 + bx + c is b^2 − 4 c.
Example. The generic complex n × n matrix is semisimple (diagonalizable). Let M ∈ Matn×n and consider the (monic) characteristic polynomial of M
χ(x) := det(xIn − M ).
We do not show this by providing an algebraic characterization of semisimplicity. Instead we observe that if a matrix M ∈ Matn×n has n distinct eigenvalues, then it is semisimple. The coefficients of the characteristic polynomial χ(x) are polynomials in the entries of M. Evaluating the discriminant at these coefficients gives a polynomial ψ which vanishes when the characteristic polynomial χ(x) of M has a repeated root.
14 CHAPTER 1. VARIETIES
We see that the set of matrices with distinct eigenvalues equals the basic open set Uψ, which is nonempty. Thus the set of semisimple matrices contains an open dense subset of Matn×n set and is therefore generic. When n = 2,
det
xI 2 −
[
a 11 a 12 a 21 a 22
])
= t^2 − t(a 11 + a 22 ) + a 11 a 22 − a 12 a 21 ,
and so the polynomial ψ is (a 11 + a 22 )^2 − 4(a 11 a 22 − a 12 a 21 ).
In each of these examples, we used the following easy fact.
Proposition 1.3.3 A set X ⊂ An^ is generic if and only if there is a nonconstant poly- nomial that vanishes on its complement.
Exercises
- Look up the definition of a topology in a text book and verify the claim that the collection of affine subvarieties of An^ form the closed sets in a topology on An.
- Prove that a closed set in the Zariski topology on A^1 is either the empty set, a finite collection of points, or A^1 itself.
- Let n ≤ m. Prove that a generic n × m matrix has rank n.
- Prove that the generic triple of points in A^2 are the vertices of a triangle.
16 CHAPTER 1. VARIETIES
Theorem 1.4.1 A product X × Y of irreducible varieties is irreducible.
Proof. Suppose that Z 1 , Z 2 ⊂ X × Y are subvarieties with Z 1 ∪ Z 2 = X × Y. We assume that Z 2 6 = X × Y and use this to show that Z 1 = X × Y. For each x ∈ X, identify the subvariety {x} × Y with Y. This irreducible variety is the union of two subvarieties,
{x} × Y =
({x} × Y ) ∩ Z 1
({x} × Y ) ∩ Z 2
and so one of these must equal {x} × Y. In particular, we must either have {x} × Y ⊂ Z 1 or else {x} × Y ⊂ Z 2. If we define
X 1 = {x ∈ X | {x} × Y ⊂ Z 1 } , and X 2 = {x ∈ X | {x} × Y ⊂ Z 2 } ,
then we have just shown that X = X 1 ∪ X 2. Since Z 2 6 = X × Y , we have X 2 6 = X. We claim that both X 1 and X 2 are subvarieties of X. Then the irreducibility of X implies that X = X 1 and thus X × Y = Z 1. We will show that X 1 is a subvariety of X. For y ∈ Y , set
Xy := {x ∈ X | (x, y) ∈ Z 1 }.
Since Xy × {y} = (X × {y}) ∩ Z 1 , we see that Xy is a subvariety of X. But we have
X 1 =
y∈Y
Xy ,
which shows that X 1 is a subvariety of X. An identical argument for X 2 completes the proof.
The geometric notion of an irreducible variety corresponds to the algebraic notion of a prime ideal. An ideal I ⊂ F[x 1 ,... , xn] is prime if whenever f g ∈ I with f 6 ∈ I, then we have g ∈ I. Equivalently, if whenever f, g 6 ∈ I then f g 6 ∈ I.
Theorem 1.4.2 An affine variety X is irredicuble if and only if its ideal I(X) is prime.
Proof. Let X be an affine variety and set I := I(X). First suppose that X is irreducible. Let f, g 6 ∈ I. Then neither f nor g vanishes identically on X. Thus Y := X ∩ V(f ) and Z := X ∩ V(z) are proper subvarieties of X. Since X is irreducible, Y ∪ Z = X ∩ V(f g) is also a proper subvariety of X, and thus f g 6 ∈ I. Suppose now that X is reducible. Then X = Y ∪ Z is the union of proper subvarieties Y, Z of X. Since Y ( X is a subvariety, we have I(X) ( I(Y ). Let f ∈ I(Y ) − I(X), a polynomial which vanishes on Y but not on X. Similarly, let g ∈ I(Z) − I(X) be a polynomial which vanishes on Z but not on X. Since X = Y ∪ Z, f g vanishes on X and therfore lies in I(X). This shows that I is not prime.
1.4. UNIQUE FACTORIZATION FOR VARIETIES 17
We have seen examples of varieties with one, two, and three irreducible components. Taking products of distinct irreducible polynomials (or dually unions of distinct hyper- surfaces), gives varieties having any finite number of irreducible components. This is all can occur as Hilbert’s Basis Theorem implies that a variety is a union of finitely many irreducible varieties.
Lemma 1.4.3 Any affine variety is a finite union of irreducible subvarieties.
Proof. An affine variety X either is irreducible or else we have X = Y ∪ Z, with both Y and Z proper subvarieties of X. We may similarly decompose whichever of Y and Z are reducible, and continue this process, stopping only when all subvarieties obtained are irreducible. A priori, this process could continue indefinitely. We argue that it must stop after a finite number of steps. If this process never stops, then X must contain an infinite chain of subvarieties, each one properly contained in the previous one,
X ) X 1 ) X 2 ) · · ·.
Their ideals form an infinite increasing chain of ideals in F[x 1 ,... , xn],
I(X) ( I(X 1 ) ( I(X 2 ) ( · · ·.
The union I of these ideals is again an ideal. Note that no ideal I(Xm) is equal to I. By the Hilbert Basis Theorem, I is finitely generated, and thus there is some integer m for which I(Xm) contains these generators. But then I = I(Xm), a contradiction.
A consequence of this proof is that any decreasing chain of subvarieties of a given variety must have finite length. There is however a bound for the length of the longest decreasing chain of irreducible subvarieties.
Combinatorial Definition of Dimension. The dimension of an irreducible variety X is given by the length of the longest decreasing chain of irreducible subvarieties of X. If
X ⊂ X 0 ) X 1 ) X 2 ) · · · ) Xm ) ∅ ,
is such a chain of maximal length, then X has dimension m.
Since maximal ideals of C[x 1 ,... , xn] necessarily have the form ma, we see that Xm must be a point when F = C. The only problem with this definition is that we cannot yet show that it is well-founded, as we do not yet know that there is a bound on the length of such a chain. Later we shall prove that this definition is correct by relating it to other notions of dimension.
1.4. UNIQUE FACTORIZATION FOR VARIETIES 19
When F = C, we will show that an irreducible variety is connected in the ususal Euclidean topology.†^ We will even show that the smooth points of an irreducible variety are connected. Neither of these facts are true over R. Below, we display the irreducible cubic plane curve V(y^2 − x^3 + x) in A^2 R and the surface V((x^2 − y^2 )^2 − 2 x^2 − 2 y^2 − 16 z^2 + 1) in A^3 R.
Both are irreducible hypersurfaces. The first has two connected components in the Eu- clidean topology, while in the second, the five components of singular points meet at the four singular points.
Exercises
- Show that the ideal of a hypersurface V(f ) is generated by the squarefree part of f , which is the product of the irreducible factors of f , all with exponent 1.
- For every positive integer n, give a decreasing chain of subvarieties of A^1 of length n+1.
- Prove that the dimension of a point is 0 and the dimension of A^1 is 1.
- Prove that the dimension of an irreducible plane curve is 1 and use this to show that the dimension of A^2 is 2.
- Write the ideal 〈x^3 − x, x^2 − y〉 as the intersection of two prime ideals. Decribe the correponding geometry.
†When and where will we show this?
20 CHAPTER 1. VARIETIES
1.5 The algebra-geometry dictionary II
The algebra-geometry dictionary of Section 1.2 is strengthened when we include regular maps between varieties and the corresponding homomorphisms between rings of regular functions. Let X ⊂ An^ be an affine variety. Any polynomial function f ∈ F[x 1 ,... , xn] restricts to give a regular function on X, f : X → F. We may add and multiply regular functions, and the set of all regular functions on X forms a ring, F[X], called the coordinate ring of the affine variety X or the ring of regular functions on X. The coordinate ring of an affine variety X is a basic invariant of X, which is, in fact equivalent to X itself. The restriction of polynomial functions on An^ to regular functions on X defines a surjective ring homomorphism F[x 1 ,... , xn] ։ F[X]. The kernel of this restriction ho- momorphism is the set of polynomials which vanish identically on X, that is, the ideal I(X) of X. Under the correspondence between ideals, quotient rings, and homomor- phisms, this restriction map gives and isomorphism between F[X] and the quotient ring F[x 1 ,... , xn]/I(X).
Example. The coordinate ring of the parabola y = x^2 is F[x, y]/〈y − x^2 〉, which is iso- morphic to F[x], the coordinate ring of A^1. To see thia, observe that substituting x^2 for y rewrites and polynomial f (x, y) as a polynomial g(x) in x alone, and y − x^2 divides the difference f (x, y) − g(x). On the other hand, the coordinate ring of the semicubical parabola y^2 = x^3 is F[x, y]/〈y^2 − x^3 〉. This ring is not isomorphic to the previous ring. For example, the element y^2 = x^3 has two factorizations into irreducible elements, while polynomials F[x] in one variable always factor uniquely
Parabola Semicubical Parabola
This quotient ring F[x 1 ,... , xn]/I(X) is finitely generated by the images of the vari- ables xi. Since I(X) is radical, the quotient ring has no nilpotent elements (elements f such that f m^ = 0 for some m) and is therefore reduced. When F is algebraically closed, these two properties characterize coordinate rings of algebraic varieties.
Theorem 1.5.1 Suppose that F is algebraically closed. Then an F algebra R is the coor- dinate ring of an affine variety if and only if R is finitely generated and reduced.
Proof. We need only show that a finitely generated reduced F algebra R is the coordinate ring of an affine variety. Suppose that the reduced F algebra R has generators r 1 ,... , rn. Then there is a surjective ring homomorphism
ϕ : F[x 1 ,... , xn] −։ R
