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Algebraic Geometry II
Notes by: Sara Lapan
Based on lectures given by: Karen Smith
Contents
Part 1. Introduction to Schemes 2
1. Modernizing Classical Algebraic Geometry 2
2. Introducing Schemes 10
3. Gluing Construction 11
4. Products 17
Part 2. Expanding our knowledge of Schemes 20
5. Properties of Schemes not derived from Rings 20
6. Quasi-Coherent Sheaves on Schemes 25
7. Quasi-Coherent Sheaves on Projective Schemes over A 31
Part 3. Introduction to Cohomology 34
8. Sheaf Cohomology 37
˘
9. Cech Cohomology 40
Part 4. Divisors and All That 44
10. Basics of Divisors 44
11. Divisors and Invertible Sheaves 46
11.1. Curves: 54
1Books recommended for this course included, but were not limited to:
(1). Basic Algebraic Geometry 2: Schemes and Complex Manifolds by Igor R. Shafarevich and M. Reid
(2). Algebraic Geometry by Robin Hartshorne
(3). The Geometry of Schemes by David Eisenbud and Joe Harris
2These notes were typed during lecture and edited somewhat, so be aware that they are not error free.
if you notice typos, feel free to email corrections to swlapan@umich.edu.
1
2 Sara W. Lapan
Part 1. Introduction to Schemes
1. Modernizing Classical Algebraic Geometry
Lecture 1. January 8, 2009
Read Shaf. II: V 1.1-1.3, Exercises p15: 1,3,4,5 (due Tuesday)
Books for the course: Hartshorne, Shaf. II, and Geometry of Schemes
Classical Algebraic Geometry:
The main object of study is an algebraic variety over a fixed algebraically closed field.
An algebraic variety, X:
• is a topological space with a cover of open sets that are affine algebraic varieties
• comes with a sheaf of rings: on each open set U, O (U) =the ring of regular
X
functions on U
Ascheme, X:
• is a topological space with a cover of open sets, each of which is an affine scheme
• comes with a sheaf of rings O
X
e e
Anaffineschemeisoftheform(Spec A,A),whereAisthesameringasAbutitisviewedas
all functions on Spec A. A scheme need not be defined over anything, whereas an algebraic
variety is defined over a fixed algebraically closed field. However, one often assumes that
the scheme is over a fixed algebraically closed field. A scheme can also be defined over a
ring, such as Z.
Local Picture of Classical Algebraic Geometry
Fix k = k. X = V({F } ) ⊆ kn is an affine variety (without loss of generality) we can
λ λ∈Λ
assume that Λ is finite and that {F } generates a radical ideal.
λ λ∈Λ
ϕ
Definition 1.1. A morphism between affine algebraic varieties X ✲Y is a map that
agrees with the restriction of some polynomial map on the ambient spaces at each point.
Definition 1.2. Given an affine algebraic variety X ⊆ kn, the coordinate ring of X,
denoted O X, is the ring of regular functions on X, which in this case is simpy functions
X
ϕ n
X ✲kthatagreewith the restriction of some polynomial on k .
Theorem 1.3 (Hilbert’s Nullstellensatz or The Fundamental Theorem of Elementary Clas-
sical Algebraic Geometry). The assignment X OX(X) defines an anti-equivalence (i.e.
a contravariant functor) of categories:
{Affine varieties over k} ✲ {Reduced, finitely-generated k-algebras}
V(I) ⊂ kn ✲ A=k[x ,...,x ]/I
1 n
X ✛ k[X]/I(X)
Since k[x ,...,x ] ✲✲ O (X) has kernel I(X) = {g ∈ k[x ,...,x ] | g| = 0}. The
1 n X 1 n X
functor is just the pull-back map:
ϕ ϕ∗
{X ✲Y}⇐⇒{O (X)✛ O (Y),g◦ϕ✛ g}
X Y
Thecategory on the right looks rather specific, so in scheme theory we want to remove some
of those “arbitrary” restrictions.
Remark 1.4. In this class, a ring is always a commutative ring with identity.
Sara W. Lapan 3
{Affine varieties over k}←→ {Reduced, finitely-generated k-algebras}
⊆ ⊆
❄ ❄
{Affine schemes over k}←→ {k−algebras}
⊆ ⊆
❄ ❄
{Affine Schemes} ←→ {Rings}
Definition 1.5. Fix A, a commutative ring with identity. As a set, Spec A is the set of
prime ideas of A. We consider it as a topological space with the Zariski topology. The
closed sets are: V({F } ) = {p ∈ Spec A | p ⊇ {F } }. Without loss of generality
λ λ∈Λ λ λ∈Λ
we can replace {F } } by any collection of elements generating the same ideal or even
λ λ∈Λ
generating any ideal with the same radical.
Example 1.6 (Spec Z). The closed points are the maximal ideals, so they are (p), where p
is a prime number. The closure of the point (0) is Spec Z. Since Z is a PID, all of the
a a
closed sets will be of the form V(n) = {(p ),...,(p )}, where n = p 1 ...p r. Note that the
1 r 1 r
topology on Spec Z is not the finite complement topology since it contains a dense point,
so we look at the maximal ideals for the open topology.
k[x ,...,x ]
Example 1.7. Let A = 1 n be reduced, where k = k. The maximal ideals of A
(F ,...,F )
1 r
are (x − a ,...,x − a ), where (a ,...,a ) ∈ V(F ,...,F ) ⊆ kn. Since closed points
1 1 n n 1 n 1 r
of Spec A are in one-to-one correspondence with maximal ideals of A, the closed points
of Spec A are the points of the affine variety V(F ,...,F ). Due to the bijection between
1 r
prime ideals in A and irreducible subvarieties, the points of Spec A correspond to irreducible
(closed) subvarieties of V(F ,...,F ).
1 r
Proposition 1.8. If N ⊆ A is the ideal of nilpotent elements, then Spec A ∼
=
Spec A/N as a topological space (but different scheme unless N = 0). homeo
Example 1.9. Spec k[x] is a topological space with one point (x).
(x2)
Example 1.10. Now let’s consider Spec (k ⊕ kx). Since k ⊕ kx ∼v.s k[x], this is the same
= 2
(x )
topological space as the previous example but not the same scheme.
k[x] k[x] ϕ k[x] ∼
Let R = and S = . The map R ✲✲ S is given by killing x. Since =k⊕kx
(x2) (x) (x2)
and k[x] ∼ k, k[x] ✲ k[x] is given by restriction:
(x = (x2)
2 2 ∂f
f = a +a x+a x +...7→f mod x =a +a x, where a =f(0),a = (0)
0 1 2 0 1 0 1 ∂x
Spec k[x] intuitively is the origin in A1 together with a “first-order neighborhood.”
(x2)
Lecture 2. January 13, 2009
Exercise 1.11. Due Tuesday: Shaf. §1: 6,7,8 and §2:1. Read up on sheaves in Hartshorne
Main Starting Point for Scheme Theory:
contravariant functor
{Rings} ✲{Topological Spaces}
ϕ aϕ
Proposition1.12. If A ✲Bisaringhomomorphism,thenSpec B ✲Spec Asending
−1
p 7→ ϕ (p) is a continuous map of topological spaces.
4 Sara W. Lapan
Proof. We need the preimage of a closed set V(I) ∈ Spec A to be closed.
a −1 a
(1) p ∈ ( ϕ) (V(I)) ⇔ ϕ(p) ∈ V(I)
(2) ⇔ϕ−1(p)∈V(I)
(3) ⇔ϕ−1(p)⊇I
(4) ⇔p⊇ϕ(I)
(5) ⇔p∈V(ϕ(I))=V(ϕ(I)B)
Where (2) ⇔ (3) since V(I) = {p ∈ Spec A | I ⊆ p}.
Corollary 1.13. (aϕ)−1(V(I)) = V(ϕ(I)B) =the ideal generated by ϕ(I) ⊆ B.
Crucial Example 1:
∼
=
ThequotienthomomorphismA ✲✲ A/I inducesahomeomorphismSpec A/I ✲V(I)⊆
Spec A. This is a quintessential example of a closed embedding of schemes.
Crucial Example 2:
The homomorphism A ✲A[1]=Alocalized at the multiplicative system {1,f,f2,...}
f
induces a homeomorphism Spec A[1] ✲D(f)={p∈SpecA|f ∈/p}=SpecA−V(f).
f
This is a quintessential example of an open embedding of schemes.
Proposition 1.14. The open sets {D(f)} form a basis for the Zariski-topology on
Spec A. f∈A
Proof. Take an arbitrary open set U. For some ideal I ⊆ A:
U =Spec A−V(I)
={p|p+I}
={p|∃f ∈I,f ∈/ p}
=∪ D(f)
f∈I
Facts/Exercises:
(1) If A is Noetherian, every open set of Spec A is a compact topological space. Even
if A is not Noetherian, Spec A is compact.
(2) Spec A is irreducible as a topological space ⇔ A/Nil(A) is a domain ⇔ Nil(A) =
{f|fn = 0} is prime
(3) Spec A is disconnected ⇔ A ∼ A ×A , where A ,A 6= ∅
= 1 2 1 2
Next we want to define the structure sheaf of an affine scheme Spec A. Given X = Spec A,
we want a sheaf of rings, O , on X. In particular, for any open set U ⊆ X we want O (U)
X ✲ X
to be a ring where for any open set V ⊆ U, the map O (U) O (V) is a ring homo-
X X
morphism given by restriction.
Classical Algebraic Geometry:
Let X ⊆ An be an affine algebraic variety over k. Then k[X] is the coordinate ring of X.
Let U ⊆ X be an open set. Then O (U) = {ϕ : U ✲k|ϕisregular at each point of
X ✛ ✛
U} and for an open set V ⊆ U, O (V) O (U) by ϕ| ϕ.
X X V
Main Features of O :
X
• OX(X)=k[X] 1
• D(f) = X −V(f) ⊆ X O (D(f))=k[X][ ]
X f
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