A course in Algebraic Geometry [Lecture notes]

84 Pages·2013·0.615 MB·English

by  Caucher Birkar, E. Belmont

A course in
Algebraic Geometry
Taught by C. Birkar
Michaelmas 2012
Last updated: May 30, 2013
1
Disclaimer
These are my notes from Caucher Birkar’s Part III course on algebraic geometry, given
at Cambridge University in Michaelmas term, 2012. I have made them public in the
hope that they might be useful to others, but these are not official notes in any way. In
particular, mistakes are my fault; if you find any, please report them to:
Eva Belmont
[email protected]
My occasional comments are in grey italics.
Contents
1 October 5 6
Definition of sheaves
2 October 8 8
Injective and surjective sheaves; exact sequences; f
∗
3 October 10 11
Basic facts about SpecA and O
X
4 October 12 13
Locally ringed spaces; morphisms of such
5 October 15 16
Examples of Schemes
6 October 18 17
Proj(S) and related definitions; irreducible, reduced (affine) schemes
7 October 19 20
Integral ⇐⇒ reduced + irreducible; generic points, definitions and examples of
open/closed immersions
8 October 22 22
Fibred product; fibre; projective, quasiprojective morphisms
9 October 24 25
M(cid:102); inverses f∗ and f−1 and their properties
10 October 26 28
Quasicoherent schemes; Noetherian schemes; quasicoherent sheaves are M(cid:102)on any
open affine; kernel and image of quasicoherent sheaves; statements about Noetherian and
coherent-ness
11 October 29 30
f∗G etc. are quasicoherent; quasicoherent ideal sheaves ⇐⇒ closed subschemes
12 October 31 33
M(cid:102); M(n); F(n); invertible sheaves; identities about tensors, pullbacks, and
inverse images of M(cid:102)and OX(n)
13 November 2 36
Locally free sheaves; Picard group; Cartier divisors
14 November 5 39
Sheaf of differential forms Ω ; exact sequence 0→Ω →(cid:76)n+1O (−1)→
X/Y X/Y 1 X
O →0
X
15 November 7 41
Cohomology of a complex; left exact functors; injective objects; right derived
functors of a left exact functor
16 November 9 44
Sh(X) has enough injectives; definition of Hi(X,F)
17 November 12 47
Injective O -modules are flasque; Hi>0(X,F)=0 for flaque F
X
18 November 14 50
Noetherian X is affine ⇐⇒ Hi>0(X,F)=0 for quasicoherent F
19 November 16 53
Cˇech cohomology
20 November 19 55
Cˇech cohomology on schemes; F flasque =⇒ Hˇi(U,F)=0 for i>0; conditions
for which Hˇi(U,F)∼=Hi(X,F)
21 November 21 58
Cohomology of Pn
22 November 23 60
Cohomology of Pn, con’t; F coherent =⇒ Hi(X,F(d))=0 for i>0 and d large,
and Hi(X,F) is a finite-dimensional V.S.
23 November 26 63
Euler characteristic; Hilbert polynomial
24 November 28 66
Duality; Weil divisors; Riemann-Roch
A Examples classes 68
*Example sheet 1 68
*Example sheet 2 70
*Example sheet 3 75
*Example sheet 4 79
Algebraic geometry Lecture 1
Lecture 1: October 5
Sheaves. Let X be a topological space.
Definition 1.1. Apresheaf isasetofalgebraicdataonthisspace: ForeveryopenU ⊂ X
we associate an abelian group F(U), such that F(∅) = 0. In addition, for every V ⊂ U
we assign a “restriction map” F(U) → F(V), such that
• if V = U, then F(U) → F(V) is the identity;
• if W ⊂ V ⊂ U then the diagram
(cid:47)(cid:47)
F(U) F(V)
(cid:36)(cid:36) (cid:15)(cid:15)
F(W)
commutes.
(Think of F(U) as a set of functions on U.)
Definition 1.2. A sheaf is a presheaf F subject to the following condition. If U = (cid:83)U
i
is an open cover, and we have a collection of s ∈ F(U ) such that s | = s | ,
i i i Ui∩Uj j Ui∩Uj
then there is a unique s ∈ F(U) such that s = s| .
i Ui
Informally, a sheaf is a system in which global data is determined by local data. (If you
have local data that is consistent, then that gives you global data.)
Usually there is another condition that says, if all the local sections are trivial, then the
globalsectionisalsotrivial. Butthisisgivenbytheuniquenessoftheprecedingdefinition.
Example 1.3. Let X be a topological space, and define U = {s : U → R continuous}.
YoucancheckthatU isasheaf. (YoucouldtakeCinsteadofR, oranytopologicalgroup.)
Stalks tell you what happens when you look at a sheaf near a point.
Definition 1.4. Let F be a presheaf on X, x ∈ X. We define the stalk of F at x to be
F = limF(U)
x −→
U(cid:51)x
Moreexplicitly,anelementofF isrepresentedbyapair(U,s)whereU ⊂ X ands ∈ F(U)
x
is a section, with the relation that (U,s) ∼ (V,t) if there is some W ⊂ U ∩V (containing
x) such that s| = t .
W W
Check that F is an abelian group.
x
Definition 1.5. Suppose that F and G are presheaves [sheaves] on X. A morphism
ϕ : F → G of presheaves [sheaves] is a collection of homomorphisms ϕ : F(U) → G(U)
U
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Algebraic geometry Lecture 1
for each open U, chosen consistently in the sense that
F(U) ϕU (cid:47)(cid:47) G(U)
(cid:15)(cid:15) (cid:15)(cid:15)
F(V) ϕV (cid:47)(cid:47) G(V)
is commutative (where the vertical maps are restriction maps).
If F and G are sheaves, then this is a morphism of sheaves.
Definition 1.6. ϕ is an isomorphism if it has an inverse morphism ψ : G → F such that
ϕ◦ψ = Id and ψ◦ϕ = Id.
For any x ∈ X, we get an induced homomorphism F → G .
x x
Given a presheaf, you can always construct an associated sheaf.
Definition 1.7 (Sheaf associated to a presheaf). Suppose F is a presheaf. The sheaf
associated to F is a sheaf F+, along with a morphism F → F+, satisfying:
• given any morphism F → G to a sheaf G, then there is a unique morphism
F+ → G such that
(cid:47)(cid:47)
F G
(cid:79)(cid:79)
(cid:32)(cid:32)
F+
Construction 1.8. F+(U) is the subset of sections s : U → (cid:70) F that satisfy the
x∈U x
following condition:
• For every x ∈ U, there is some neighborhood W ⊂ U of x, and a section t ∈
F(W), such that s(y) = t for every y ∈ W. (Here t denotes the image of t in
y y
the stalk F .)
y
Notethatthismeanss(x) ∈ F foreveryx, andmoreover, everypointhasaneighborhood
x
W in which all of these elements t of the germ come from the same section t ∈ F(W).
y
The idea is that all the “bad” sections vanish in the stalk.
Exercise 1.9. ProvethatthereisanaturalmapF → F+ satisfyingtheconditionsabove,
and that F+ is a sheaf.
It is “obvious” that if F is a sheaf, then F → F+ is an isomorphism.
Remark 1.10. For every x ∈ X, then the map F → F+ is an isomorphism. This is not
x x
surprising, because F and F+ has the same local data. The proof of this is routine, and
you should do this.
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Algebraic geometry Lecture 2
Definition 1.11 (Imageandkernelofamorphism). SupposethatF andG arepresheaves
on some topological space X, and suppose ϕ : F → G is a morphism.
• The presheaf kernel of ϕ is defined kerpre(ϕ)(U) = ker(F(U) → G(U)).
• Thepresheafimageofϕissimilarlydefinedasimpre(ϕ)(U) = im(F(U) → G(U)).
Now assume that F and G are sheaves.
• The kernel is kerpre(ϕ) (i.e. you have to show that the presheaf kernel is actually
a sheaf, if F and G are sheaves).
• This does not work for the image. So define im(ϕ) = (impre(ϕ))+.
Definition 1.12. We say that ϕ is injective if ker(ϕ) = 0. We say that ϕ is surjective if
im(ϕ) = G.
Remark 1.13. For ϕ to be surjective, all the ϕ(U) do not need to be surjective. (They
need to be “locally surjective”.)
Example 1.14. Let X = {a,b}, U = {a}, V = {b}. (Together with the empty set, we
have four open sets). Define a presheaf as follows:
F(X) = Z,F(U) = F(V) = 0
The stalks at both points vanish, so F+ = 0.
Example 1.15. Let X be the same as before. Now define a presheaf as follows:
G(X) = 0,G(U) = G(V) = Z
The smallest open set containing a is U, so G = Z, and similarly G = Z. Then G+(X) =
a b
Z⊕Z (the points are completely independent, hence functions have an independent choice
ofvalueonaorb),andG+(U) = G+(V) = Z,andtherestrictionmapsarejustprojections.
Lecture 2: October 8
Let X be a topological space, ϕ : F → G is a morphism of sheaves. Then
(1) ϕ is injective ⇐⇒ ϕ : F → G is injective for all x ∈ X
x x x
(2) ϕ is surjective ⇐⇒ ϕ : F → G is surjective for all x ∈ X
x x x
(3) ϕ is an isomorphism ⇐⇒ ϕ : F → G is an isomorphism for all x ∈ X
x x x
Proof. Suppose ϕ is injective. Assume ϕ (U,s) → 0. By definition, ϕ (U,s) = (U,ϕ (s)).
x x U
In order for this to be zero on the stalk, it is locally zero: there is some W ⊂ U such that
ϕ (s)| = 0. By diagram chasing in
U W
(cid:47)(cid:47)
F(U) G(U)
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47)
F(W) G(W)
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Algebraic geometry Lecture 2
ϕ (s| ) = 0. Since ϕ is injective, s| = 0. Thus, (U,s) = (W,s| ) = 0. Therefore, ϕ
W W W W x
is injective.
Now suppose ϕ is injective at every x. Suppose ϕ (s) = 0 for some s ∈ F(U). Since ϕ
x U x
is injective, and ϕ (U,s) = 0 in F for all x, then (U,s) = 0 in F . That is, for all x ∈ U,
x x x
there is some W ⊂ U such that s| = 0. By definition of a sheaf, s = 0 in F(U). Thus
x Wx
ϕ is injective for all U.
U
Assume ϕ is surjective. Pick (U,t) ∈ G . We can’t necessarily get a preimage of t, but
x
there is some small neighborhood in which t| has a preimage. Let s ∈ F(W) be the
W
preimage of t| . In particular, ϕ (W,s) = (W,t| ) = (U,t). So ϕ is surjective.
W x W x
Now assume that ϕ is surjective for all x. I don’t understand why we can’t just use the
x
fact that imϕ has the same stalks as the preimage version (the actual image), so showing
imϕ = G is the same as showing G = (impreϕ) for all x.
x x
We have a factorization
ϕ (cid:47)(cid:47)
F G
(cid:79)(cid:79)
(cid:34)(cid:34) (cid:31)(cid:63)
Im(ϕ)
By replacing F by its image Im(ϕ) we could assume that ϕ is injective. By the first part
of the theorem, we can assume that ϕ is injective, hence an isomorphism for all x. We
x
need to now show that (the new) ϕ is an isomorphism.
Pick t ∈ G(U). We will find local preimages and glue them. For each x ∈ U there is some
W ⊂ U (containing x) and s ∈ F(W ) such that ϕ (s ) = t| . Since ϕ is injective,
x x x Wx x W
s | = s | . Now by the definition of sheaves, there is some
x Wx∩Wy y Wx∩Wy
s ∈ F(U)
such that ϕ (s) = t. This means that the ϕ are isomorphisms for all U, as we wanted.
U U
Suppose that ϕ is an isomorphism. By (1) and (2) ϕ is an isomorphism. Now assume
x
that ϕ are all isomorphisms. Use the same argument in (2).
x
(cid:35)
Definition 2.1. Suppose that X is a topological space. Then a complex of sheaves on X
is a sequence
··· ϕ→−1 F →ϕ0 F →ϕ1 F → F → ···
−1 0 1 2
such that im(ϕ ) ⊂ ker(ϕ ).
i i+1
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Algebraic geometry Lecture 3
Wesaythatthiscomplexisanexactsequenceifim(ϕ ) ⊂ ker(ϕ )foralli. Inparticular,
i i+1
a short exact sequence is an exact sequence of the form
0 → F → F → F → 0
1 2 3
Exercise 2.2. Acomplex0 → F → G → H → 0isashortexactsequenceiffthesequences
0 → F → G → H → 0 is an exact sequence.
x x x
Example 2.3. Let X be a topological space, A an abelian group. Start by defining
F(U) = A for every U. But this isn’t a sheaf. Instead define the constant sheaf at A to
be F+.
Proposition 2.4. The constant sheaf at A is isomorphic to G, defined as
G(U) = {α : U → A : α is continuous}
where A is given the discrete topology.
Proof. G is a sheaf, and we have a natural morphism F → G where the function F(U) →
a
G(U) is just the constant map a (cid:55)→ (U → A). This uniquely determines a morphism
ϕ : F+ → G.
I claim that this is an isomorphism. It suffices to show that ϕ is an isomorphism, and
x
since F+ ∼= F , it suffices to show that ψ : F → G is an isomorphism.
x x x x x
ψ is injective: If ψ (U,s) = 0 = (U,ψ (s)) = (U,α ) then α = 0 on some point, and
x x U s s
hence s = 0.
ψ is surjective: Pick (U,t) ∈ G . (t : U → A where x (cid:55)→ t(x).) Put W = t−1{t(x)},
x x
which is open because A has the discrete topology. Now (U,t) = (W,t| ). Note that t|
W W
is a constant function. If we put s = t(x), then ψ (W,s) = (W,t| ).
x W
(cid:35)
Definition 2.5. Suppose that we have a continuous map f : X → Y. If F is a sheaf
on X, then define a presheaf f F where (f F)(U) = F(f−1U). This is called the direct
∗ ∗
image of F.
Proposition 2.6. f F is a sheaf.
∗
(cid:83)
Proof. Assume U = U is an open cover of U, and there are sections s ∈ (f F)(U ) that
i i ∗ i
agree on intersections. By definition, (f F(U )) = F(f−1U ). Since F is a sheaf, there is
∗ i i
some s ∈ F(f−1U) such that s| = s . So s ∈ (f F) is the thing you want.
Ui i ∗
(cid:35)
Example 2.7. Let Y be a topological space, X = {x} ⊂ Y and f is the inclusion map.
Suppose A is an abelian group. A defines a sheaf F on X. We call f F the skyscraper
∗
sheaf on Y at X: this is
(cid:40)
A if x ∈ U
(f F)(U) =
∗
0 otherwise
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