Modern Algebraic Geometry: from Algebraic Sets to Algebraic Varieties [Lecture notes]

228 Pages·2005·0.92 MB·English

by  Andreas Hermann

Modern
Algebraic Geometry
from Algebraic Sets to Algebraic Varieties
by Andreas Hermann
May 31, 2005 (227 pages)
Contents
0.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.3 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Affine Geometry 15
1.1 Algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Zariski-Topology . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3 Spaces with Functions . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.5 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . 44
1.6 Singularities and Dimension . . . . . . . . . . . . . . . . . . . 48
2 Projective Geometry 52
3 The Language of Schemes 53
3.1 Prime Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Zariski Topology . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Spectral Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 Appendix - Algebra 74
4.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4 Krull Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5 Graded Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.6 Regular Local Rings . . . . . . . . . . . . . . . . . . . . . . . 100
4.7 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Appendix - Topology 109
5.1 Noetherian Spaces . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3 Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.5 Etale Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
2
5.6 Gluing Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6 Proofs 141
3
0.1 Symbols
A, B, C algebras
C, D divisors, closed sets
E, F fields
G, H groups and monoids
I, J irreducible sets, index sets
L, M, N modules
P, Q prime divisors, open sets
R, S, T rings and sheafs
U, V multiplicatively closed sets, open sets
W constructible sets
X, Y, Z topological spaces, varieties
a, b, c elements of rings and fields
d, e degree of polynomials, dimension
e neutral element of a monoid
f, g, h polynomials, elements of algebras
i, j, k, l integers and indices
m, n natural numbers
p, q, r residue classes of polynomials
s, t, u polynomial variables
u, v, w elements of spaces and varieties
x, y, z elements of spaces and varieties
α, β, γ multi-indices, sections
ι, κ (canonical) monomorphisms
η, ζ unusual (homo)morphisms
λ, µ, ν morphisms of varieties
%, σ (canonical) epimorphisms
%, σ, τ regular functions
ϕ, ψ homomorphisms
Φ, Ψ isomorphisms
∆ diagonal of a bundle
A, B, C categories
B basis of a topology
P family of open sets
S, T topologies (all open sets)
Z Zariski-closed sets
a, b, c ideals
p, q prime ideals
m, n maximal ideals
A, B, C bundles of ?s
R, S, T sheaves of ?s
4
0.2 Notations
N natural numbers {0,1,2,...}
Z the integers N∪(−N)
Q the rationals quotZ
R the field of real numbers
C complex numbers R+iR
F the standard base field
V(a) algebraic set of the ideal a
I(X) vanishing ideal of X
B(p) open base set {x ∈ X | p(x) 6= 0}
B(a) open base set {p ∈ specR | a 6∈ p}
O(X) regular functions on X
A[X] coordinate algebra F[t1,...,tn]/I(X)
a(x) maximal ideal mx/I(X)
A[X] localisation of A[X] in a(x)
x
m[X] maximal ideal a(x)A[X]
x x
A(X) coordinate field quotA(X)
m(x) maximal ideal kn(% 7→ %(x))
M(X) rational functions on X
O(X) local ring {(U,%) | x ∈ U }
x
m(X) maximal ideal {(U,%) | x ∈ U,%(x) = 0}
x
5
0.3 Fundamentals
Algebraic Geometry
Algebraic Geometry is an ancient discipline of mathematics that has un-
dergone several revolutions throughout its history. Nowadays one should
distinguish between two different kinds of algebraic geometry: elementary
and scheme-theoretic algebraic geometry. The elementary theory studies
geometric objects defined by polynomial equations by identifying them with
algebraic objects (this is possible as polynomials are algebraic objects). The
scheme-theoretic approach is technically more demanding and one of the
most sophisticated theories of contemporary mathematics. It started out
as an attempt to merge number theory, commutative algebra and of course
elementary algebraic geometry into a single theory. We leave it to history to
determine wether this attempt has been a success. But the unquestionable
benefit of this modern approach to algebraic geometry is that it tranfers
geometric ideas and insights into commutative algebra.
What we wish to present here is an introduction into the theory of
schemes. From a strictly formal point of view we could do this right away.
But sadly the geometric insights of this theory are very cryptic and can only
be understood by expressing them in terms of elementary algebraic geome-
try. Hence we will first deal with elementary algebraic geometry. Only then
will we introduce the basic notions of schemes and reinterpret the elemen-
tary objects in the new-forged language of schemes. As several problems
from commutaive algebra and number theory can be expressed in the lan-
guage of schemes as well, this will enable us to transfer geometric ideas to
these problems: write down the algebraic problem in terms of schemes and
interpret what this would mean in elementary algebraic geometry.
Prerequesites
We will start with elementary algebraic geometry but this alrady assumes
familiarity with basic notions of algebra and topology. And to further struc-
turize the theory we will sometimes also employ some very mild category
theory (e.g. to enlighten the correspondence between algebraic and geomet-
ric objects). Later on we will also require more advanced techniques from
commutative algebra (when it comes to singularities and dimension). And
in order to introduce schemes we will have to use the language of sheaves
(and bundles). In short: algebraic geometry does have its prerequesites!
But in order to spare the reader to study all these theories we have included
an appendix that summarizes (oftenly with proves) all the methods needed
throughout this text. However we have included it as an appendix in order
to suggest to read these sections as needed and not in advance. Apart from
thisappendixwewishtosortlyclarify(resp.introduce)somebasicnotions:
6
Topology
• By a topological space we understand an ordered pair (X,T) where
X is an arbitary set and T ⊆ P(X) is a collection of subsets, such
that∅andX arecontainedinT andT isclosedunderarbitaryunions
and finite intersections, formally that is iff:
∅,X ∈ T
(cid:91)
P ∈ T (i ∈ I) =⇒ {P | i ∈ I} ∈ T
i i
P ,...,P ∈ T =⇒ P ∩···∩P ∈ T
1 n 1 n
In this case T is said to be a topology on X and the P ∈ T are called
theopensetsofX. ThecomplementsCP = X\P ofopensetsP ∈ T
are said to be closed subsets of X.
• NowasubsetL ⊆ X ofatopologicalspace(X,T)issaidtobelocally
closediffitistheintersectionofaclosedandanopensubset, formally
P ⊆ X open :⇐⇒ P ∈ T
C ⊆ X closed :⇐⇒ CC ∈ T
L ⊆ X locally closed :⇐⇒ ∃P,Q ∈ T : L = Q\P
And a subset W ⊆ X is said to be constructible, iff it is contained
in the boolean algebra generated by T in P(X). In other words W is
constructible iff it is a finite union of locally closed subsets, formally
∃L ,...,L ⊆ X locally closed
W ⊆ X constructible :⇐⇒ 1 k
such that W = L ∪···∪L
1 k
• Since open sets are closed under arbitary unions we may define the
interior of an arbitary subset A ⊆ X to be the largest open subset
contained in A. Formally that is
(cid:91)
A◦ := {P ⊆ X P ⊆ A, P ∈ T }
By going to complements closed sets are closed under arbitary inter-
sections. ThisallowstodefinetoclosureofA ⊆ X tobethesmallest
closed subset of X containing A. Formally that is
(cid:92)
A := {C ⊆ X A ⊆ C, CC ∈ T }
In particular A◦ ⊆ A ⊆ A and A◦ ⊆ A is an open, A ⊆ X is a
closed subset of X. Finally A = A◦ holds true, if and only if A ⊆ X
is open, likewise A = A is equivalent to A ⊆ X being closed.
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• This allows to define the border of an arbitary subset A ⊆ X to be
the closure of A minus its interior. Formally that is
∂A := A\A◦
Note that the closure A and complement of the interior CA◦ both are
closed sets of X. And hence the border ∂A = A∩CA◦ is closed, too.
• Let again (X,T) be a topological space and A ⊆ X be a subset of X,
then A inherits a topology T ∩A called the relative topology from
X, by virtue of
T ∩A := {Q∩A | Q ∈ T }
AndwewillautomaticallyequipanysubsetAofX withthistopology,
withoutmentioningitseperately. ThatisP ⊆ Aisdefinedtobeopen
if and only if there is some open set Q ⊆ X such that P = Q∩A.
• It is a standard exercise to verify the following: if (X,T) is a topo-
logical space and A ⊆ S ⊆ X are arbitary subsets, then we consider
(S,T ∩S) in the relative topology. Hence we can regard the closure
of A in S (say A ) or in X (say A ) respectively. Then we get
S X
A = A ∩S
S X
• Now a system B ⊆ P(X) of subsets of X is said to be a basis of the
topology T, iff the open sets of X are precisely the unions of the sets
contained in B, i.e. iff
(cid:40) (cid:41)
(cid:91)
T = B I 6= ∅,B ∈ B
i i
i∈I
It is an easy exercise to verify that a family of open sets B ⊆ T is a
basis of the topology T if and only if it satisfies the following property
∀P ∈ T ∀x ∈ P ∃B ∈ B : x ∈ B ⊆ P
• Now consider two topological spaces (X,T ) and (Y,T ). Then a
X Y
mapping f : X → Y is said to be continuous, if the preimages of
open sets are open again, that is
∀Q ∈ T : f−1(Q) ∈ T
Y X
And a function f : X → Y is said to be a homeomorphism iff
it is bijective and both f and f−1 are continuous. In this case we
will write f : X ≈ Y. Finally (X,T ) and (Y,T ) are said to be
X Y
homeomorphic if there is a homoemorphism f : X ≈ Y.
8
Algebra
• We assume that the reader is familiar with the notion of commuta-
tive rings and the standard notations employed for such. In short we
assume all rings to be associative and have a unit element. Explictly
we understand a commutative ring to be a triple (R,+,·) where
R 6= ∅ is a nonempty set and + : R ×R → R : (a,b) 7→ a+b and
· : R×R → R : (a,b) 7→ ab are binary operations that satisfy all of
the following properties:
∀a,b,c ∈ R : a+(b+c) = (a+b)+c
∀a,b,c ∈ R : a(bc) = (ab)c
∀a,b ∈ R : a+b = b+a
∀a,b ∈ R : ab = ba
∃0 ∈ R ∀a ∈ R : a+0 = a
∃1 ∈ R ∀a ∈ R : a·1 = a
∀a ∈ R ∃n ∈ R : a+n = 0
a·(b+c) = (a·b)+(a·c)
(a+b)·c = (a·c)+(b·c)
For ease of notation we will refer to R as a commutative ring though
we would have to refer to (R,+,·) from a formal point of view. Let
now R and S be two (commutative) rings, then a mapping ϕ : R → S
is said to be a ring homomorphism, iff it satisfies
∀a,b ∈ R : ϕ(a+b) = ϕ(a)+ϕ(b)
∀a,b ∈ R : ϕ(ab) = ϕ(a)ϕ(b)
ϕ(1 ) = 1
R S
• Letnow(R,+,·)beacommutativering,thenthequadrupel(A,+,·,ƒ)
is said to be an R-algebra, iff (A,+,·) is a commutative ring and
ƒ : R × A → A : (a,f) 7→ af is an exterior operation of R on A
satisfying the following properties:
∀a ∈ R ∀f,g ∈ A : a(f +g) = (af)+(ag)
∀a,b ∈ R ∀f ∈ A : (a+b)f + (af)+(bf)
∀a,b ∈ R ∀f ∈ A : (ab)f + a(bf)
∀a ∈ R ∀f,g ∈ A : (af)g = a(fg)
∀f ∈ A : 1 f = f
R
9
For ease of notation we will refer to A as a commutative R-algebra
though we would have to refer to (A,+,·,ƒ) from a formal point of
view. Let now A and B be two commutative R-algebras, then a map-
ping ϕ : A → B is said to be an R-algebra homomorphism, iff it is
a ring homomorphism that also satisfies
∀a ∈ R ∀f ∈ A : ϕ(af) = aϕ(f)
• Let now R be any commutative ring, then a subset a ⊆ R is said to
be an ideal of R iff it satisfies the following list of properties
0 ∈ a
∀a,b ∈ a : a+b ∈ a
∀a ∈ a : −a ∈ a
∀a ∈ a ∀b ∈ R : ab ∈ a
And we will abbreviate this by writing a (cid:163) R. Note that in case
i
A is a commutative R-algebra and a (cid:163) A is an ideal of A, then a
i
also is an R-submodule of A (as for any a ∈ R and any f ∈ a ⊆ A
we get af = a(1 f) = (a1 )f ∈ a). Hence the residue ring A/a is a
A A
commutative R-algebra again.
• Let again R be any commutative ring and a (cid:163) R be an ideal of R.
i
Then a direct proof shows that we obtain another ideal - called the
√
radical of a and written as a - containing a, by letting
√ (cid:110) (cid:111)
a ⊆ a := a ∈ R | ∃k ∈ N : ak ∈ a (cid:163)i R
√
It is a standard exercise to show that the assignment R : a 7→ a is a
projection (in the sense that R2 = R). And a is said to be a radical
√
ideal (which is sometimes also called perfect) iff a = a. That is the
radical ideals are the image of R. Finally R satisfies the identity
(cid:112) (cid:112) √ (cid:112)
ab = a∩b = a∩ b
√
• Now a comuutative ring is said to be reduced iff 0 = 0 in other
words iff for any a ∈ R we get: ak = 0 (for some k ∈ N) already
implies a = 0. Finally recall that there is the following famous list of
equivalencies (for any ideal a (cid:163) R)
i
a maximal ⇐⇒ R/a field
a prime ⇐⇒ R/a integral domain
a radical ⇐⇒ R/a reduced
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