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ALGEBRAIC GEOMETRY

J.S. MILNE

Abstract. These are the notes for Math 631, taught at the University of Michigan,

Fall 1993. They are available at www.math.lsa.umich.edu/∼jmilne/

Please send comments and corrections to me at jmilne@umich.edu.

v2.01 (August 24, 1996). First version on the web.

v3.01 (June 13, 1998). Added 5 sections (25 pages) and an index. Minor changes

to Sections 0–8.

Contents

Introduction 2

0. Algorithms for Polynomials 4

1. Algebraic Sets 14

2. Aﬃne Algebraic Varieties 30

3. Algebraic Varieties 44

4. Local Study: Tangent Planes, Tangent Cones, Singularities 59

5. Projective Varieties and Complete Varieties 80

6. Finite Maps 101

7. Dimension Theory 109

8. Regular Maps and Their Fibres. 117

9. Algebraic Geometry over an Arbitrary Field 131

10. Divisors and Intersection Theory 137

11. Coherent Sheaves; Invertible Sheaves. 143

12. Diﬀerentials 149

13. Algebraic Varieties over the Complex Numbers 151

14. Further Reading 153

Index 156

1996, 1998 J.S. Milne. You may make one copy of these notes for your own personal use.

2J.S.MILNE

Introduction

Just as the starting point of linear algebra is the study of the solutions of systems

of linear equations,



j=1

= d

,i=1,... ,m, (*)

the starting point for algebraic geometry is the study of the solutions of systems of

polynomial equations,

,... ,X

)=0,i=1,... ,m, f

∈ k[X

,... ,X

Note immediately one diﬀerence between linear equations and polynomial equations:

theorems for linear equations don’t depend on which ﬁeld k you are working over,

but those for polynomial equations depend on whether or not k is algebraically closed

and (to a lesser extent) whether k has characteristic zero. Since I intend to emphasize

the geometry in this course, we will work over algebraically closed ﬁelds for the major

part of the course.

A better description of algebraic geometry is that it is the study of polynomial func-

tions and the spaces on which they are deﬁned (algebraic varieties), just as topology

is the study of continuous functions and the spaces on which they are deﬁned (topo-

logical spaces), diﬀerential geometry (=advanced calculus) the study of diﬀerentiable

functions and the spaces on which they are deﬁned (diﬀerentiable manifolds), and

complex analysis the study of holomorphic functions and the spaces on which they

are deﬁned (Riemann surfaces and complex manifolds). The approach adopted in

this course makes plain the similarities between these diﬀerent ﬁelds. Of course, the

polynomial functions form a much less rich class than the others, but by restricting

our study to polynomials we are able to do calculus over any ﬁeld: we simply deﬁne



i−1

Moreover, calculations (on a computer) with polynomials are easier than with more

general functions.

Consider a diﬀerentiable function f(x, y, z). In calculus, we learn that the equation

f(x, y, z)=C (**)

deﬁnes a surface S in R

, and that the tangent space to S at a point P =(a, b, c)has

equation



∂f

∂x



(x − a)+



∂f

∂y



(y −b)+



∂f

∂z



(z − c)=0. (***).

The inverse function theorem says that a diﬀerentiable map α : S → S



of surfaces is

a local isomorphism at a point P ∈ S if it maps the tangent space at P isomorphically

onto the tangent space at P



= α(P ).

For example, suppose that the system (*) has coeﬃcients a

∈ k and that K is a ﬁeld containing

k. Then (*) has a solution in k

if and only if it has a solution in K

, and the dimension of the

space of solutions is the same for both ﬁelds. (Exercise!)

Think of S as a level surface for the function f, and note that the equation is that of a plane

through (a, b, c) perpendicular to the gradient vector (f)

at P .)

Consider a polynomial f(x, y, z) with coeﬃcients in a ﬁeld k.Inthiscourse,we

shall learn that the equation (**) deﬁnes a surface in k

, and we shall use the equation

(***) to deﬁne the tangent space at a point P on the surface. However, and this is

one of the essential diﬀerences between algebraic geometry and the other ﬁelds, the

inverse function theorem doesn’t hold in algebraic geometry. One other essential

diﬀerence: 1/X is not the derivative of any rational function of X;norisX

np−1

characteristic p = 0. Neither can be integrated in the ring of polynomial functions.

Some notations. Recall that a ﬁeld k is said to be algebraically closed if every

polynomial f(X) with coeﬃcients in k factors completely in k.Examples:C,orthe

subﬁeld Q

of C consisting of all complex numbers algebraic over Q.Everyﬁeldk

is contained in an algebraically closed ﬁeld.

A ﬁeld of characteristic zero contains a copy of Q, the ﬁeld of rational numbers. A

ﬁeld of characteristic p contains a copy of F

, the ﬁeld Z/pZ.ThesymbolN denotes

the natural numbers, N = {0, 1, 2,...}. Given an equivalence relation, [∗] sometimes

denotes the equivalence class containing ∗.

“Ring” will mean “commutative ring with 1”, and a homomorphism of rings will

always carry 1 to 1. For a ring A, A

is the group of units in A:

= {a ∈ A |∃b ∈ A such that ab =1}.

A subset R of a ring A is a subring if it is closed under addition, multiplication, the

formation of negatives, and contains the identity element.

We use Gothic (fraktur)

letters for ideals:

abcmnpqABCMNPQ

abcmnpqABCMNP Q

We use the following notations:

X ≈ YXand Y are isomorphic;

∼

YXand Y are canonically isomorphic (or there is a given or unique isomorphism);

= YXis deﬁned to be Y ,orequalsY by deﬁnition;

X ⊂ YXis a subset of Y (not necessarily proper).

The deﬁnition on page 2 of Atiyah and MacDonald 1969 is incorrect, since it omits the condition

that x ∈ R ⇒−x ∈ R — the subset N of Z satisﬁes their conditions, but it is not a subring of Z.

0. Algorithms for Polynomials

In this section, we ﬁrst review some basic deﬁnitions from commutative algebra,

and then we derive some algorithms for working in polynomial rings. Those not

interested in algorithms can skip the section.

Throughout the section, k will be a ﬁeld (not necessarily algebraically).

Ideals. Let A be a ring. Recall that an ideal a in A is a subset such that

(a) a is a subgroup of A regarded as a group under addition;

(b) a ∈ a, r ∈ A ⇒ ra ∈ A.

The ideal generated by a subset S of A is the intersection of all ideals A containing

a — it is easy to verify that this is in fact an ideal, and that it consists of all ﬁnite

sums of the form



with r

∈ A, s

∈ S.WhenS = {s

,... ,s

}, we shall write

,... ,s

) for the ideal it generates.

Let a and b be ideals in A.Theset{a + b | a ∈ a,b∈ b} is an ideal, denoted

by a + b. The ideal generated by {ab | a ∈ a,b∈ b} is denoted by ab.Notethat

ab ⊂ a ∩ b. Clearly ab consists of all ﬁnite sums



with a

∈ a and b

∈ b,and

if a =(a

,... ,a

)andb =(b

,... ,b

), then ab =(a

,... ,a

Let a be an ideal of A.Thesetofcosetsofa in A forms a ring A/a,anda → a+a is

a homomorphism ϕ : A → A/a.Themapb → ϕ

−1

(b) is a one-to-one correspondence

between the ideals of A/a and the ideals of A containing a.

An ideal p if prime if p = A and ab ∈ p ⇒ a ∈ p or b ∈ p.Thusp is prime if and

only if A/p is nonzero and has the property that

ab =0,b=0⇒ a =0,

i.e., A/p is an integral domain.

An ideal m is maximal if m = A and there does not exist an ideal n contained

strictly between m and A.Thusm is maximal if and only if A/m has no proper

nonzero ideals, and so is a ﬁeld. Note that

m maximal ⇒ m prime.

The ideals of A × B are all of the form a × b,witha and b ideals in A and B.To

see this, note that if c is an ideal in A ×B and (a, b) ∈ c,then(a, 0) = (a, b)(1, 0) ∈ c

and (0,b)=(a, b)(0, 1) ∈ c. This shows that c = a × b with

a = {a | (a, b) ∈ c some b ∈ b}

and

b = {b | (a, b) ∈ c some a ∈ a}.

Proposition 0.1. The following conditions on a ring A are equivalent:

(a) every ideal in A is ﬁnitely generated;

(b) every ascending chain of ideals a

⊂ a

⊂··· becomes stationary, i.e., for some

m, a

= a

m+1

= ···.

properly contained in any other ideal in the set.

Algebraic Geometry: 0. Algorithms for Polynomials 5

Proof. (a) ⇒ (b): If a

⊂ a

⊂··· is an ascending chain, then ∪a

is again an

ideal, and hence has a ﬁnite set {a

,... ,a

} of generators. For some m,allthea

belong a

and then

= a

m+1

= ···= a.

(b) ⇒ (c): If (c) is false, then there exists a nonempty set S of ideals with no

maximal element. Let a

∈ S; because a

is not maximal in S, there exists an

ideal a

in S that properly contains a

. Similarly, there exists an ideal a

in S

properly containing a

, etc.. In this way, we can construct an ascending chain of

ideals a

⊂ a

⊂··· in S that never becomes stationary.

generated. Let c =(a

,... ,a

) be a maximal element of S.Ifc = a, so that there

exists an element a ∈ a, a/∈ c,thenc



=(a

,... ,a

,a) ⊂ a and properly contains c,

which contradicts the deﬁnition of c.

AringA is Noetherian if it satisﬁes the conditions of the proposition. Note that,

in a Noetherian ring, every ideal is contained in a maximal ideal (apply (c) to the

set of all proper ideals of A containing the given ideal). In fact, this is true in any

ring, but the proof for non-Noetherian rings requires the axiom of choice (Atiyah and

MacDonald 1969, p3).

Algebras. Let A be a ring. An A-algebra is a ring B together with a homomorphism

: A → B.Ahomomorphism of A-algebras B → C is a homomorphism of rings

ϕ: B → C such that ϕ(i

(a)) = i

(a) for all a ∈ A.

An A-algebra B is said to be ﬁnitely generated (or of ﬁnite-type over A)ifthere

exist elements x

,... ,x

∈ B such that every element of B can be expressed as

a polynomial in the x

with coeﬃcients in i(A), i.e., such that the homomorphism

A[X

,... ,X

] → B sending X

to x

is surjective.

A ring homomorphism A → B is ﬁnite,andB is a ﬁnite A-algebra, if B is ﬁnitely

generated as an A-module

Let k be a ﬁeld, and let A be a k-algebra. If 1 =0inA, then the map k → A is

injective, and we can identify k with its image, i.e., we can regard k asasubringof

A. If 1 = 0 in a ring R,theR is the zero ring, i.e., R = {0}.

Polynomial rings. Let k be a ﬁeld. A monomial in X

,... ,X

is an expression of

the form

···X

∈ N.

The total degree of the monomial is



. We sometimes abbreviate it by X

,α=

,... ,a

) ∈ N

The elements of the polynomial ring k[X

,... ,X

] are ﬁnite sums



···a

···X

···a

∈ k, a

∈ N.

with the obvious notions of equality, addition, and multiplication. Thus the mono-

mials from a basis for k[X

,... ,X

]asak-vector space.

The term “module-ﬁnite” is used in this context only by the English-insensitive.

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