Mine J. S. - Algebraic Geometry.pdf - Geometry - Kuya

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

0. Algorithms for Polynomials

1. Algebraic Sets

2. Ane Algebraic Varieties

3. Algebraic Varieties

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

5. Projective Varieties and Complete Varieties

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.

J.S. MILNE

Introduction

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

of linear equations,

a ij X j = d i , i =1 ,... ,m, (*)

j =1

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

polynomial equations,

f i ( X 1 ,...,X n )=0 , i =1 ,... ,m, f i ∈ k [ X 1 ,... ,X n ] .

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, 1

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

deﬁnes a surface S in

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

equation 2

∂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

S if it maps the tangent space at P isomorphically

onto the tangent space at P = α ( P ).

∈

k and that K is a ﬁeld containing

k . Then (*) has a solution in k n if and only if it has a solution in K n , and the dimension of the

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

2 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 (

1 For example, suppose that the system (*) has coe6cients a ij ∈

f ) P at P .)

dX a i X i = ia i X 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 (**)

a local isomorphism at a point P

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

shall learn that the equation (**) deﬁnes a surface in k 3 , 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 ;noris X np− 1 in

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 coecients in k factors completely in k .Examples:

,orthe

subﬁeld

al of

consisting of all complex numbers algebraic over

.Everyﬁeld k

is contained in an algebraically closed ﬁeld.

A ﬁeld of characteristic zero contains a copy of

, the ﬁeld of rational numbers. A

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 × =

p , the ﬁeld

.Thesymbol

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. 3 We use Gothic (fraktur)

letters for ideals:

{

∈

|∃

∈

A such that ab =1

}

abcmnpqABCMNPQ

We use the following notations:

YX and Y are isomorphic;

X = YX and Y are canonically isomorphic (or there is a given or unique isomorphism);

X d = YX is deﬁned to be Y ,orequals Y by deﬁnition;

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

≈

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

that x

∈

⇒−

∈

R — the subset

satisﬁes their conditions, but it is not a subring of

ﬁeld of characteristic p contains a copy of

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

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

sums of the form r i s i with r i

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

∈

A , s i

∈

S .When S =

{

s 1 ,... ,s m

}

, we shall write

( s 1 ,...,s m ) for the ideal it generates.

Let

and

be ideals in A .Theset

{

a + b

∈ a

∈ b }

is an ideal, denoted

. The ideal generated by

consists of all ﬁnite sums a i b i with a i

{

∈ a

∈ b }

is denoted by

.Notethat

ab ⊂ a ∩ b

. Clearly

∈ a

and b i

∈ b

,and

=( a 1 b 1 ,... ,a i b j ,... ,a m b n ).

Let a be an ideal of A .Thesetofcosetsof a in A forms a ring A/ a ,and a → a + a is

a homomorphism ϕ : A → A/ a

=( b 1 ,...,b n ), then

) is a one-to-one correspondence

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

An ideal

.Themap

b → ϕ − 1 (

if prime if

= A and ab

∈ p ⇒

∈ p

or b

∈ p

.Thus

is prime if and

only if A/

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 .Thus m is maximal if and only if A/ m has no proper

nonzero ideals, and so is a ﬁeld. Note that

maximal

⇒ m

prime.

The ideals of A

B are all of the form

a × b

,with

and

ideals in A and B .To

see this, note that if

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

a × b

with

{

( a, b )

∈ c

some b

∈ b }

and

{

( 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 1 ⊂ a 2 ⊂··· becomes stationary, i.e., for some

m, a m = a m +1 = ···.

properly contained in any other ideal in the set.

if a =( a 1 ,...,a m )and

Algebraic Geometry: 0. Algorithms for Polynomials

Proof.

(a)

⇒

(b):If

a 1 ⊂ a 2 ⊂···

is an ascending chain, then

∪ a i is again an

ideal, and hence has a ﬁnite set

{

a 1 ,... ,a n }

of generators. For some m ,allthe a i

belong

a m and then

a m = 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 1 ∈ S ; because

a 1 is not maximal in S , there exists an

ideal

a 2 in S that properly contains

a 1 . Similarly, there exists an ideal

a 3 in S

properly containing

a 2 , etc.. In this way, we can construct an ascending chain of

ideals

a 1 ⊂ a 2 ⊂ a 3 ⊂···

in S that never becomes stationary.

(c)

⇒

(a):Let

be an ideal, and let S be the set of ideals

b ⊂ a

that are ﬁnitely

generated. Let

=( a 1 ,... ,a r ) be a maximal element of S .If

, so that there

exists an element a

∈ a

, a/

∈ c

,then

c =( a 1 ,... ,a r ,a )

⊂ a

and properly contains

which contradicts the deﬁnition of

Aring A 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

i B : A

→

B .A homomorphism of A-algebras B

→

C is a homomorphism of rings

A .

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

exist elements x 1 ,... ,x n

→

C such that ϕ ( i B ( a )) = i C ( a ) for all a

∈

B such that every element of B can be expressed as

a polynomial in the x i with coecients in i ( A ), i.e., such that the homomorphism

A [ X 1 ,... ,X n ]

∈

B sending X i to x i is surjective.

A ring homomorphism A

→

B is ﬁnite ,and B is a ﬁnite A -algebra, if B is ﬁnitely

generated as an A -module 4 .

Let k be a ﬁeld, and let A be a k -algebra. If 1 =0in A , 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 ,the R is the zero ring, i.e., R =

{

}

Polynomial rings. Let k be a ﬁeld. A monomial in X 1 ,...,X n is an expression of

the form

The total degree of the monomial is a i . We sometimes abbreviate it by X α ,α =

( a 1 ,... ,a n ) ∈ N

X a 1

···

X a n

n , j

∈ N

n .

The elements of the polynomial ring k [ X 1 ,... ,X n ] are ﬁnite sums

c a 1 ···a n X a 1

···

X a n

n , a 1 ···a n ∈

k, a j ∈ N

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

mials from a basis for k [ X 1 ,...,X n ]asa k -vector space.

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

ϕ : B

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