algebraic geometry lecture notes -...
TRANSCRIPT
Algebraic GeometryLecture Notes
following the Part III course Algebraic Geometryby Dr C. Birkar
Michaelmas term 2009Cambridge
written by Clemens Koppensteiner
Version 1.0.1August 10, 2010
These notes are unofficial lecture notes and are not endorsed by Dr. Birkar.
The current version of this text as well as the LATEX source code can be found at http://caramdir.at/math.
This work is licensed under the Creative Commons Attribution-Share Alike 3.0 Austria License. Toview a copy of this license, visit http://creativecommons.org/licenses/by-sa/3.0/at/ or senda letter toCreative Commons171 Second Street, Suite 300San FranciscoCalifornia, 94105USA.
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Contents
1. General Abstract Nonsense 51.1. Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2. Ringed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3. Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2. Schemes 132.1. Affine Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2. Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3. Proj and Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4. Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5. Sheaves of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6. Cartier Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.7. Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3. Cohomology 373.1. Cohomology of Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2. Cech Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3. Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4. The Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A. Example Sheets 52A.1. Sheet 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.2. Sheet 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56A.3. Sheet 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A.4. Sheet 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
B. Some Mathematicians 61
C. Changes 62
Introduction
The content of these notes is mostly identical to the content of the course “Algebraic Geometry” heldby Dr. C Birkar in Michaelmas term 2009 in Cambridge. Some parts have however been slightlyreorganized. Further, I added some additional remarks and useful lemmas. These are marked by *.Notes in the margins usually refer to texts where additional information or omitted proofs can befound. Knowledge of commutative algebra will often be silently assumed in the text.
The document is split into three parts. The first one discusses topics that, while mostly developedfor use in algebraic geometry, are used in other branches of mathematics as well. The second chapterwill introduce the main objects of algebraic geometry – schemes and the morphisms between them.The final chapter discusses cohomological methods in algebraic geometry. I plan to add solutions tothe example sheets in an appendix, but this is not done yet.
The course covered subsets of chapters II and III of Hartshorne’s Algebraic Geometry [Har77].Therefore that book serves as the basic reference for this text. Most of chapters 1 and 2 of these
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notes can also be found in Liu’s Algebraic Geometry and Arithmetic Curves [Liu02], which sometimesprovides a good second view on a topic. To get some more geometric intuition about schemes andsome of the notions we will develop, I can highly recommend reading the examples provided in TheGeometry of Schemes by Eisenbud and Harris [EH00].
If you find any error or have any comments how the text might be improved, please write me a mail [email protected]. Updated versions of this document can be found at http://caramdir.at/math.
Preliminaries
Conventions All rings are commutative with one and 0 6= 1. All ring homomorphisms take 1 to 1. Agraded ring is always of the form S =
⊕d≥0 Sd while a graded module is of the form M =
⊕d∈ZMd.
A graded homomorphism of graded rings is a ring homomorphism that preserves degrees.
Definition 0.1. A homomorphism f : A→ B of local rings (A,mA) to (B,mB) is a local homomor-phism if f(mA) ⊆ mB .
Definition 0.2. If M =⊕Md is a graded module, we define M(n) for n ∈ Z to be the graded module
M(n) =⊕M(n)d with M(n)d = Mn+d. In particular, from a graded ring S we get S-modules S(n).
Definition 0.3. If S is a graded ring, M a graded S-module, p a homogeneous prime ideal andb ∈ S a homogeneous element, we write S(p), S(b), M(p), M(b) for the degree zero elements in Sp, Sb,Mp and Mb respectively.
In a expression containing a list of things, elements that are marked with a hat are omitted. Forexample, t1 · · ·“tj · · · tn means the product of the t1, . . . , tn except for tj .
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1. General Abstract Nonsense
Before beginning to study proper algebraic geometry, we will introduce some objects and ideas that –while they were primarily developed with algebraic geometry in mind – are really of more generaluse in mathematics. First will introduce sheaves, which essentially clarify any situation where thereis a dichotomy of global versus local properties of a geometric space. Then we will discuss ringedspaces. These are an abstraction of what a “geometric space” is. Here I collected some definitionswhich make sense in this more general setting – listeners of the Part III Complex Manifolds lectureswill note that the many definitions here were also made in the complex manifold class by simplyreplacing “ringed space” by “complex manifold”. The finial section of this chapter will be devoted tothe fundamentals of homological algebra.
1.1. Sheaves
Definition 1.1. Let X be a topological space. A presheaf F on X consists of the following data:
• For each open set U ⊆ X, an Abelian group F (U), the elements of which are called sectionsover U ;
• For each inclusion of open sets V ⊆ U ⊆ X a restriction homomorphism F (U)→ F (V ), s 7→s|V .
Further, it has to fulfill the following conditions:
1. F (∅) = 0 and for each open U the restriction map F (U)→ F (U) is the identity map;
2. If W ⊆ V ⊆ U are open sets and s ∈ (U), then s|W = (s|V )W .
A presheaf F on X is called a sheaf if it satisfies the sheaf condition:
3. For all open U ⊆ X: If U =⋃Ui is an open cover and we are given si ∈ F (Ui) such that
si|Ui∩Uj = sj |Ui∩Uj for all i, j, then there exists a unique s ∈ F (U) with s|Ui = si for all i.
Example 1.2. For any topological space X setting
F (U) = s : U → R continuous
with the usual restriction of functions gives a sheaf F .
Definition 1.3. If U is an open subset of X and F is a (pre)sheaf on X, then F |U is the (pre)sheafon U given by V 7→ F (V ) for V ⊆ U open. It is called the restriction of F to U .
Lemma* 1.4. Let Ui be family of open subsets on X and set U =⋃Ui. Let Uij = Ui ∩ Uj. For [Liu02, lemma 2.2.7]
any presheaf F on X there is a complex of Abelian groups
0→ F (U)d0−→∏i
F (Ui)d1−→∏i,j
F (Uij),
where d0 : s 7→ (s|Ui)i and d1 : (si)i 7→(si|Uij − sj |Uij
)i,j
.
Then F is a sheaf if and only if this complex is exact for every family of open subsets of X.
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Proof. This is just a reformulation of the sheaf condition.
Definition 1.5. The stalk Fx of a (pre)sheaf F at a point x ∈ X is
Fx = lim−→U open,x∈U
F (U).
Remark 1.6. Every element of Fx is represented by a pair (U, s), where x ∈ U open and s ∈ F (U).Two such elements (U, s), (V, t) are equal in Fx if there is an open set W ⊆ U ∩ V with s|W = t|V .
Definition 1.7. Let F , G be (pre)sheafs on X. A morphism of (pre)sheavesφ : F → G is a collection
F (U) G(U)
F (V ) G(V )
φU
φV
of homomorphisms φU : F (U)→ G(U) such that φ commutes with the restriction maps: for everyinclusion of open sets U ⊆ V , we have φU (·)|V = φV (·|V ). A morphism of (pre)sheaves is anisomorphism if it has a two-sided inverse that is also a morphism.
Remark 1.8. For each x ∈ X, φ : F → G induces a homomorphism φx : Fx → Gx by (U, s) 7→(U, φU (s)).
Definition 1.9. Let F be a presheaf on a topological space F . A sheaf F + together with a morphism
F F +
G
α
φ ψ
of presheaves α : F → F + is called the sheaf associated to F if the following universal property holds:For every sheaf G and every morphism of presheaves φ : F → G there exists a unique morphism ofsheaves ψ : F + → G such that φ = ψ α.
Theorem 1.10. Let F be a presheaf on a topological space X. Then the associated sheaf (F +, α)exists and is unique up to unique isomorphism.
Proof. Uniqueness follows directly from the universal property.
Let F +(U) be the sets : U →
∐x∈U
Fx :∀x ∈ U : s(x) ∈ Fx and there is x ∈ V ⊆ U openand t ∈ F (V ) such that ∀y ∈ V : s(y) = (V, t) ∈ Fy
.
(NB: This means that the sections of F + are locally sections of F .) The natural restriction mapsmake F + a sheaf. For U ⊆ X and s ∈ F (U), let αU (s) ∈ F +(U) be the function x 7→ (U, s) ∈ Fx.
Now suppose we are given φ : F → G . We need to construct ψ : F + → G . Let s ∈ F +(U). Byconstruction there exists an open cover U =
⋃Ui and si ∈ F (Ui) such that αUi(si) = s|Ui . Let
ti = φ(si). We easily see that ti|Ui∩Uj = tj |Ui∩Uj . Therefore, since G is a sheaf, there exists a uniquet ∈ G(U) such that t|Ui = ti and we have to put ψU (s) = t.
Remark 1.11. For each x ∈ X, the natural homomorphism Fx → F +x is an isomorphism.
Definition 1.12. Let φ : F → G be a morphism of presheaves. The presheaf kernel kerφ of φ isgiven by (kerφ)(U) = kerφU . The presheaf image imφpre of φ is given by (imφpre)(U) = imφU .
Now, let φ : F → G be a morphism of sheaves. Then kerφ is a sheaf, called the kernel of φ. Theimage imφ of φ is the sheaf associated to imφpre. (Note that it can be seen as a subsheaf of G .) Themorphism φ is injective if kerφ = 0 and surjective if imφ = G .
Remark 1.13. Even if F → G is a surjective morphism, the maps F (U)→ G(U) are in general notsurjective (see Exercise A.1).
Theorem 1.14. Let φ : F → G be a morphism of sheaves on X. Then φ is injective, surjective oran isomorphism if and only if for each x ∈ X, φx : Fx → Gx has this property.
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Proof.
• Injectivity : First suppose that φ is injective and pick any x ∈ X. We have to show that φx isinjective. Suppose we have s ∈ F (U) (with x ∈ U) such that φx(U, s) = (U, φU (s)) = 0 ∈ Gx.By definition this means that there exists W ⊆ U with x ∈W such that
0 = φU (s)|W = φW (s|W ).
Hence s|W ∈ kerφW . Since φ is injective, s|W = 0, so (U, s) = (W, s|W ) = 0 ∈ Fx. Thus φx isinjective.
Now suppose that φx is injective for all x. Let s ∈ F (U) with φU (s) = 0. Then φx(U, s) =(U, φU (s)) = 0 ∈ Gx for all x in U . Since φx is injective (U, s) = 0 in Fx. In other words, for allx ∈ U there exists an open neighborhood W ⊆ U of x with s|W = 0. By the sheaf conditions = 0 in F (U), so φU is injective.
• Surjectivity : Suppose that φ is surjective and pick any x ∈ X. Let (U, t) be any element inGx. Since φ is surjective, there exists an open neighborhood W ⊆ U of x and s ∈ F (W ) withφW (s) = t|W . Hence φx(W, s) = (W,φW (s)) = (W, t|W ) = (U, s) ∈ Gx, i.e. φx is surjective.
Conversely suppose all φx are surjective and pick t ∈ G(U). For all x ∈ U , by the surjectivity ofφx, there exists an open neighborhood W ⊆ U of x and s ∈ F (W ) with φx(W, s) = (U, t) ∈ Gx.So for all x ∈ U , there exists an open neighborhood V ⊆W of x such that φW (s)|V = tV . Inother words, sections of G are locally the image of sections of F . Hence im F = G .
• A morphism φ is an isomorphism if and only if all φU are bijective.
If φ is an isomorphism, then we have just shown that all φx are bijective and hence isomorphisms.
Suppose that all φx are isomorphisms. We have already shown that all φU are injective. Lett ∈ G(U). As above, for all x ∈ U there exists a neighborhood V ⊆ U of x and sV ∈ F (V )with φV (sV ) = t|V . By the injectivity of φ, the sections sV are uniquely determined and arecompatible. Hence the sheaf condition gives s ∈ F (U) with φU (s) = t, i.e. φU is surjective.
Definition 1.15. A sequence of sheaves is a family Fi (i ∈ Z) of sheaves on X and morphismsφi : Fi → Fi+1. It is usually written in the form
· · · −→ F−2φ−2−→ F−1
φ−1−→ F0φ0−→ F1
φ1−→ F2 −→ · · · .
A complex of sheaves is a sequence with imφi ⊆ kerφi+1 for all i ∈ Z. It is an exact sequence ifimφi = kerφi+1 for all i ∈ Z. A short exact sequence is an exact sequence of the form
0→ F1 → F2 → F3 → 0. (1.1)
Remark 1.16. A sequence of the form (1.1) is exact if and only if
0→ (F1)x → (F2)x → (F3)x → 0
is exact for all x ∈ X.
Example 1.17. Let X be a topological space and A a fixed Abelian group. Define a presheaf F pre
on X by F pre(U) = A for all U 6= ∅ with the identities as restriction maps. Then F = (F pre)+
is called the constant sheaf defined by A on X. It is isomorphic to the sheaf G given by G(U) =s : U → A locally constant.
Definition 1.18. Assume that f : X → Y is a continuous map of topological spaces and F is asheaf on X. The direct image sheaf (or push-down) f∗F is the sheaf on Y given by
(f∗F )(U) = F (f−1(U)) U ⊆ X open.
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Remark 1.19. Let X = y ⊆ Y and f : X → Y . A sheaf F on X is is completely determined byF (X) = A. The direct image sheaf is given by
(f∗F )(U) =
®A, y ∈ U0, otherwise
.
It is called the skyscraper sheaf on Y at y given by A. Note that (f∗F )y′ = 0 for y′ /∈ y.
Definition 1.20. Suppose f : X → Y is a continuous map of topological spaces and G is a sheaf onY . Then the sheaf on X associated to the presheaf
U 7→ lim−→V⊇f(U)
G(V )
is called the inverse image of G and is denoted f−1G .
Remark* 1.21. For every x ∈ X, (f−1G)x = Gf(x). Also, if V is an open subset of Y and ι : V → Yis the injection, then ι−1G = G |V .
Remark* 1.22. There are natural maps f−1f∗F → F and G → f∗f−1G .[Har77, Exercise
II.1.18]Lemma* 1.23. Let B be a base of the topological space X and let F and G be sheaves on X. Given
[EH00, Proposition
I-12] a collection of maps φU : F (U)→ G(U) for all U ∈ B that commute with the restriction maps, there
is a unique morphism φ : F → G of sheaves such that φU = φU for all U ∈ B.
1.2. Ringed Spaces
Definition 1.24. A ringed space (X,OX) is a topological space X together with a sheaf of rings,called the structure sheaf , OX on X. A morphism of ringed spaces f : (Y,OY )→ (X,OX) consists ofa continuous map f : Y → X and a morphism of sheaves f# : OX → f∗OY .
A locally ringed space is a ringed space (X,OX) such that all stalks (OX)P , P ∈ X, are local rings.A morphism of locally ringed spaces (Y,OY )→ (X,OX) is a morphism in the above sense such thatall induced homomorphisms (OX)f(Q) → (OY )Q, Q ∈ Y are local homomorphisms.
In both cases, an isomorphism is a morphism with a two-sided inverse.
Remark* 1.25. The notation f# was not used in the lecture, but is used in many books (like[Har77, Liu02, EH00]). If the space X is clear, we will sometimes just write O for OX . For the stalk(OX)x (x ∈ X) we will usually write OX,x and sometimes just Ox.
Definition 1.26. Let (X,O) be a locally ringed space and x ∈ X. The maximal ideal of the localring Ox is denoted mx. The residue field at x is k(x) = Ox/mx.
Definition 1.27. Let (X,OX) be a ringed space. An OX-module is a sheaf F on X such thatfor every open U ⊆ X, F (U) is an OX(U)-module and such that for every inclusion of open setsV ⊆ U in X and s ∈ OX(U), m ∈ F (U), (sm)|V = sVmV . A morphism of OX-modules φ : F → Gis a morphism of sheaves such that for every open U ⊆ X the map φU : F (U) → G(U) is anOX(U)-module homomorphism. The set of all OX -module morphisms from F to G is denoted byHomOX (F ,G).
Remark 1.28. The direct sum, direct limit and inverse limit of OX -modules are again OX -modules. Thekernel and image of a map of OX -modules are OX -modules. If f : (X,OX)→ (Y,OY ) is a morphismof ringed spaces and F is an OX -module, then f∗F is an OY -module (via f# : OY → f∗OX).
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Definition 1.29. If F and G are OX -modules, we define F ⊗OX G to be the sheaf associated to thepresheaf U 7→ F (U)⊗OX(U) G(U).
If f : (X,OX)→ (Y,OY ) is morphism of ringed spaces and we have an OY -module G , then f−1G isan f−1OY -module, but not an OX -module. The idea is to tensor it with OX to obtain an OX -module.For this note that we have a morphism of sheaves f# : OY → f∗OX , which induces a morphismf−1OY → f−1f∗OX . We also have a natural morphism f−1f∗OX → OX (see remark 1.22). Togetherthey give a morphism f−1OY → OX .
Definition 1.30. Let f : (X,OX) → (Y,OY ) be a morphism of ringed spaces and let G be anOY -module. The inverse image of G is
f∗G = f−1G ⊗f−1OY OX .
Definition 1.31. Let (X,OX) be a ringed space. An OX -module L is called invertible, if for everypoint x of X, there exists an open neighborhood U of x with L |U = OX |U .
Theorem 1.32. The set of isomorphism classes of invertible sheaves on a ringed space X forms agroup under ⊗OX . The neutral element is OX and the inverse of an invertible sheaf L is given byHomOX (L ,OX).
Definition 1.33. The group of isomorphism classes of invertible sheaves over a ringed space X iscalled the Picard group and denoted by Pic(X).
Proof of theorem 1.32. Let L and M be invertible sheaves on (X,OX). For every x ∈ X thereexists an open neighborhood U of x such that L |U ∼= OX |U and M |U ∼= OX |U . So (L ⊗OX M )|U ∼=OX |U ⊗OX |U OX |U ∼= OX |U . Thus the product of two invertible sheaves is again invertible.
Since L ⊗OX OX ∼= L for every invertible sheaf L , the neutral element has to be OX .We will now define an isomorphism φ : L ⊗OX H omOX (L ,OX)→ OX as the associated morphism
of sheaves to the presheaf morphism given for each open subset W ⊆ X by
L(W )⊗OX(W ) H omOX (L ,OX)(W )→ OX(W )
l ⊗ ϑ 7→ ϑW (l)
Pick x ∈ X and a neighborhood U of x such that L |U ∼= OX |U . Then H omOX (L ,OX)|U ∼= OX (inparticular this means that H omOX (L ,OX) is invertible) and φU is an isomorphism. This can be donefor each x ∈ X, so φ is an isomorphism.
1.3. Derived Functors
No proofs are given in this section. References can be found in [Har77, Section III.1].
Definition 1.34. An Abelian category is a category which behaves like the category of Abeliangroups.
Example 1.35. The following categories are Abelian:
• The category of Abelian groups, denoted Ab.
• The category of modules over a ring R.
• The category of sheaves of Abelian groups on a topological space X, denoted Sh(X).
• The category of OX -modules on a ringed space (X,OX), denoted M(X).
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Definition 1.36. Let A be an Abelian category.
• A complex A• in A is a sequence
· · · −→ Ai−1 di−1
−−−→ Aidi−→ Ai+1 −→ · · · ,
where the Ai are objects in A and the di are morphisms in A with di+1 di = 0 for all i.
• The i-th cohomology of A• is hi(A•) = ker di/ im di−1.
• A morphism of complexes f : A• → B• is a commutative diagram
· · · Ai−1 Ai Ai+1 · · ·
· · · Bi−1 Bi Bi+1 · · ·
fi−1 fi fi+1
• A short exact sequence of complexes
0→ A• → B• → C• → 0
is a sequence of complexes such that all induced sequences
0→ Ai → Bi → Ci → 0
are exact.
Theorem 1.37. Let 0→ A• → B• → C• → 0 be a short exact sequence of complexes. Then thereexists natural maps that give a long exact sequence of cohomology
· · · → hi(A•)→ hi(B•)→ hi(C•)→ hi+1(A•)→ · · · .
Definition 1.38. Let F : A→ B be a (covariant) functor between Abelian categories.
• F is additive if the maps : Hom(A,A′)F−→ Hom(FA,FA′) are homomorphisms of Abelian
groups.
• F is called left exact if it is additive and takes every short exact sequence
0→ A′ → A→ A′′ → 0
in A to an exact sequence0→ FA′ → FA→ FA′′
in B.
Definition 1.39.
• An object I in an Abelian category A is called injective if for every commutative diagram
0 A A′
I
∃g
in A with an exact first row there exists a morphism g : A′ → I which fits into the diagram.
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• An injective resolution of an object A of an Abelian category A is an exact sequence
0→ A→ I0 → I1 → I2 → · · · ,
such that all Ij are injective in A.
• An Abelian category is said to have enough injectives is every object has an injective resolution.
Definition 1.40. Let F : A → B be a left exact functor between Abelian categories and assumethat A has enough injectives. For each i ∈ N, define the right derived functor RiF : A→ B of F inthe following way: For each object A of A pick an injective resolution
0→ A→ I0 → I1 → I2 → · · ·
and apply F to get a complex
D• : 0→ FI0 → FI1 → FI2 → · · · .
Then RF i(A) = hi(D•).
Theorem 1.41. With the notation of the above definition the following statements hold:
1. RiF (A) does not depend on the choice of injective resolution (up to natural isomorphism).
2. R0F ∼= F .
3. Each short exact sequence 0→ A′ → A→ A′′ → 0 in A induces (in a canonical way) a longexact sequence
· · · → RiF (A′)→ RiF (A)→ RiF (A′′)→ Ri+1F (A′)→ · · ·
in B.
4. Each commutative diagram
0 A′ A A′′ 0
0 B′ B B′′ 0
with exact rows induces a commutative diagram
· · · RiF (A′) RiF (A) RiF (A′′) Ri+1F (A′) · · ·
· · · RiF (B′) RiF (B) RiF (B′′) Ri+1F (B′) · · ·
with exact rows.
Definition 1.42. With the notation above: An object J ∈ A is called F -acyclic if RiF (J) = 0 forall i ≥ 1.
Theorem 1.43. Again using the above notation, if
0→ A→ J0 → J1 → J2 → · · ·
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is an exact sequence such that all J i are F -acyclic (this is called an acyclic resolution of A) and
E• : 0→ FJ0 → FJ1 → FJ2 → · · ·
is the corresponding complex in B, then RF i(A) = hi(E•) for all i ≥ 0.
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2. Schemes
In this chapter we will begin the study of schemes – the main objects of algebraic geometry – andthe morphisms between them. The first section will introduce affine schemes, which are the buildingblocks of general schemes which we will introduce next. As a special case we will discuss projectivespace. Then we will take a look at some special classes of morphisms between schemes. Already inthe first section we will introduce the sheaf of modules on an affine scheme associated to a module.In section 5 we will study sheaves of modules more closely. We will finish this chapter by giving shortintroductions to the notions of divisors and differentials.
2.1. Affine Schemes
Definition 2.1. Let A be a ring. Then SpecA is the set of prime ideals of A. If a is a ideal of A wedefine V (a) = p ∈ SpecA : a ⊆ p.
Lemma 2.2. Let A be a ring and let a, b, ai be ideals of A.
1. V (ab) = V (a ∩ b) = V (a) ∪ V (b)
2. V (∑i ai) =
⋂i V (ai)
3. V (a) ⊆ V (b)⇔√a ⊇√b
Proof.
1. For a prime ideal p:
p ∈ V (ab)⇔ ab ⊆ p⇔ a ⊆ p or b ⊆ p⇔ p ∈ V (a) ∪ V (b).
A similar argument works for V (a ∩ b).
2. Again for p prime:
p ∈ VÄ∑
aiä⇔∑
ai ⊆ p⇔ ∀i : ai ⊆ p⇔ ∀i : p ∈ V (ai)⇔ p ∈⋂V (ai).
3. Follows immediately from √a =
⋂p∈V (a)
p.
Definition 2.3. Taking the sets of the form V (a) for some ideal a ⊆ A to be the closed sets definesa topology in SpecA, called the Zariski topology . The sets D(f) = SpecA \ V ((f)), f ∈ A are calledprincipal open sets.
Remark 2.4. The principal open sets form a basis for the Zariski topology: Let U = SpecA \ V (a) beopen. If a = (bi : i ∈ I), then V (a) =
⋂i∈I V ((bi)) and hence U =
⋃D(bi).
Lemma* 2.5. Let A be a ring. Then all principal open sets are quasi-compact. In particular SpecAis quasi-compact.
13
Proof. Let D(b) be covered by open sets Ui. Since the principal open sets form a basis of the topologywe may without loss of generality assume that all Ui are principal open sets contained in D(b); sayUi = D(di).
From D(b) =⋃D(di), we obtain V ((b)) =
⋂V ((di)) = V (
∑(di)), so
√(b) =
√∑(di). Hence
there exists l ≥ 1 such that bl ∈∑
(di). In particular, there exist finitely many di, say d1, . . . , dn,and bi ∈ A with bl =
∑ni=1 bidi. Therefore V (b) ⊇ V (
∑nj=1(dj)) ⊇ V (
∑(di)) = V (b). Hence we have
equality and D(b) =⋃ni=1D(di).
Definition 2.6. Let A be a ring, X = Spec(A) and M an A-module. We will define the sheaf
associated to M , denoted M , of A-modules on X: For an open subset U ⊆ X, define M(U) to be theset of functions
s : U →∐p∈U
Mp
such that for each p ∈ U , s(p) ∈Mp and such that s is locally a fraction mb with m ∈M , b ∈ A, i.e.
for each p ∈ U there exist a neighborhood V ⊆ U of p and elements m ∈M and b ∈ A such that forall q ∈ V , b /∈ q and s(q) = m
b in Aq.
Definition 2.7. Let A be a ring, X = SpecA. We make X into a ringed space by taking thestructure sheaf to be OX = A.
Remark 2.8. Since this definition is so fundamental, we will write it out: For an open U ⊆ X, defineOX(U) to be the set of functions
s : U →∐p∈U
Ap
such that for each p ∈ U , s(p) ∈ Ap and such that s is locally a quotient of elements in A, i.e. for eachp ∈ U there exist a neighborhood V ⊆ U of p and elements a, b ∈ A such that for all q ∈ V , b /∈ qand s(q) = a
b in Aq. Then OX with the obvious restriction maps (which are ring homomorphisms) isa sheaf on X making (X,OX) into a ringed space.
Theorem 2.9. Let A be a ring, X = SpecA and M and A-module. Then:
1. M is an OX-module.
2. For each p ∈ X,ÄMäp
= Mp.
3. For each b ∈ A, M(D(b)) = Mb.
4. M(X) = M .
Proof.
1. As Mp is an Ap-module this is immediately obvious.
2. We define a homomorphism of Ap-modules f :ÄMäp→ Mp by (U, s) 7→ s(p) ∈ Mp. By
construction of M , this is well-defined.
First, we will show that f is injective: Let f((U, s)) = 0 and let s be locally given by mb , m ∈M ,
b ∈ A. In particular this means that mb = 0 in Mp. So there exists c ∈ A \ p with cm = 0. Thus
mb = 0 at every point of D(b) ∩D(c) 3 p. Therefore the restriction of s to U ∩D(b) ∩D(c) 6= ∅
is 0 and so (U, s) = 0.
Now we will show that f is surjective: Pick any element mb ∈ Mp. Let W = D(b) 3 p and
take s ∈ M(W ) to be the “constant function” s : W →∐
q∈W Mq : q 7→ mb ∈ Mq. Then
f((W, s)) = mb ∈Mp.
14
3. We define a function g : Mb → M(D(b)) by sending mbn to the “constant function” defined by
mbn (as above).
The function g is well-defined: If mbn = m′
bn′in Mb, then bl(mbn
′ −m′bn) = 0 in M for some l
and hence mbn = m′
bn′∈Mq for every q in D(b).
First we show that g is injective. Assume that g( abn ) = 0 in M(D(b)). This means thatabn = 0 ∈ Mq for each q ∈ D(b). Hence there are elements h(q) ∈ A \ q such that h(q)m = 0.
Let a be annihilator of m. Then h(q) ∈ a, but h(q) /∈ q. So a * q for all q ∈ D(b). ThereforeV (a) ∩ D(b) = ∅, i.e. V (a) ⊆ V (b). By lemma 2.2, b ∈
√a, so bl ∈ a for some l ≥ 1. Thus
blm = 0 and mbn = 0 in Mb.
Proving the surjectivity of g is harder. Let s ∈ M(D(b)). By definition of M there exists anopen cover Ui of D(b) such that s|Ui is given by mi
ei. Since the principal open sets are a base
of the topology we may assume that Ui = D(di) for some di ∈ A. Principal open subsets arequasi-compact (Lemma 2.5), so we may assume that the cover is finite.
For the fractions to be well-defined we must have D(di) ⊆ D(ei), i.e.√
(di) ⊆√
(ei). So there
exist ni ≥ 1 and e′i ∈ A with dnii = e′iei. This implies miei
=e′imidnii
. The upshot of this is that
by replacing di by dnii (since D(di) = D(dnii )) and mi by e′imi we may assume that D(b) iscovered by open subsets Ui = D(di) such that s|Ui is given by mi
di.
We will change the elements di such that the compatibility conditions become simpler. Noticethat D(di) ∩D(dj) = D(didj). So by construction, mi
di|D(didj) =
mjdj|D(didj), i.e. mi
di=
mjdj
in
Mdidj . So there exists nij ≥ 1 with (didj)nij (midj −mjdi) = 0 in M . Since there are only
finitely many possibilities to pick the pair (i, j), we can take n′ to be larger than all of the nij .
Rewriting the equality condition, we have dn′+1j (dn
′
i mi)− dn′+1i (dn
′
j mj) = 0. By replacing di by
dn′+1i and mi by dn
′
i mi we can assume that dimj − djmi = 0 while still having s representedas mi
dion D(di).
From D(b) =⋃D(di) we obtain
√(b) =
√∑(di) and hence bl =
∑ni=1 bidi for some l ∈ N and
bi ∈ A. Set m =∑ni=1 bimi. Then
djm =∑
bidjmi =∑
bidimj =Ä∑
bidiämj = blmj .
Therefore mbl
=mjdj
on D(dj). Hence g(mbl
) = s everywhere. So g is surjective.
4. Obtained from the previous point with b = 1.
Corollary 2.10. Let A be a ring, X = Spec(A). Then:
1. For each p ∈ X, OX,p = Ap.
2. For each b ∈ A, OX(D(b)) = Ab.
3. OX(X) = A.
Proof. This is just the special case M = A of theorem 2.9.
In particular the corollary asserts that (X,OX) is a locally ringed space. Actually it is a veryimportant one, giving rise to the following definition:
Definition 2.11. An affine scheme is a locally ringed space that is isomorphic to (SpecA, A) forsome ring A.
We usually write SpecA both for the topological space and the affine scheme.
15
Theorem 2.12. Let α : A→ B be a homomorphism of rings. Write X = SpecA and Y = SpecB.Then α induces (naturally) a morphism (Y,OY )→ (X,OX) of locally ringed spaces. Every morphism(of locally ringed spaces) (Y,OY )→ (X,OX) comes from a homomorphism A→ B.
Proof. Given α : A → B, we define f : SpecB → SpecA by f(p) = α−1(p). Since f−1(V (a)) =V (α(a)B), f is continuous. We need to define f# : OX → f∗OY . Let f(p) = q. Then we have a localhomomorphism αp : Aq → Bp. By definition, each element s ∈ OY (U) is a functions U →
∐q∈U Aq.
Define f#U (s) = t ∈ OY (f−1(U)) as t : f−1U →
∐p∈f−1U Bp with t(p) = αp(s(f(p))). If s is locally
given by a quotient ab , then t is locally given by the quotient α(a)
α(b) . So t is indeed an element of f∗OY .
The induced maps on the stalks (OX)q → (f∗OY )q → (OY )p are just the local homomorphisms αq,so (f, f#) is indeed a morphism of locally ringed spaces.
Conversely let (f, f#) be a morphism (Y,OY ) → (X,OX). By 2.10.3 we have a homomorphism
of rings α = f#X : A → B. We will show that α induces the given morphism. For any p ∈ Y , put
q = f(p). Then we have a commutative diagram
A = OX(X) OY (Y ) = B
Aq = (OX)q (OY )p = Bp
β
α
γ
f#p
From this we obtain:
q = β−1(qq) = β−1((f#p )−1(pp)) = α−1(γ−1(pp)) = α−1(p).
Hence f = α−1 : SpecB → SpecA. Also f#p is given by αp. Now, for any open set U ⊆ X with
q ∈ U we have a commutative diagram
OX(U) OY (f−1(U))
Aq = (OX)q (OY )p = Bp
f#U
f#p = αp
A section s ∈ OX(U) is determined by all values s(q), q ∈ U . But f#p (s(q)) = αp(s(q)). So f#
U (s) isdetermined by the values αp(s(q)), hence f# coincides with the map defined in the first step.
2.2. Schemes
Definition 2.13. A scheme is a locally ringed space (X,OX) such that every point x ∈ X has anopen neighborhood U such that (U,OX |U ) is an affine scheme. An (iso)morphism of schemes is an(iso)morphism of locally ringed spaces.
Examples 2.14. Let A be a ring and X = SpecA.
• If A is a field, then X has only one point.
• If A is a DVR, then X has only two points: 0 and mA (the maximal ideal of A). The latter is aclosed point (i.e. mA is a closed subset of X), while 0 is not a closed point. Actually it isdense, i.e. 0 = X (0 is a generic point).
Definition 2.15. Let A be a ring. The affine scheme Spec(A[t1, . . . , tn]) is called affine n-space overA and is denoted AnA.
16
Remark 2.16. If k is an algebraically closed field, then Ank is essentially the classical n-dimensional see also [EH00,section II.1.1]affine space. Notice that the new affine space Ank has more points than the classical one. E.g., the
points of A1k are the prime ideals (t− a), a ∈ k plus the zero ideal.
Example 2.17. The space X = SpecZ consists of the prime ideals (p) with p ∈ Z a prime numbertogether with 0. We have (OX)0 = Z0 = Q and (OX)(p) = Z(p) and the residue field k ((p)) = Fp.Example 2.18. For any ring A and ideal a C A[t1, . . . , tn], we have a scheme Spec(A[t1, . . . , tn]/a).In particular take A = k to be an algebraically closed field. Classically, two ideals a, b define thesame subset of affine space if and only if
√a =√b. However, as schemes, Spec(A[t1, . . . , tn]/a) is not
necessarily isomorphic to Spec(A[t1, . . . , tn]/b).For instance, if a = (t) C k[t], b = (tm) C k[t], then
√a =√b, but the two spectra differ. In an see also [EH00,
section II.3.1]informal sense Spec(k[t]/b) corresponds to the point (t) ∈ A1k with multiplicity m.
A1k
(t) 0
Example 2.19. Take the ideal a = (t21 + t22 + 1) of R[t1, t2]. Classically, the affine set V (a) is see also [EH00,section II.2]empty. However the scheme X = Spec(R[t1, t2]/a) is not empty. Also we have a homomorphism
R[t1, t2]/a→ C[t1, t2]/a which induces a morphism of schemes Spec(C[t1, t2]/a)→ Spec(R[t1, t2]/a).
Definition 2.20. Let (X,OX) be a scheme.
• X is reduced , if OX(U) is reduced (i.e. has no nilpotent elements) for all open U ⊆ X.
• X is irreducible, if X is irreducible as a topological space (i.e. if X = X1 ∪X2 for closed Xi,then X1 = X or X2 = X or equivalently if no two open subsets of X are disjoint).
• X is integral , if OX(U) is an integral domain for all U ⊆ X.
Example 2.21. Let X = Spec(A). Then:
• X is reduced if and only if A is reduced (i.e. Nil(A) = 0).
• X is irreducible if and only if Nil(A) is a prime ideal. [X is irreducible ⇔ if X = V (a)∪V (b) =V (ab), then X = V (a) or X = V (b) ⇔ if ab ⊆
√o, then a ⊆
√0 or b ⊆
√0 ⇔
√0 = Nil(A) is
prime.]
• X is reduced and irreducible, if and only if Nil(A) = 0 is a prime ideal, which is the case if anonly if A is an integral domain.
Lemma 2.22. Let X be a scheme and s ∈ OX . Then the set
Ws = x ∈ X : s /∈ mx ⊆ (OX)x
is an open subset of X.
Proof. By covering X with open affine sets, we can assume that X is affine. Let X = SpecA andpick x = p ∈ X. As mx is the localization of p, we have s ∈ mx ⊆ (OX)x = Ap if and only if s ∈ p(where we write s both for the section and its image in the stalk). This in turn holds if and only ifp ∈ V ((s)). Therefore X \Ws is closed, so Ws is open.
Theorem 2.23. A scheme is integral if and only if it is reduced and irreducible.
Proof. Let (X,OX) be the scheme.First assume that X is integral. So by definition OX(U) is an integral domain for each open
set U . As integral domains are reduced rings, X must be reduced. We will now show that Xis also irreducible. Suppose that this was not true, i.e. that there exist proper closed subsetsX1, X2 of X with X = X1 ∪X2. Then (X \X1) ∩ (X \X2) = ∅, where X \Xi are nonempty open
17
sets. Let Ui ⊆ X \ Xi be open affine sets. Since U1 ∩ U2 = ∅, the sheaf condition implies thatOX(U1 ∪ U2) ∼= OX(U1)×OX(U2) which is not an integral domain. This is a contradiction to theassumption that X is integral, so X must be irreducible.
Now assume that X is reduced and irreducible. We have to show that X is integral. Let U be anopen subset of X and suppose we have elements a, b ∈ OX(U) with ab = 0 in OX(U). By the lemma,Wa,Wb ⊆ U are open sets.
Let V ⊆ U be an open affine set. Since both reduced and irreducible are properties that areinherited by V , example 2.21 implies that OX(V ) is an integral domain. Since 0 = (ab)|V = a|V b|V ,either a|V = 0 or b|V = 0.
Hence for every x ∈ U there exists an open (affine) neighborhood of x where a = 0 or an openneighborhood, where b = 0. So either x /∈ Wa or x /∈ Wb. In other words Wa ∩Wb = ∅. So byirreducibility, one of the sets is empty; say Wa. But then a would lie in the intersection of all primeideals, i.e. in the nilradical which is assumed to be just 0. Thus a = 0 and OX(U) is integral asrequired.
Definition 2.24. Let X be an integral scheme. By exercise A.20, the local ring OX,η at the genericpoint η of X (i.e. η is dense in X) is a field. It is called the function field of X and denoted K(X).
Definition 2.25. A scheme X is called Noetherian if it can be covered by finitely many open affinesubsets Ui = SpecAi such that all Ai are Noetherian rings.
Definition 2.26. Let (X,OX) be a scheme. An open subscheme of X is an open subset U of X withstructure sheaf OX |U . A morphism f : Y → X is called an open immersion if f is an isomorphism ofY with an open subscheme U of X.
Definition 2.27. Let (X,OX) be a scheme. A closed immersion is a morphism of schemes f : Y → X
Y Y ′
X
g
f f ′
such that f is a homeomorphism of Y with a closed subset of X and OX → f∗OY is surjective. Aclosed subscheme of X is an equivalence class of closed immersions, where we say that two closedimmersions f : Y → X and f ′ : Y ′ → X are equivalent if there exist an isomorphism g : Y → Y ′ withf = f ′ g.
Example 2.28. Let X = SpecA and take an ideal aCA. Consider the scheme Y = Spec(A/a). Thecanonical homomorphism A→ A/a corresponds to a morphism f : Y → X. Obviously f(Y ) = V (a)and one can check that f is a homeomorphism onto the closed set V (a).
For any point x ∈ f(Y ) (say x = f(y)) we have a homomorphism of local rings Ax = (OX)x →(f∗OY )x = (OY )y = (A/a)x/a which is surjective. For x /∈ f(Y ), the map (OX)x → (f∗OY )x = 0 istrivially surjective. Hence the morphism OX → f∗OY is surjective, so f is a closed immersion
Also note that for each n ∈ N, an defines a closed immersion fn : Yn = Spec(A/an) → X. Butfn(Yn) = V (an) = V (a). Hence V (a) ⊆ X can carry infinitely many different subscheme structures.
Remark* 2.29. If Y is any closed subset of X, then, as we just saw, in general there exist manydifferent closed subscheme structures on Y . However there exits a unique reduced closed subschemestructure on Y . This structure is called the induced reduced structure on Y . See [Liu02, proposition2.4.2c] or [Har77, example II.3.2.6].
Definition 2.30. Let S be a scheme. A scheme over S or an S-scheme is a scheme X together witha morphism X → S. A morphism X → Y is a morphism of schemes over S if the diagram
X Y
S
commutes.
18
Remark* 2.31. If S = SpecA for some ring A, we say that a scheme over SpecA is a scheme over Aor an A-scheme. By exercise A.8, SpecZ is the terminal object in the category of schemes, so everyscheme can be seen as a scheme over Z.
X ×S Y Y
X S
q
p
Definition 2.32. Let S be a scheme and X,Y two S-schemes. The fibred product of X and Yover S is an S-scheme X ×S Y together with two morphism of S-schemes p : X ×S Y → X andq : X ×S Y → Y (called projection maps), such that the following universal property holds:
Whenever Z is an S-scheme and f : Z → X, g : Z → Y are two morphisms of S-schemes, thereexists a unique morphism φ : Z → X ×S Y of S-schemes such that the following diagram commutes:
Y
Z X ×S Y S
X
g
φ
f
q
p
Theorem 2.33. The fibred product exists and is unique up to unique isomorphism. [Har77, theoremII.3.3], [Liu02,proposition 3.1.2]Proof. Uniqueness is clear form the universal property.
Suppose S = SpecA, X = SpecB, Y = SpecC. We will show that X ×S Y = Spec(B ⊗A C). LetZ be another scheme such that we have a commutative diagram (using theorem 2.12)
Y
Z X ×S Y S
X
g
f
q
p
We need to show that there exists a unique morphism Z → X×SY fitting into the diagram. By exerciseA.8, this is the same as showing that there is a unique homomorphism of rings B ⊗A C → OZ(Z)fitting into the diagram
B
OZ(Z) B ⊗A C A
C
g
f
q
p
But this is just the universal property of the tensor product.For the general case, we need to cover X, Y and S by affine schemes and construct X ×S Y locally
in the affine open sets and finally glue all of them together. See the references.
Lemma* 2.34. Let S be a scheme and let X and Y be S-schemes. Then [Liu02, proposition3.1.4]
1. For any S-scheme Z, there are canonical isomorphisms
X ×S S ∼= X, X ×S Y ∼= Y ×S X, (X ×S Y )×S Z ∼= X ×S (Y ×S Z).
2. Let Z be a Y -scheme and consider it as an S-scheme via Z → Y → S, then there exists acanonical isomorphism
(X ×S Y )×Y Z ∼= X ×S Z,
19
where X ×S Y is a Y -scheme via the second projection map.
3. For morphisms f : X → X ′ and g : Y → Y ′ of S-schemes, there exists a unique morphism ofS-schemes f × g : X ×S Y → X ′ ×S Y ′ such that the diagram
X X ′
X ×S Y X ′ ×S Y ′
Y Y ′
f
p p′
q q′
f × g
g
commutes.
4. Let i : U → X and j : V → Y be open subschems. Then the momrphism i × j gives anisomorphism
U ×S V ∼= p−1(U) ∩ q−1(V ) ⊆ X ×S Y.
If y is any point of a scheme Y , then the composition OY (Y )→ (OY )y → k(y) induces a canonicalmorphism Spec k(y)→ Y (see exercise A.9).
Definition 2.35. Let f : X → Y be a morphism of schemes. The fibre of f over y ∈ Y is
Xy = X ×Y Spec k(y).
Example 2.36. Let X be a scheme. There is a unique morphism f : X → SpecZ which correspondsto Z→ OX(X). Set Y = SpecZ. For the point y = (0) ∈ Y , the residue field is k((0)) = Z/m0 = Q.Then Xy = X ×SpecZ SpecQ (if Y = SpecA, then Xy = Spec(A⊗Z Q)).
If y = (p) for a prime number p, then k(y) = Zp/m = Fp (the finite field of p elements). ThusXy = X ×SpecZ SpecFp (if X = SpecA, then Xy = Spec(A⊗Z Fp)). This is called reduction mod p.
Example 2.37. Let be X a scheme over a field K (i.e. over SpecK) and let K ⊂ L be a field extension.The fibred product X ×SpecK SpecL is a scheme over L. In particular, AnK ×SpecK SpecL ∼= AnL, asK[t1, . . . , tn]⊗K L ∼= L[t1, . . . , tn].
Example 2.38. Let f : X → Y be a morphism of schemes with Y integral. Then Y has a uniquegeneric point η. The generic fibre of f is defined to be Xη. By definition Xη is a scheme overk(η) = K(Y ), the function field of Y .
2.3. Proj and Projective Space
Definition 2.39. Let S =⊕
d≥0 Sd be a graded ring. We set
ProjS = p ∈ SpecS : p is homogeneous and⊕d≥1
Sd * p.
For any homogeneous ideal a of S we define V+(a) = p ∈ ProjS : a ⊇ p and for any homogeneousb ∈ S, we set D+(b) = ProjS \ V+((b)).
Lemma 2.40. Let S be a graded ring. Then:
1. For any homogeneous ideals a, bC S: V+(a) ∪ V+(b) = V+(a ∩ b) = V+(ab).
2. For any family ai of homogeneous ideals of S:⋂V+(ai) = V+(
∑ai).
20
Proof. As in the affine case.
By taking the V+(a) to be the closed sets, ProjS becomes a topological space.
Definition 2.41. Let S =⊕Sd be a graded ring, set X = ProjS and let M =
⊕Md be a graded
S-module. For any open U ⊆ X, define M(U) to be the set of functions
s : U →∐p∈U
M(p)
such that for each p ∈ U , s(p) ∈ M(p) and such that s is locally a quotient mb with m ∈ M , b ∈ S
homogeneous and of the same degree, i.e. for each p ∈ U there exist a neighborhood V ⊆ U of p andhomogeneous elements b ∈ S, m ∈M of the same degree such that for all q ∈ V , b /∈ q and s(q) = m
bin M(q).
Remark 2.42. We use the same notation M both in the affine case and in the case of a graded moduleover ProjS. It should usually be clear from the context which construction is meant.
Definition 2.43. Let S be a ring, X = ProjS. We set OX = S. So, for an open U ⊆ X, defineO(U) to be the set of functions
s : U →∐p∈U
S(p)
such that for each p ∈ U , s(p) ∈ S(p) and such that s is locally a quotient of homogeneous elementsin S of the same degree, i.e. for each p ∈ U there exist a neighborhood V ⊆ U of p and homogeneouselements a, b ∈ S of the same degree such that for all q ∈ V , b /∈ q and s(q) = a
b in S(q). Then OXwith the obvious restriction maps (which are ring homomorphisms) is a sheaf on X making (X,OX)into a ringed space.
Theorem 2.44. Let S be a graded ring, X = ProjS and M a graded S-module. Then:
1. For any p ∈ X, Mp = M(p). In particular, (OX)p = S(p).
2. For any homogeneous b ∈⊕
d≥1 Sd there exists an isomorphism of locally ringed spaces
(D+(b), OX |D+(b)) ∼= SpecS(b)
which takes M |D+(b) to fiM(b).
3. In particular, (X,OX) is a scheme.
Proof. not given
Definition 2.45. Let A be any ring. The scheme Proj(A[t1, . . . , tn]) is denoted PnA and called theprojective n-space over A.
Example 2.46. If k is an algebraically closed field, then Pnk is essentially the same as the classicaln-dimensional projective space over k.
Definition 2.47. Let Y be a scheme. The projective n-space over Y is PnY = Y ×SpecZ PnZ. Amorphism f : X → Y is called projective, if there exists n such that f factors into a closed immersionX → PnY followed by the projection PnY → Y . A quasi-projective morphism is a morphism that factorsinto an open immersion followed by a projective morphism.
Remark* 2.48. If Y = SpecA, then PnY = PnA (see exercise A.26).
21
2.4. Morphisms
Remark 2.49. The following properties are all defined for morphisms of schemes. However, if X is ascheme over S, X is said to have one of these properties if the morphisms X → S has it. If no basescheme S is given, then X is considered to be a scheme over Z (see remark 2.31).
Definition 2.50. A morphism of schemes f : X → Y is of finite type if there exists a covering ofY be open affine subschemes Vi = SpecBi, such that for each i, f−1(Vi) can be covered by finitelymany open affine subsschems Uij = SpecAij , where each Aij is a finitely generated Bi-algebra.
Definition 2.51. Let f : X → Y be a morphism of schemes. The diagonal morphism is the uniquemap ∆: X → X ×Y X whose composition with both projection maps is the identity.
X
X X ×Y X Y
X
id
∆
id
f
f
The morphism f is called separated if ∆ is a closed immersion.
Remark* 2.52. Separatedness is somewhat related to the Hausdorff property of topological spaces[Liu02, proposition
3.3.1] (note that schemes are in general not Hausdorff): A topological space X is Hausdorff if and only ifthe diagonal ∆ = (x, x) : x ∈ X is closed in the product space X ×X. See also [EH00, exerciseIII-3].
Example 2.53. Let K be a field. Then A1K → SpecK is a separated morphism.
Theorem 2.54. Every morphism of of affine schemes is separated.
Proof. Let f : X = SpecA→ SpecB = Y be a morphism of affine schemes, given by a homomorphismB → A. Then X×Y X = Spec(A⊗BA) and the diagonal morphism corresponds to the homomorphismA⊗B A→ A given by a⊗ a′ 7→ aa′. This is surjective, so by example 2.28 the diagonal morphism ∆is a closed immersion.
Theorem 2.55. An arbitrary morphism f : X → Y is separated if and only if the image of thediagonal morphism is a closed subset of X ×Y X.
Proof. The “only if”-direction is true by definition. So assume that ∆(X) is a closed subset of X. Wehave to show that ∆ is actually a closed immersion. Consider the first projection p : X ×Y X → X.Then p ∆ = idX ; hence ∆ is a homeomorphism onto its image, which is closed. It remains to beshown that OX×YX → ∆∗OX is surjective. This is a local question. For Q /∈ ∆(X), the induced mapon the stalk at Q is certainly surjective. So let P ∈ X and choose an open affine neighborhood U ofP in X such that f(U) is contained in an open affine subset of V of Y . Then U ×V U is an openaffine neighborhood of ∆(P ). By theorem 2.54 above, ∆: U → U ×V U is separated. Therefore themap of sheaves is surjective in a neighborhood of ∆(P ). Since P was arbitrary, OX×YX → ∆∗OX issurjective as required.
Remark 2.56. A local ring R inside a field L is a valuation ring if for any other local ring A with
Rα−→ A → L and α local, R = A.
Theorem 2.57 (Valuative Criterion for Separatedness). Let X be a Noetherian scheme. A morphismf : Y → X is separated if and only if the following holds:
22
For every valuation ring R with fraction field K and every commutative diagram
SpecK X
SpecR Y
ι fg
(2.1)
where ι is induced by the inclusion R → K, there exists at most one morphism g : SpecR → Xfitting into the diagram.
Proof. Not given. (See [Har77, theorem II.4.3].)
Example 2.58. Let k be a field and U = V = A1k. We glue U and V along the open subschemes
U \ (t), V \ (t) and call the result X.
x
x′
Let x, x′ be the two origins. Let K be the fraction field of Ox ∼= O′x. By exercise A.10, there aremorphisms g, g′ making the following diagram commutative:
SpecK X SpecK
Spec Ox Spec k Spec Ox′
g
∼=
g′
∼=
If we set R = Ox ∼= Ox′ , we have two different morphism from SpecR to X making (2.1) commute.Hence X is not separated over Spec k.
Lemma 2.59. Closed immersions of Noetherian schemes are separated. This is also truefor non-Noetherianschemes, see [Liu02,proposition 3.3.9]Proof. Let f : X → Y be a closed immersion. Suppose that we have a commutative diagram like
(2.1). We have to show that there is at most one g that fits into it.
Let y be the image of mR in Y and y′ the image of the generic point of SpecR. Since the diagramis commutative, y′ ∈ f(X). As f(X) is closed, it must also contain y. Take any open affine subset Uof Y with y ∈ U and set V = f−1U . This gives a commutative diagram
SpecK V X
SpecR U Y
f
h
f
Since h−1U is open, it contains the generic point and thus y′ ∈ U . Hence it is enough to look atthe left side of the diagram. Let U = SpecA. Since V → U is a closed immersion, V = SpecA/afor some ideal aCA (see exercise A.21 or corollary 2.87). By theorem 2.12 we get a commutative
23
diagram on the level of rings:
K A/a
R A
Since A → A/a is surjective, there is at most one homomorphism A/a → R that fits into thediagram.
Definition 2.60. A morphism of schemes f : X → Y is universally closed if for every morphismS → Y and every closed subset T of S ×Y X, the image of T under the projection map is closed in S.
T S ×Y X X
p(T ) S Y
p f
Remark* 2.61. By setting S = Y and using lemma 2.34 we see that if f is universally closed, then inparticular f is closed.
Definition 2.62. A morphism is called proper if it is separated, of finite type and universally closed.
Remark* 2.63. In general topology a continuous function f : X → Y is called proper if the preimageof every compact set is compact. If X is Hausdorff and Y locally compact, then f is proper if andonly if for any space Z, the map Z ×X → Z × Y given by (z, x) 7→ (z, f(x)) is closed.
Example 2.64. Let X = A1k for a field k and consider f : X → Spec k. We will show that f is
not universally closed and hence is not proper. Take S = A1k and T ⊆ A1
k ×Spec k A1k = A2
k to beV ((t1t2 − 1)). The image of T under the projection p is A1
k \ (t), which is not closed.
S = A1k
Tp
A2k
Theorem 2.65 (Valuative Criterion for Properness). Let X be Noetherian and f : X → Y a morphismof finite type. Then f is proper if and only if the following holds:
24
For every valuation ring R with fraction field K and every commutative diagram
SpecK X
SpecR Y
ι fg
(2.2)
where ι is induced by the inclusion R → K, there exists exactly one morphism g : SpecR→ X fittinginto the diagram.
Proof (Sketch). First suppose that f is proper and that we are given a diagram as in the statement. see also [Har77,theorem 4.7]Since f is separated, there exists at most one g fitting into it (theorem 2.57). So we only have to
show the existence of g. Set U = SpecK and Z = SpecR. Consider the fibred product
U
Z ×Y X X
Z Y
h
p f
where h exists by the universal property. Put a = h(U), T = a ⊆ Z ×Y X. The proper map f isuniversally closed, so p(T ) is closed in Z. Thus p(T ) contains the maximal ideal mR. Choose b ∈ Tsuch that p(b) = mR. Take the reduced induced subscheme structure on T . Obviously T is irreducibleand hence integral. There is a local homomorphism R = (OZ)mR → OT,b. Also, a is the generic pointof T and by construction we have a homomorphism k(a) → K. By integrality, OT,b → k(a). Thissituation induces a morphism Z → T and so a morphism g : Z → X which fits into the diagram(without proof).
Now suppose that for every diagram like (2.2) we have a unique morphism g. We have to provethat f is proper. By assumption, f is of finite type and by theorem 2.57 it is separated. So we onlyneed to verify that f is universally closed. Hence suppose we are given a morphism S → Y andsuppose that T ⊆ S ×Y X is closed. We put the reduced induced structure on T .
T S ×Y X X
S Y
p f
It turns out that in order to prove that p(T ) is closed, it is enough to show that a ⊆ p(T ) for alla ∈ p(T ). So pick any a ∈ p(T ) with p(t) = a for some t ∈ T . Set S′ = a and let b ∈ S′. We haveto show that b ∈ p(T ). Put the reduced induced structure on S′ (it is integral as above). By exerciseA.10, there exists a morphism Spec OS′,b → S′. Also, a is the generic point of S′, so that there areinclusions OS′,b → k(a) → k(t). By commutative algebra [?], there exists a valuation ring R for k(t)such that OS′,b → R is a local homomorphism. This gives a diagram
Spec k(t) T S ×Y X X
SpecR Spec OS′,b S′ S Y
p fg
h
25
By the hypothesis, there exists a unique morphism g : SpecR→ X fitting into the diagram. By theuniversal property of the fibred product S×YX, g determines a unique morphism h : SpecR→ S×YX.The morphism h has to send the generic point of SpecR into T . Since T is closed that impliesh(SpecR) ⊆ T . In particular, there exists t′ ∈ T such that h(mR) = t′. The commutativity of thediagram ensures that p(t′) = b, so that b ∈ p(T ) as required.
Lemma 2.66. For Noetherian schemes the following statements hold:This is also truefor non-Noetherianschemes, see [Liu02,proposition 3.3.16]
1. Closed immersions are proper.
2. Compositions of proper morphisms are proper.
Proof.
1. Proceed as in the proof of lemma 2.59. Note that there must always exist a homomorphismA/a→ R that fits into the last diagram. Hence we only have to show that V → U is of finitetype. But A/a is obviously a finitely generated A-algebra.
2. Let f : X → Y and g : Y → Z be proper morphisms. The composition g f is of finite type, sowe can use the valuative criterion. Suppose we have a commutative diagram
X
SpecK Y
SpecR Z
f
gf
gh
e
Since g is proper there exists a unique morphism h and since f is proper we get a uniquemorphism e. So g f is proper.
Theorem 2.67. Every projective morphism of Noetherian schemes is proper.[Har77, thm II.4.9]
Again, this is also truefor non-Noetherianschemes, see [Liu02,proposition 3.3.30] or[GD61a, thm. 5.5.3].
Proof. Let f : X → Y be a projective morphism of Noetherian schemes. Since f is the compositionof a closed immersion and the projection PnY → Y , it is of finite type. So we can apply the valuativecriterion for properness. Suppose we have a commutative diagram (R a valuation ring of K)
SpecK X PnY PnZ
SpecR Y SpecZ
g
fp
q
As g is a closed immersion, it is proper by lemma 2.66. Hence it is enough to prove that p is proper.By the universal property of fibred products, we see that there exists a unique map SpecR→ PnYfitting into the diagram if and only if there exists a unique fitting map SpecR→ PnZ. So it suffices toshow that q : PnZ → SpecZ is proper. We have reduced the situation to the following commutativediagram:
SpecK PnZ
SpecR SpecZ
e
q
26
The projective n-space PnZ = ProjZ[t0, . . . , tn] is covered by the open subsets Ui = D+(ti) fori = 0, . . . , n and by theorem 2.44 these are
Ui = D+(ti) = SpecZ[t0, . . . , tn](ti) = SpecZït0ti, . . . ,
tnti
ò.
Let a = e(SpecK). If a /∈ Ui for some i, then a ∈ V+(ti) = Pn−1Z , so that we can do induction. (The
base case n = 0 is trivial.)Therefore we can assume that a ∈
⋂i Ui. In that case each
tjti
is an element of Oa. Further, since titj
is also in Oa, these elements are all invertible in Oa. The morphism SpecK → PnZ gives an inclusion
k(a)→ K. Let sji be the image oftjti
under this inclusion (and the projection Oa → k(a)).By commutative algebra [?], there exists a valuation v : K → G (where G is a totally ordered
Abelian group) such that
v(µϑ) =v(µ) + v(ϑ) R =µ ∈ K : v(µ) ≥ 0v(µ+ ϑ) ≥minv(µ), v(ϑ) mR =µ ∈ K : v(µ) > 0
Note that v(sjl) = v(sji) + v(sil). Fix an index j such that v(sj0) is minimal among s10, . . . , sn0.So, for each i we have v(sij) = v(si0) − v(sj0) ≥ 0, i.e. sij ∈ R for all i. This gives a natural
homomorphism Zît0tj, . . . , . . . , tntj
ó→ R which in turn gives a morphism SpecR→ Uj ⊆ PnZ that fits
into the diagram.It remains to be shown that there is at most one such morphism. If there was another such
morphism, then the image had to lie in another Ui. But then both sji and sij would lie in R∗ whichis not possible.
Definition 2.68. A morphism f : X → Y is quasi-compact if there exists a cover of Y by openaffines Yi such that f−1(Yi) is quasi-compact for each i.
Remark* 2.69. f is quasi-compact if and only if for every cover of Y by open affines Yi the preimages [Har77, ExerciseII.3.2]f−1(Yi) are quasi-compact for all i.
2.5. Sheaves of Modules
Theorem 2.70. Let X = SpecA and Y = SpecB be affine schemes. Let M , Mi be A-modules, N a [Har77, Prop. II.5.2]
B-module and f : X → Y a morphism of schemes. Then:
1. ‰M1 ⊗AM2 = ›M1 ⊗OX›M2
2. ‡⊕Mi =⊕›Mi
3. f∗M = fiBM , where BM is M considered as a B-module.
4. f∗‹N = ‚N ⊗B AProof.
1. Let P be the presheaf given by P (U) = M(U) ⊗OX(U)‹N(U). We will define a morphism
φ : P → ‚M ⊗A N . Let U be an open subset of X, s ∈ M(U), t ∈ ‹N(U). Define an element
r ∈ ‚M ⊗A N(U) as
r : U →∐p∈U
(M ⊗A N)p ∼=∐p∈U
Mp ⊗ApNp, r(p) = s(p)⊗ t(p).
27
If s is locally given by ma and t by n
b , then r is locally given by m⊗nab . The association (s, t) 7→ r
is bilinear and hence descends to a homomorphism φU : P (U)→ ‚M ⊗A N(U). This gives the
morphism (of presheaves) φ : P → ‚M ⊗A N , which in turn gives a morphism (of sheaves)
φ+ : P + = M ⊗OX‹N → ‚M ⊗A N.
For any p ∈ X the homomorphism
φ+p = φp : P +
p = Pp = Mp ⊗ApNp −→ ‚M ⊗A Np
∼= (M ⊗A N)p
is an isomorphism (see also exercise A.28). Hence φ+ is an isomorphism.
2. The proof is like the one of the first statement, only simpler.
3. We will define a morphism ψ : fiBM → f∗M . Let U ⊆ Y be open. Pick an s ∈fiBM(U). Thens is by definition a function U →
∐q∈U (BM)q. If f(p) = q, then we have a homomorphism
βp : (BM)q →Mp (as we have a ring homomorphism B → A). Define ψU (s) to be the function
f−1U →∐
p∈f−1U
Mp, p 7→ βp (s (f (p))) ∈Mp.
Let α : B → A be the ring homomorphism corresponding to f : X → Y . If s is locally givenby m
b (m ∈ BM , b ∈ B), then ψU (s) is locally given by mα(b) . Observe that if b ∈ B, then
f−1D(b) = D(α(b)). Now
BMb∼= fiBM(D(b))
ψD(b)−−−→ (f∗M)(D(b)) = M(f−1D(b)) = M(D(α(b))) ∼= Mα(b)
is the natural B-module isomorphism between the two outer modules. Thus ψ is an isomorphism.
4. Proof ommited.
Remark 2.71. For A-modules M and N and X = SpecA, there exists a natural isomorphismHomA(M,N) ∼= HomOX (M, ‹N) (see exercise A.29).
Remark 2.72. Let X be SpecA. Any sequence of A-modules
· · · →M−1 →M0 →M1 → . . . (2.3)
induces a sequence of OX -modules
· · · →fiM−1 → ›M0 → ›M1 → . . . . (2.4)
The sequence (2.4) is exact if and only if (2.3) is exact. Indeed, the latter is exact if and only if alllocalized sequences
· · · → (M−1)p → (M0)p → (M1)p → . . .
are exact. These are just the sequences on the stalks induced by 2.4.
Definition 2.73. Let X be a scheme. An OX -module F is quasi-coherent if X can be covered by
open affine schemes Ui = SpecAi such that F |Ui = ›Mi for some Ai-modules Mi. It is coherent ifadditionally all Mi are finitely generated Ai-modules.
Example 2.74. For any scheme X, the structure sheaf OX is coherent. Indeed for any open affinesubscheme U = SpecA we have OX |U = A.
28
Example 2.75. Let A be a DVR with maximal ideal m and set X = SpecA. This scheme has onlythree open sets: ∅, X and U = X \ m = η, where η is the generic point of X (corresponding tothe zero ideal). Let K = k(η). Let F be the sheaf defined by F (X) = 0, F (U) = K. For any openaffine covering X =
⋃Ui, there must be one Ui with Ui = X, otherwise the maximal ideal would not
be contained in any of the open sets. Since M(X) = M there is no A-module M such that M = F .So F is not quasi-coherent.
Lemma 2.76. If X is an affine scheme and F an OX-module, then there exists a natural morphism‡F (X)→ F .
Proof. To ease notation set M = F (X). To define a morphism of sheaves M → F , it suffices to
define compatible maps M(D(b))→ F (D(b)) for each b ∈ A. By theorem 2.9, M(D(b)) ∼= Mb. Thus
we can define the map M(D(b))→ F (D(b)) by sending mbr to 1
br ·m|D(b) (this is defined since 1br is
an element of OX(D(b))). It is easy to check that these maps are well-defined and compatible.
Corollary 2.77. Let X be SpecA and let M be an A-module. Then M |D(b)∼= ›Mb.
Proof. By theorem 2.9 and the preceding lemma, there is a morphism φ : ›Mb → M |D(b). For any
point p ∈ D(b), φp : (›Mb)p ∼= Mp →Mp is an isomorphism. Hence φ is an isomorphism.
Theorem 2.78. Let X be a scheme. An OX-module F is quasi-coherent if and only if for every
open cover of X by open affine schemes Ui = SpecAi, there are Ai-modules Mi such that F |Ui ∼=›Mi.
Remark 2.79. If X is Noetherian, a similar statement holds for coherent sheaves with Mi finitelygenerated.
Proof. One direction is obvious. So suppose F is quasi-coherent and choose any open affine subsetU = SpecA of X. We have to find an A-module M such that F |U ∼= M . By definition, for each
point p ∈ U there exists an open affine neighborhood W = SpecB of p such that F |W ∼= ‹N for someB-module N . By the corollary we can replace W by any principal open subset of itself. In particular,we can assume that W ⊆ U and W = D(b) for some b ∈ A. Hence we can reduce to the situation
X = U is affine, where we have to show that F ∼= ‡F (X). Using the corollary again, we see that
in this case X is covered by finitely many D(bi) such that for all i we have F |D(bi)∼= ›Mi for an
Abi -module Mi.
Let fi : D(bi)→ X and fij : D(bibj)→ X be the inclusion maps. Note that D(bi)∩D(bj) = D(bibj).Set
G =⊕
fi∗(F |D(bi)
)and L =
⊕fij∗
(F |D(bibj)
).
Define a map φ : F → G by sending s ∈ F (V ) to (s1, . . . , sn) with si = s|D(bi)∩V . Further defineψ : G → L by sending (si)i ∈ G(V ) to (si|D(bibj)∩V − sj |D(bibj)∩V )i,j . By lemma 1.4 (i.e. the sheafcondition) the sequence
0→ F φ−→ G ψ−→ L (2.5)
is exact. By theorem 2.70.3 we have fi∗(F |D(bi)
)= fi
AMi and fij∗(F |D(bibj)
)= flAMij for some
Abibj -modules Mij . Taking global sections in (2.5) and using the sheaf condition (lemma 1.4) weobtain the exact sequence
0→ F (X)→⊕
AMi →⊕
AMij .
Hence (by remark 2.72 and theorem 2.70.2) the sequence
0→‡F (X)→⊕fi
AMi →⊕fl
AMij (2.6)
29
is exact. From the sequences (2.5) and (2.6) we deduce that both F and ‡F (X) are isomorphic to
kerψ. Thus F ∼=‡F (X) as required.
Theorem 2.80. Let X be a scheme and φ : F → G a morphism of quasi-coherent sheaves on X.Then kerφ and imφ are also quasi-coherent. If X as Noetherian and F ,G are coherent, then kerφand imφ are coherent.
Proof. Suppose U = SpecA is an open affine subset of X. Then there exist A-modules M and Nsuch that F |U = M and G |U = ‹N . So φU gives an A-module homomorphism M → N . Let L be thekernel of this homomorphism. This gives an exact sequence
0→ L→ M → ‹N.Thus on U we have kerφ ∼= L. Letting U range over an open affine cover we see that kerφ isquasi-coherent. The argument for imφ is similar, but involves an additional step; see Exercise A.30.
For the case that X is Noetherian, suppose that M and N are finitely generated and A is Noetherian.Then L is a submodule of a Noetherian module and hence finitely generated. Hence kerφ is coherentand a similar argument shows that imφ is coherent.
Theorem 2.81. Let f : X → Y be a morphism of schemes.
1. If G is a quasi-coherent (resp. coherent) sheaf on Y , then f∗G is quasi-coherent (resp. coherent)on X.
2. If X is Noetherian (or if f is separated and quasi-compact) and F is a quasi-coherent sheaf onsee also [GD60, thm.
9.2.1] X, then f∗F is quasi-coherent.
Proof.
1. Let G be a quasi-coherent OY -module. Since the problem is local, we may assume that Y is
affine, say Y = SpecA. Then there exists an A-module M such that G = M . Let U = SpecB
be an open affine subset of X. By theorem 2.70, (f∗G)|U ∼= ‚M ⊗A B. So f∗G is quasi-coherent.
Now assume that G is coherent. Then M is finitely generated over A and therefore M ⊗A B isfinitely generated over B. So f∗G is indeed coherent.
2. The problem is again local (in Y ), so that we may assume that Y = SpecA is affine. Under[Har77, Prop. II.5.8c]
either hypothesis we can cover X with finitely many open affine subsets Ui. If f is separated,then Ui ∩Uj is affine. If X is Noetherian, Ui ∩Uj can be covered with finitely many open affinesubsets Uijk. (Include the separated case in this notation.) By the sheaf condition, we have anexact sequence
0→ f∗F →⊕i
f∗(F |Ui)→⊕i,j,k
f∗(F |Uijk).
The last two sheaves in this sequence are quasi-coherent by theorem 2.70. Thus f∗F isquasi-coherent by theorem 2.80.
Remark* 2.82. If F is coherent, then f∗F need not be coherent: Consider for example the inclusion
SpecQ → SpecZ and the sheaf ‹Q. However, if f is proper, then f∗F is coherent if F is coherent[GD61b, Theoreme 3.2.1].
Definition 2.83. Let X be a scheme. An ideal sheaf on X is an OX -module I ⊆ OX If f : Y → Xis a closed immersion, then the kernel of f# : OX → f∗OY is called the ideal sheaf of Y and denotedIY .
30
Remark 2.84. Recall that a closed subscheme is given by an equivalence class of closed immersions.The ideal sheaves of all the closed immersions in a class are the same. So really the ideal sheaf is anobject associated with a closed subscheme.
Theorem 2.85. Let X be a scheme. Then there exists a natural bijection
closed subschemes of X ←→ quasi-coherent ideal sheaves.
Remark 2.86. If X is Noetherian, all these quasi-coherent ideal sheaves are in fact coherent.
Proof. To each closed subscheme Y of X we associate the ideal sheaf IY : The closed immersionf : Y → X is obviously quasi-compact and it is separated by lemma 2.59. Thus f∗OY is quasi-coherent(theorem 2.81) as is the kernel of OX → f∗OY (theorem 2.80), i.e. IY is a quasi-coherent ideal sheafon Y .
Conversely, suppose that I is a quasi-coherent ideal sheaf on X. Consider the exact sequence
0→ I → OX → OX/I → 0
Let Γ be the set of points p of X with (OX/I )p = 0. We will show that Γ is an open subset of X.Let U = SpecA be an open affine subset of X. Then (by theorem 2.78) I |U = a for some ideal a of
A. Also, (OX/I )|U = fiA/a. A prime ideal p of A gives a point in U ∩ Γ if and only if its image inA/a is 0. Thus U ∩ Γ = U \ V (a) which is open. Letting U range over an open affine cover, we seethat Γ is open. Hence Y = X \ Γ is closed.
Let f : Y → X be the inclusion and consider Y with the structure sheaf f−1(OX/I ). Then
f∗OY = OX/I (on any affine U we have f∗OY = fiA/a) and if we let f# : OX → f∗OY be theprojection, then IY = ker f# = I .
Corollary 2.87. If X = SpecA, then the closed subschemes of X are in one-to-one correspondencewith the ideals of A.
Proof. In this case there is a one-to-one correspondence between quasi-coherent ideal sheaves I on Xand ideals a of A, which is given by I = a. Thus the statement is a direct consequence of 2.85.
Theorem 2.88. If M is a graded module over a graded ring S, then M is a quasi-coherent sheafover ProjS. If S is Noetherian and M finitely generated over S, then M is coherent.
Proof. Not given.
Definition 2.89. Let S be a graded ring and X = ProjS. Then the OX -module fiS(n) is denoted byOX(n) (so that OX(0) = OX). For any OX -module F we put F (n) = F ⊗OX OX(n).
Remark* 2.90. OX(1) is called the twisting sheaf of Serre and F (n) is the twisted sheaf .
Theorem 2.91. Suppose that S =⊕
d≥0 Sd is a graded ring that is generated by S1 as an algebraover S0. Let X = ProjS and let M,N be graded S-modules. Then
1. M ⊗OX‹N ∼= ‚M ⊗S N
2. M(n) = flM(n). In particular, OX(m)⊗OX OX(n) = OX(m+ n).
3. Let α : S → T be a surjective graded homomorphism of graded rings and let f : ProjT → ProjS
be the corresponding morphism (see exercises A.14 and A.22). Then f∗M ∼= ‚M ⊗S T . If K is
a T -module, then f∗‹K ∼= SK.
31
Remark 2.92. The condition “generated by S1 as an algebra over S0” is fulfilled by polynomial ringsand their quotients.
Proof.
1. We will define a morphism φ : M ⊗OX‹N → ‚M ⊗S N . Since the open sets of the form D+(b)
for homogeneous b ∈⊕
d≥1 Sd form a base of the topology of X, it suffices to define φ on thesesets.
Pick a homogeneous b ∈⊕
d≥1 Sd. By theorems 2.44 and 2.70 we have
M |D+(b)∼= fiM(b), ‹N |D+(b)
∼= fiN(b)ÄM ⊗OX
‹Nä∣∣∣D+(b)
∼= M |D+(b) ⊗OX |D+(b)‹N |D+(b)
∼= ÂM(b) ⊗S(b)N(b)
and ‚M ⊗S N |D+(b)∼= Â(M ⊗S N)(b).
Therefore we have to define maps ϑb : M(b) ⊗S(b)N(b) → (M ⊗S N)(b). For this we use the
natural map given bym
br⊗ n
br′7→ m⊗ n
br+r′.
The maps ϑb are compatible and thus glue to define φ.
If b ∈ S1, then ϑb is an isomorphism. Hence φ|D+(b) is an isomorphism for every b ∈ S1 byremark 2.71. But since S1 generates S as an S0-algebra, the D+(b) with b ∈ S1 cover X.Therefore φ is an isomorphism.
2. Using the identities M(n) ∼= M ⊗S S(n) and S(n) ⊗S S(m) ∼= S(m + n) this is a directconsequence of part 1.
3. Pick c ∈ T homogeneous and of positive degree. Pick b ∈ S homogeneous of the same degreewith α(b) = c. One easily sees that f−1D+(b) = D+(c).
We will define a morphism ψ : f∗M → ‚M ⊗S T . Similar to the first part, we define it on theopen sets D+(b). We have (using theorem 2.70 for the last step)Ä
f∗Mä∣∣∣D+(c)
= f∗ÄM |D+(b)
ä= f∗
ÄfiM(b)
ä∼= ÂM(b) ⊗S(b)
T(c).
Also ‚M ⊗S T |D+(c)∼= Â(M ⊗S T )(c). Together with the homomorphism
µ : M(b) ⊗S(b)T(c) → (M ⊗S T )(c),
m
br⊗ t
cr′7→ m⊗ t
cr+r′
this defines ψ. If c ∈ T1, then µ is an ismorphism. Hence ψ is an isomorphism over D+(c) andso, like above, it is a global isomorphism.
[The second statement was not proved in the lecture.]
Remark 2.93. Under some mild conditions, if F is a quasi-coherent sheaf on X = ProjS, then
F = ‡Γ∗(F ), where Γ∗(F ) =⊕
n∈Z F (n)(X).
32
2.6. Cartier Divisors1
Motivation Let X be a smooth variety over an algebraically closed field k. A subvariety D ⊆ X iscalled a prime divisor if dimD = dimX − 1. A divisor is a linear combination D =
∑niDi with
ni ∈ Z and the Di prime divisors.
For every x ∈ X there exists a rational function f on X such that D|U = (f)|U for some openneighborhood U of x. So a divisor is given by a collection (Ui, fi) (where the fi have to satisfy somecompatibility conditions). To extend the notion of divisor to arbitrary schemes, we use this lastproperty for the definition of a divisor. In order to avoid some technicalities we will assume the ourschemes are integral. For a more general definition see [Har77, Section II.6] or [Liu02, Section 7.1].
Note that if X is an integral scheme with function field K(X), then for every open subset U ⊆ Xthere is an injection OX(U) → K(X) [Liu02, Proposition 2.4.18].
Definition 2.94. Let X be an integral scheme. A Cartier divisor D on X is an equivalence class ofsystems (Ui, fi), where the open sets Ui cover X and fi ∈ K(X) such that for all i, j
fifj∈ O∗X(Ui ∩ Uj) = invertible elements of OX(Ui ∩ Uj) ⊆ K(X)
where two systems (Ui, fi) and (Vα, gα) are equivalent if for all i, α
figα∈ O∗X(Ui ∩ Vα).
Definition 2.95. Let D,D′ be two Cartier divisors on an integral scheme X, given by (Ui, fi) and(U ′j , f
′j). Their sum D+D′ is defined to be the system (Ui∩U ′j , fif ′j) (as X is integral the intersection
is always nonempty). The inverse −D of D is given by (Ui, f−1i ). Divisors of the form (X, f) for
some f ∈ K(X) are called principal . The group of Cartier divisors is
Div(X) =(group of all Cartier divisor,+)
principal divisors.
Remark* 2.96. Two divisor D, D′ are called linearly equivalent, denoted D ∼ D′, if their differenceis principal. So Div(X) is the group of all Cartier divisors modulo linear equivalence. Also note thatmany authors (e.g. [Har77, Liu02]) denote the full group of divisors by Div(X) and the group ofdivisors modulo linear equivalence by CaCl(X) and call the latter the (Cartier) divisor class group.
Definition 2.97. Let D be a Cartier divisor on an integral scheme X given by (Ui, fi). To D weassociate the sheaf OX(D) given by (for each U ⊆ X open)
OX(D)(U) = h ∈ K(X) : hfi ∈ OX(Ui ∩ U) ⊆ K(X) for all i.
Note that this is well-defined: Suppose that D is also given by the system (Vα, gα). For h ∈OX(D)(U) we have hfi ∈ OX(Ui ∩ U), so
hgα|U∩Ui∩Vα = hfi︸︷︷︸∈OX(U∩Ui∩Vα)
· gαfi︸︷︷︸
∈OX(U∩Ui∩Vα)
∈ OX(U ∩ Ui ∩ Vα).
Since the sets U ∩ Ui ∩ Vα cover U ∩ Vα, the sheaf condition implies hgα ∈ OX(U ∩ Vα). Therefore ifh ∈ OX(D)(U) as defined by (Ui, fi), then h is also in OX(D)(U) defined by (Vα, gα). By a symmetricargument the converse inclusion holds too.
1The proofs in this section are non-examinable.
33
Theorem 2.98. Let X be an integral scheme and let D,D′ be Cartier divisors on X.
1. OX(D) is an invertible sheaf.
2. OX(D +D′) ∼= OX(D)⊗OX OX(D′).
3. OX(−D) ∼= OX(D)−1.
4. D −D′ is principal if and only if OX(D) ∼= OX(D′).
Therefore the map D 7→ OX(D) induces an injective homomorphism Div(X)→ Pic(X).
Proof.
1. Let D be given by (Ui, fi). Then OX(D)|Ui = OUi · 1fi
as modules over OUi . So OX(D)|Ui ∼= O|Uias OUi-modules. Thus OX(D) is invertible.
2. Define a morphism
OX(D)⊗OX OX(D′)ψ−→ OX(D +D′)
in the following way: For any open set V ⊆ X define
OX(D)(V )⊗OX(V ) OX(D′)(V )→ OX(D +D′)(V ), h⊗ h′ 7→ hh′.
We can assume that D and D′ are given by (Ui, fi) and (Ui, f′i) respectively. So D + D′ is
given by (Ui, fif′i). Hence if hfi and h′f ′i are in OX(V ∩Ui) for all i, the hh′fif
′i ∈ OX(V ∩Ui),
so hh′ ∈ OX(D +D′)(V ), i.e. ψ is well-defined.
Now check that it is an isomorphism.
3. “exercise or ignore”
2.7. Differential Forms2
Definition 2.99. Let A be a ring, B an A-algebra and M a B-module. An A-derivation from Binto M is a map d: B →M with
1. d(b+ b′) = db+ db′ for all b, b′ ∈ B;2. d(bb′) = bdb′ + b′ db for all b, b′ ∈ B;3. da = 0 for all a ∈ A.
The module of relative differential forms of B over A is a B-module ΩB/A together with an A-derivation d: B → ΩB/A such that for any B-module M and A-derivation d′ : B →M there exists aunique B-module homomorphism f : ΩB/A →M such that the following diagram commutes:
B ΩB/A
M
d
d′ ∃!f
Proposition 2.100. The pair (ΩB/A, d) exists and is unique up to unique isomorphism. It can beconstructed in the following way: ΩB/A is the free B-module generated by symbols of the form df ,f ∈ B, modulo the relations
1. d(f + g) = df + dg for all f, g ∈ B;2. d(fg) = g df + f dg for all f, g ∈ B;
2The proofs in this section are non-examinable.
34
3. da = 0 for all a ∈ A.
Example 2.101. For B = A[t1, . . . , tn], we get ΩB/A∼= B⊕n: every element of ΩB/A can be written
as a sum∑ni=1 fi dti with fi ∈ B in a unique way (this needs some work to prove).
Definition 2.102. Let f : X → Y be a morphism of affine schemes, say X = SpecB and Y = SpecA.
The sheaf of relative differential forms on X over Y is ΩX/Y = flΩB/A.More generally, if f : X → Y is any morphism of schemes, the sheaf of relative differential forms [Liu02, Proposition
6.1.17]on X over Y , again denoted ΩX/Y is defined as follows: Cover Y with open affine schemes Ui andcover each f−1Ui with open affine schemes Vi,α. Then for each pair i, α the restriction of f gives thesheaf ΩVi,α/Ui . These sheaves can be glued together to form the sheaf ΩX/Y on X.
Example 2.103. If X = AnA, Y = SpecA and B = A[t1, . . . , tn], then ΩX/Y = flΩB/A = ‹B⊕n.
Theorem 2.104. Let f : X → Y and g : S → Y be morphisms of schemes and let p : X ×Y S → Xbe the projection. Then
p∗ΩX/Y = ΩX×Y S/S .
Theorem 2.105. If f : X → Y and g : Y → Z are morphisms of schemes, then the sequence
f∗ΩY/Z → ΩX/Z → ΩX/Y → 0
is exact.
Theorem 2.106. Suppose that A is a ring, X = PnA and Y = SpecA. Then there is an exactsequence
0→ ΩX/Y →n⊕i=0
OX(−1)→ OX → 0.
Proof. Let S = A[t0, . . . , tn] and let T be the S-module S(−1)⊕(n+1) with free basis e0, . . . , en.Further let α : T → S be the (graded) S-module homomorphism given by ei 7→ ti and let M be itskernel. This gives an exact sequence of S-module
0→M → T → S.
Passing to sheaves, we get an exact sequence
0→ M → T → S.
By definition, T = OX(−1)⊕(n+1) and S = OX . The map α : T → S is not surjective, but one canshow that the corresponding map of sheaves is surjective. So the above sequence is really the exactsequence
0→ M → OX(−1)⊕(n+1) → OX → 0.
We will now show that M ∼= ΩX/Y . In order to do this, we cover X by the open affines Ui = D+(ti)
and define φi : ΩX/Y |Ui → M |Ui as follows: Let Bi = S(ti) = A[ t0ti , . . . ,tnti
]. Then ΩX/Y |Ui = ΩUi/Y =‡ΩBi/A and ΩBi/A is generated (as a Bi-module) by d( tkti ) (k = 0, . . . , n). We define a homomorphism
hi : ΩBi/A →M(ti) by d( tkti ) 7→ 1t2i
(tiek − tkei). (tiek − tkei ∈ kerα and the 1t2i
is needed to obtain an
element of degree 0.) It turns out that hi is an isomorphism. We define φi to be the morphism ofsheaves induced by hi. So φi is also an isomorphism.
To obtain an isomorphism ΩX/Y → M we will glue the φi: Because of tkti
= tktj
tjti
on Ui ∩ Uj we
have
d(tkti
) =tktj
d(tjti
) +tjti
d(tktj
),
which one can use to show that the φi are compatible.
35
Remark 2.107. Let X is a variety over an algebraically closed field k (i.e. an integral and separatedscheme of finite type over Spec k). Then X is smooth (i.e. all local rings OX,x are regular) if andonly if ΩX/ Spec k is locally free (i.e. locally it is isomorphic to a direct sum of copies of OX).
If X is a smooth variety over k, the tangent sheaf to X is defined as TX = H omOX (ΩX/k,OX).
Further the sheaf ωX = ΛdimXΩX/ Spec k is called the canonical sheaf. It is invertible and theassociated divisor is called the canonical divisor. It plays an important role in the Riemann-Rochtheorem (see 3.40).
36
3. Cohomology
3.1. Cohomology of Sheaves
Theorem 3.1. Let (X,OX) be a ringed space. Then the category M(X) of OX-modules on X hasenough injectives.
Proof. Let F be any OX -module. We need to find an injective OX -module I and an injectivemorphism F → I .
For each x ∈ X, the stalk Fx is an OX,x-module. By commutative algebra [Eis95, Corollary A3.9],there exists a monomorphism of OX,x-modules Fx → Ix such that Ix is injective. We can consider Ixas a sheaf on the one-point space x. Let fx : x → X be the inclusion map. Then fx∗Ix is anOX -module. We set
I =∏x∈X
fx∗Ix.
We will first show that there is an injective morphism F → I . Consider any OX -module G . By theuniversal property of direct products,
HomOX (G , I ) ∼=∏x∈X
HomOX (G , fx∗Ix).
We will show that for each x ∈ X, we have HomOX (G , fx∗Ix) ∼= HomOX,x(Gx, Ix): If φ : G → fx∗Ixis a morphism of OX -modules, then φx : Gx → Ix is in HomOX,x(Gx, Ix). Conversely, if we are givenan OX,x-module homomorphism Gx → Ix, then then for any open U ⊆ X with x ∈ U we have thecomposition
G(U)→ Gx → Ix = (fx∗Ix)(U)
(and for x /∈ U , (fx∗Ix)(U) = 0, so that in this case there is a unique map G(U)→ (fx∗Ix)(U)) andthese give a morphism G → (fx∗Ix)(U).
In particular, by construction of I , for each point x ∈ X we have an injection Fx → Ix and thesegives a morphism F → I , which has to be injective as it is injective on stalks.
Finally we will show that I is injective: Suppose we are given an injective morphism G → Hof OX -modules and a morphism G → I . By the above discussion this situation corresponds tocommutative diagrams (with the first row exact)
0 Gx Hx
Ix
∃
for each x ∈ X. Since Ix is injective, there exists a homomorphism Hx → Ix fitting into the abovediagram. Again by the above discussion, these homomorphism give a morphism H → I such that
37
the diagram
0 G H
I
commutes. Now we can inductively construct an injective resolution for F as follows:
0 F I I 1 I 2 · · ·
I 0/
F I 1/(
I 0/
F)
0 0
Corollary 3.2. Let X be a topological space. Then the category Sh(X) of sheaves of Abelian groupson X has enough injectives.
Proof. We can consider X as a ringed space by setting OX to be the constant sheaf defined by Z.Then Sh(X) = M(X) and we can apply the preceding theorem.
Definition 3.3. Let X be a topological space and let F : Sh(X)→ Ab be the global sections functorF 7→ F (X). Then the i-th cohomology group of F is Hi(X,F ) = RiF (F ).
Definition 3.4. Let X be a topological space. A sheaf F on X is called flasque if for every opensubset U ⊆ X the restriction map F (X)→ F (U) is a surjection.
Remark* 3.5. If F is flasque, then for every inclusion V ⊆ U of open sets, the restriction mapF (U)→ F (V ) is surjective.
Theorem 3.6. Let (X,OX) be a ringed space. Then every injective I ∈M(X) is flasque.
Proof. Pick any open subset U of X. We have to show that the restriction I (X)→ I (U) is surjective.Define a presheaf of OX -modules F on X by
F (W ) =
®0, if W * U
OX(W ), if W ⊆ U
for every open W ⊆ X. Pick any element t ∈ I (U). We have to find s ∈ I (X) with s|U = t. In orderto do this define a morphism φ+ : F + → I by
φW : F (W )→I (W )
1 7→®
0, if W * U
t|W , if W ⊆ U.
Since I is injective, we can find ψ : OX → I fitting into the following commutative diagram:
0 F + OX
I
φ+ ψ
38
Put s = ψX(1) ∈ I (X). Then
s|U = ψX(1)|U = ψU (1|U ) = ψU (1) = φ+U (1) = t.
Lemma 3.7. Let X be a topological space and
0→ F → G → H → 0
a short exact sequence of sheaves with F flasque. Then the map G(X)→ H (X) on global sections issurjective.
Proof. Denote the map G → H by α. Note that locally α is surjective and we have to prove that itis surjective on global sections. Pick any t ∈ H (X). We have to find a section s ∈ G(X) that mapsto t. Let S be the set of all pairs (U, s) such that s ∈ G(U) maps to t|U . Define a partial order on Sby setting (U1, s1) ≤ (U2, s2) if U1 ⊆ U2 and s2|U1
= s1. By the sheaf condition every chain in S isbounded. Thus Zorn’s lemma provides a maximal element (U ′, s′) of S.
Suppose U ′ 6= X. Let x ∈ X \ U ′. Then, by local surjectivity, there exists an open neighborhoodV of x and a section s ∈ G(V ) such that (V, s) ∈ S. Then
s′|U ′∩V − s|U ′∩V ∈ ker (G(U ′ ∩ V )→ H (U ′ ∩ V )) = F (U ′ ∩ V ).
Since F is flasque there exists a global section r ∈ F (X) ⊆ I (X) such that r|U ′∩V = s′|U ′∩V − s|U ′∩V .Then (r|V + s) and s′ restrict to the same section in G(U ′ ∩ V ). Hence they glue to give an element(U ′ ∪ V, s) of S. This is a contradiction to the maximality of (U ′, s′). Therefore U ′ = X and s′ mapsto t. So αX is surjective.
Lemma 3.8. Let X be a topological space and
0→ F → G → H → 0
a short exact sequence of sheaves with F and G flasque. Then H is also flasque.
Proof. Applying Lemma 3.7 to the sheaves restricted to any open U ⊆ X we see that all sequences
0→ F (U)→ G(U)→ H (U)→ 0
are exact. For any open U ⊆ X consider the commutative diagram
0 F (X) G(X) H (X) 0
0 F (U) G(U) H (U) 0
where the rows are exact and the vertical arrows are the restriction maps. Applying the four lemmawe see that the restriction map H (X)→ H (U) is also surjective, so that H is flasque.
Theorem 3.9. Let X be a topological space and F a flasque sheaf on X. Then the cohomologygroups Hi(X,F ) are zero for all i ≥ 1. In other words, every flasque sheaf is acyclic with respect tothe global sections functor on Sh(X).
Proof. Let I be the first step in an injective resolution if F and set G = I/F so that we have theshort exact sequence
0→ F → I → G → 0. (3.1)
Note that F is flasque by assumption and I is flasque by theorem 3.6 hence G is flasque by thelemma.
39
We apply the long exact sequence of (3.1):
0 H0(X,F ) H0(X, I ) H0(X,G)
H1(X,F ) H1(X, I ) H1(X,G)
H2(X,F )
Since I is injective, Hi(X, I ) = 0 for i ≥ 1. Also the sequence
0→ H0(X,F )→ H0(X, I )→ H0(X,G)→ 0
is exact by Lemma 3.7 and hence H1(X,F ) = 0. The same argument applied to the flasque sheaf Gshows that H1(X,G) = 0. Now an easy induction shows that Hi(X,F ) (and hence also Hi(X,G))is zero for i ≥ 0.
Corollary 3.10. Let X be a topological space and F a sheaf on X. Then cohomology can be computedusing “flasque resolutions”: For any exact sequence
0→ F → J 0 → J 1 → J 2 → · · · ,
with all J i flasque, the cohomology groups of
0→ J 0(X)→ J 1(X)→ J 2(X)→ · · ·
are isomorphic to Hi(X,F ).
Proof. This immediately follows from the last theorem and the general fact that derived functors canbe computed using an “acyclic resolution” (theorem 1.43).
Corollary 3.11. Let (X,OX) be a ringed space. Then the derived functors of the global sectionsfunctor M(X)→ Ab, F → F (X) coincide with the cohomology functors Hi(X,−) (computed withrespect to Sh(X)).
Proof. To compute the derived functors of the global sections functor of an OX -module F , we haveto chose an injective resolution of F in M(X). By theorem 3.6, injective OX -modules are flasque andby the corollary 3.10 a flasque resolution gives the cohomology groups Hi(X,F ).
Definition 3.12. Let X be a topological space. The dimension of X is the supremum of the allintegers n such that there is a chain ∅ 6= X0 ( X1 ( X2 ( · · · ( Xn = X of irreducible closed subsetsof X.
Theorem 3.13. Suppose X is a Noetherian topological space of dimension d. Then for every sheaf[Har77, Theorem
III.2.7] F on X the cohomology groups Hi(X,F ) vanish for all i > d.
Proof. Omitted.
Lemma 3.14. Let A be a Noetherian ring and I and injective A-module. Then I is a flasque sheaf[Har77, Proposition
III.3.4] on SpecA.
Proof. Omitted.
Theorem 3.15. Let X be a Noetherian scheme. Then the following are equivalent:
1. X is affine;
40
2. Hi(X,F ) = 0 for all quasi-coherent sheaves F on X and all i ≥ 1;
3. H1(X,F ) = 0 for all coherent ideal sheaves F on X.
Proof.
(1) implies (2) Since X is affine, we can assume X = SpecA with A Noetherian. Let F be a
quasi-coherent sheaf on X. By theorem 2.78, F = M for some A-module M . Any injective resolution
0→M → I0 → I1 → I2 → · · ·
of M gives an exact sequence (by remark 2.72)
0→ F = M → ‹I0 → ‹I1 → ‹I2 → · · · .
By the preceding lemma this sequence is a flasque resolution of X; so we can use it to calculate thecohomology groups. But applying the global sections functor to this sequence recovers the originalsequence of modules which is exact. So the cohomology groups Hi(X,F ) vanish for i ≥ 1.
(2) implies (3) trivial
(3) implies (1) This is the hardest part. We are going to use the criterion of exercise A.27. Inorder to do so we need to construct elements b ∈ OX(X) such that D(b) = x ∈ X : b /∈ mx is anopen affine subscheme.
Pick any closed point x ∈ X and any open affine neighborhood U of x. Set Y = X \ U . LetIY be a quasi-coherent ideal sheaf corresponding to some subscheme structure on Y . We can alsoconsider x as the closed subscheme of X given by Spec k(x) (cf. exercise A.9). Further, let IY ∪xbe the ideal sheaf corresponding to the subscheme structure on the closed subset Y ∪ x. Since X isNoetherian both IY and IY ∪x are coherent. This gives a short exact sequence
0→ IY ∪x → IY → k(x)→ 0
where k(x) is identified with the skyscraper sheaf at x given by k(x). The long exact sequence of thissequence is
0→ IY ∪x(X)→ IY (X)α−→ k(x)→ H1(X, IY ∪x)→ · · · .
By assumption H1(X, IY ∪x) = 0, so α is surjective. Hence there exists b ∈ IY (X) ⊆ OX(X) withα(b) = 1. Thus x ∈ D(b). Also, since b ∈ IY (X), D(b) ∩ Y = ∅, i.e. D(b) ⊆ U . But U is affine, soD(b) = D(b|U ) ⊆ U is an open affine subset.
In this way, we can cover X with open affine sets of the form D(b). Since X is Noetherian, it canbe covered by only finitely many, say D(b1), . . . , D(bn). We have to show that b1, . . . , bn generateOX(X).
Define a morphism φ : O⊕nX → OX by φWÄ(δij)
nj=1
ä= bi|W for all open subsets W ⊆ X. For
affine subsets U of X, b1|U , . . . , bn|U generate OX(U), so φ is surjective. We have to show that φX issurjective.
Set F = kerφ, so that we have a short exact sequence
0→ F → O⊕nX → OX → 0. (3.2)
Consider the filtrations
G0 = 0 ⊆ G1 = OX ⊕ 0⊕ · · · ⊕ 0 ⊆ G2 = OX ⊕ OX ⊕ 0⊕ · · · ⊕ 0 ⊆ · · · ⊆ Gn = O⊕nX
41
andF0 = 0 ⊆ F1 = G1 ∩ F ⊆ · · · ⊆ Fn = Gn ∩ F .
This gives injectionsFi+1
Fi=
F ∩Gi+1
F ∩ Gi→ Gi+1
Gi∼= OX ,
so that Fi+1/Fi is an ideal sheaf of X. As X is Noetherian, Fi+1/Fi is coherent. So by assumption,H1(X,Fi+1/Fi) = 0. From the long exact sequence of
0→ Fi → Fi+1 → Fi+1/Fi → 0
and F0 = 0 we can inductively show that H1(X,Fi) = 0. In particular, H1(X,F ) = 0. Now the longexact sequence of (3.2) is
0→ H0(X,F )→ H0(X,O⊕nX )φX−−→ H0(X,OX)→ H1(X,F ) = 0.
Therefore φX is surjective as required.
3.2. Cech Cohomology
Definition 3.16. Let X be a topological space. Let U = (Ui)i∈I be an open covering of X indexedby a well-ordered set I. Let F be a sheaf on X. We will write Ui0,...,ip for Ui0 ∩ · · · ∩ Uip .
• For p ≥ 0 define
Cp(U,F ) =∏
i0<···<ip
F (Ui0,...,ip).
By convention, the empty product is 0, i.e. if |I| < p+ 1, then Cp(U,F ) = 0.
• For p ≥ 0 define a homomorphism
dp : Cp(U,F )→ Cp+1(U,F )
by sending (si0,...,ip)i0<···<ip to (tj0,...,jp+1)j0<···<ip+1
with
tj0,...,jp+1=
p+1∑k=0
(−1)k sj0,...,jk,...,jp+1
∣∣∣Uj0,...,jp+1
.
• It is easy to check that dp+1 dp = 0. The complex
0d−1
−−→ C0(U,F )d0−→ C1(U,F )
d1−→ C2(U,F )d2−→ · · ·
is called Cech complex.
• The pth Cech cohomology group of F with respect to the covering U is
Hp(U,F ) =ker dp
im dp−1.
Theorem 3.17. Using the notation of the definition,
H0(U,F ) ∼= H0(X,F ) = F (X).
42
Proof. By definition,
C0(U,F ) =∏i∈I
F (Ui),
d0(si)i∈I =(sj |Ui∩Uj − si|Ui∩Uj
)i<j
.
ThereforeH0(U,F ) = ker d0 =
(si) ∈ C0(U,F ) : sj |Ui∩Uj = si|Ui∩Uj∀i, j
.
By the sheaf condition such a family (si) uniquely determines a section s ∈ F (X), so ker d0 ∼=F (X).
Example 3.18. Let X = P1K = ProjK[t0, t1], where K is a field. Let U = (U0, U1) with Ui = D+(ti).
We will calculate H•(U,OP1K
). The intersection U0 ∩ U1 is equal to D+(t0t1), so by theorem 2.44 wehave
C0(U,OX) =OX(U1)⊕ OX(U2) = K[t0, t1](t0) ⊕K[t0, t1](t1)
C1(U,OX) =OX(U0 ∩ U1) = K[t0, t1](t0t1)
Since X is integral we can consider all sections as elements of the function field K(X) = K(t0, t1).The Cech complex looks like
0 K[t0, t1](t0) ⊕K[t0, t1](t1) K[t0, t1](t0t1) 0
(p, q) q − p
d−1 d0 d1
First we want to calculate H0(U,OX) = ker d0 = (p, q) : p = q. We can write p = F
td00
, q = G
td11
with
degF = d0, degG = d1. So we have F
td00
= G
td11
in K(t0, t1), i.e. Ftd11 = Gtd00 in K[t0, t1]. The last
ring is a UFD, so td00 |F and td11 |G. Hence there exists a0, a1 ∈ K such that F = a0td00 and G = a1t
d11 .
This gives a0 = p = q = a1, so that
OX(X) = H0(X,OX) = H0(U,OX) = ker d0 = K.
Next we need to calculate im d0. Now, im d0 is the sub-K-vector space of K[t0, t1](t0t1) generatedby all elements of K[t0, t1](t0) and K[t0, t1](t1). We want to show that this all of K[t0, t1](t0t1). Pick
any r ∈ K[t0, t1](t0t1) and write r = E(t0t1)d
for some E ∈ K[t0, t1] with degE = 2d. By linearity, we
may assume that E is a monomial, say E = tm0 tn1 . Because m+ n = 2d, either m ≥ d or n ≥ d. If
m ≥ d, then r =tm0 t
n1
(t0t1)d=
tm−d0 tn1td1
∈ K[t0, t1](t1). Similarly for n ≥ d. Thus im d0 = K[t0, t1](t0t1) and
H1(U,OX) = 0.
Obviously Hp(U, X) = 0 for p ≥ 2.
Example 3.19. Like in the last example, let X = P1K = ProjK[t0, t1], where K is a field and cover X
by U = (U0, U1) with Ui = D+(ti). We will calculate H•(U,ΩP1K/ SpecK).
We have U0∼= SpecK[t0, t1](t0) = SpecK[ t1t0 ]. Therefore
ΩX/ SpecK(U0) = ΩOX(U0)/K = K[t0, t1](t0) d
Åt1t0
ã.
Similarly,
ΩX/ SpecK(U1) = ΩOX(U1)/K = K[t0, t1](t1) d
Åt0t1
ã.
43
Further, from U1 ∩ U2 = D( t0t1 ) ⊆ U1∼= A1
K ,
ΩX/ SpecK(U0 ∩ U1) =(ΩX/ SpecK(U1)
)(t0t1
) = K
ït0t1,t1t0
òd
Åt0t1
ã.
So the Cech complex is given by
0→ K[t0, t1](t0) d
Åt1t0
ã⊕K[t0, t1](t1) d
Åt0t1
ãd0−→ K
ït0t1,t1t0
òd
Åt0t1
ã.→ 0
d0
Åpd
Åt1t0
ã, q d
Åt0t1
ãã= q d
Åt0t1
ã− pd
Åt1t0
ã.
On U1 ∩ U2 we have t0t1· t1t0 = 1, so that t0
t1dÄt1t0
ä+ t1
t0dÄt0t1
ä= d(1) = 0. Using this relation we can
simplify d0 to
d0
Åp d
Åt1t0
ã, q d
Åt0t1
ãã=
Åq +
t21t20p
ãd
Åt0t1
ã.
To calculate the kernel of d0, we need to determine all p, q with(q +
t21t22p)
= 0. A Calculation similar
to the one in the precessing example show that this is the case if and only if p = q = 0. HenceH0(U,ΩP1
K/ SpecK) = ker d0 = 0.
It is easy to see that
Kît0t1, t1t0
ódÄt0t1
äim d0
∼= K d
Åt0t1
ã.
So H1(U,ΩP1K/ SpecK) ∼= K.
Example 3.20. Let X = P1K and let F be the constant sheaf given by Z. Let U again be (U0 =
D+(t0), U1 = D+(t1)). Since X is integral (and hence irreducible), F (U) = Z for all nonempty opensubsets U of X. The Cech complex is
0 Z⊕ Z Z 0
(m,n) n−m
d0
Therefore we have
ker d0 = (m,m) : m ∈ Z ∼= Z,im d0 = Z.
Thus the cohomology groups are H0(U,F ) = Z and Hp(U,F ) = 0 for p ≥ 1.
Example 3.21. Let X be the circle S1 with the usual topology induced by R2. Again let F be theconstant sheaf defined by Z. Chose to distinct points a and b on the circle and set U = S1 \ a andV = S1 \ b. Cover S1 with U = (U, V ). Since U ∩ V = S1 \ a, b has two components, the Cechcomplex is
0 F (U)⊕ F (V ) F (U ∩ V ) 0
Z⊕ Z Z2
(m,n) (n−m,n−m)
d0
Therefore H0(U,F ) ∼= Z and H1(U,F ) ∼= Z.
44
Definition 3.22. Let X be a topological space. Let U = (Ui)i∈I be a finite open covering of X. LetF be a sheaf on X. Let fi0,...,ip : Ui0,...,ip → X be the inclusion maps. Set
Fi0,...,ip =(fi0,...,ip
)∗ F |Ui0,...,ip ∈ Sh(X).
For p ≥ 0 define
Cp(U,F ) =∏
i0<···<ip
Fi0,...,ip .
Define dp : Cp(U,F )→ Cp(U,F ) analogous to definition 3.16. The complex of sheaves
0 −→ C 0(U,F )d0−→ C 1(U,F )
d1−→ C 2(U,F )d2−→ · · ·
is called Cech complex.
Remark 3.23. C (U,F )(X) = C(U,F ).
Lemma 3.24. With the notation of the definition, there exists a morphism F → C 0(U,F ) such thatthe sequence
0 −→ F −→ C 0(U,F )d0−→ C 1(U,F )
d1−→ C 2(U,F )d2−→ · · ·
is exact.
Proof. The map F → C 0(U,F ) is defined by sending s ∈ F (W ) to (s|W∩Ui)i for each open subsetW of X. The exactness of
0→ F → C 0(U,F )→ C 1(U,F )
is just the sheaf condition for F as formulated in lemma 1.4.To prove that the rest of the sequence is exact, we will show exactness on stalks. Pick any x ∈ X.
We can assume that x ∈ U1,...,n where I = 1, . . . , n because if x /∈ Uj we can just ignore the indexj as it has no contribution to the stalk at x.
We will define maps
Cp(U,F )xep−→ Cp−1(U,F )x.
for all p ≥ 1: Let (W, s) ∈ Cp(U,F )x, i.e. x ∈W and
s ∈ Cp(U,F )(W ) =∏
i0<···<ip
F (W ∩ Ui0,...,ip).
Since x ∈ U1,...,n, we can assume that W ⊆ U1,...,n. Write s = (si0,...,ip). Define an elementt = (ti0,...,tp−1
) ∈ Cp−1(U,F )x by
ti0,...,ip−1=
®s1,i0,...,ip−1 , if i0 6= 1
0, if i0 = 1,
where we assume that I = 1, 2, . . . . Set ep(s) = t. It turns out that on Cp(U,F )x we have theformula
ep+1 dp + dp−1 ep = id .
Cpx Cp+1x
Cp−1x Cpx
dp
ep
ep+1
dp−1
Now if dp sends an element (W, s) ∈ Cp(U,F )x to zero in Cp+1(U,F ), then dp−1ep(W, s) = (W, s).Therefore (W, s) is in the image of dp−1. Thus the sequence is exact.
45
Theorem 3.25. Let X be a topological space, U a finite open covering of X and F a flasque sheafon X. Then Hp(U,F ) = 0 for p ≥ 1.
Proof. Since F is flasque, all F |Ui0,...,ip are flasque which in turn implies that each Fi0,...,ip is flasque.Therefore their direct sum Cp(U,F ) is flasque for all p. Thus lemma 3.24 provides a flasque resolutionof F :
0 −→ F −→ C 0(U,F ) −→ C 1(U,F ) −→ C 2(U,F ) −→ · · · (3.3)
But cohomology can be calculated using flasque resolutions (corollary 3.10), so if we apply the globalsections functor to the sequence (3.3)
0d−1
−−→ C 0(U,F )(X)d0−→ C 1(U,F )(X)
d1−→ C 2(U,F )(X)d2−→ · · · , (3.4)
then ker dp
im dp−1 = Hp(X,F ) = 0 for p ≥ 1, since the homology groups for flasque sheaves vanish (theorem3.9).
On the other hand the groups in the sequence (3.4) are just the groups Cp(U,F ) of the usual Cechcomplex, which calculate the Cech cohomology groups. So,
Hp(U,F ) =ker dp
im dp−1= Hp(X,F ) = 0 for p ≥ 1.
Theorem 3.26. Let X be a separated (over SpecZ) Noetherian scheme, F a quasi-coherent sheafon X and U a finite covering of X by open affine subschemes. Then for all p,
Hp(X,F ) = Hp(U,F ).
Proof. Let U = (Ui)i∈I . For each i ∈ I there exists an OX(Ui)-module Mi such that F |Ui = ›Mi. Let
Ji be an injective module such that 0 → Mi → Ji is exact. Then ‹Ji is flasque and 0 → ›Mi → ‹Jiremains exact. Let Gi be the direct image of ‹Ji under the inclusion map Ui → X. Set G =
∏Gi.
Then G is a flasque, quasi-coherent sheaf on X and F injects into G . Let H = G/F . Then H isquasi-coherent and there is a short exact sequence
0→ F → G → H → 0.
Consider any sequence i0 < i1 < · · · < ip. Since X is separated, Ui0,...,ip is affine (exercise A.24).The long exact sequence of cohomology gives
0→ F (Ui0,...,ip)→ G(Ui0,...,ip)→ H (Ui0,...,ip)→ H1(Ui0,...,ip ,F |Ui0,...,ip )︸ ︷︷ ︸=0 by theorem 3.15
The direct sum of all these sequences gives an exact sequence
0→ C•(U,F )→ C•(U,G)→ C•(U,H )→ 0
By theorem 1.37 we get a long exact sequence.
· · · → Hp(U,F )→ Hp(U,G)→ Hp(U,H )→ Hp+1(U,F )→ · · ·
Since G is flasque, Hp(U,G) = 0 = Hp(X,G) for all p ≥ 1 (theorems 3.9 and 3.25). In particular, wehave
H0(U,F ) H0(U,G) H0(U,H ) H1(U,F ) 0
H0(X,F ) H0(X,G) H0(X,H ) H1(X,F ) 0
46
We already know that H0(U,F ) = H0(X,F ) = F (X) (Theorem 3.17; the same is true for G andH ). The maps on the 0-th cohomology groups are just the maps on global sections. So the cokernelsare the same, i.e. H1(U,F ) ∼= H1(X,F ) and the analogous statement holds for H . Parts of the longexact sequence look like
0 Hp(U,H ) Hp+1(U,F ) 0
0 Hp(X,H ) Hp+1(X,F ) 0
This allows us to do induction to show the result for all p.
3.3. Projective Space
Lemma* 3.27. Let π : Y → X be a closed subscheme. Let F be a sheaf on Y . Then Hi(Y,F ) = [Har77, lem. III.2.10]
Hi(X,π∗F ).
Proof. See exercise A.34.
Theorem 3.28. Let k be a field. Let X = Pnk = Proj k[t0, . . . , tn]. Let d ∈ Z. Then:
1. H0(X,OX(d)) is isomorphic to the k-vector space generated by monomials of degree d int0, . . . , tn. In particular, if d < 0, then it is 0.
2. Hn(X,OX(d)) ∼= H0(X,OX(−d− n− 1)).
3. Hp(X,OX(d)) = 0 if 0 < p < n or p > n.
Proof. Note that X is Noetherian and separated over Spec k (and hence over Z). Put U = (Ui)i=0,...,n
with Ui = D+(ti), so that we can apply theorem 3.26 and calculate cohomology via the Cech complex.Set S = k[t0, . . . , tn].
1. We always have H0(X,OX(d)) = OX(d)(X). Every element of s of OX(d)(X) is uniquelydetermined by a system (si) with si ∈ OX(d)(Ui) = S(d)(ti) such that si|Ui∩Uj = sj |Ui∩Uj . Each
si can be written in the form fi/tlii where fi is a homogeneous polynomial with deg fi = li + d.
The condition si|Ui∩Uj = sj |Ui∩Uj translates to fi/tlii = fj/t
ljj in k(t0, . . . , tn). In particular,
fitljj = fjt
lii in S and hence tl
i
i divides fi and tljj divides fj . Thus fi/t
lii is in S, is independent
of i and has degree d. So OX(d)(X) is generated as a k-vector space by the homogeneouspolynomials of degree d.
2. The Cech complex ends with
· · · −−−→ Cn−1(U,OX(d))dn−1
−−−→ Cn(U,OX(d))︸ ︷︷ ︸=OX(d)(U0∩···∩Un)
=S(d)(t0···tn)
−−−→ 0
Every element of Cn(U,OX(d)) is of the form α = f(t0···tn)l
such that deg f = (n+ 1)l+d (in S),
where we choose l such that it is minimal. Assume that f is a monomial, say f = tm00 · · · tmnn
If mi ≥ l for some i, then α =tm00 ···t
mi−li
···tmnn(t0···ti···tn)l
∈ im dn−1. So assume that all mi < l. Also, at
least one mi is zero as otherwise l would not be minimal. Then (n+ 1)l+ d = deg f ≤ (l− 1)n,so d ≤ −n− l ≤ −n− 1. So, if d > −n− 1, then f is always in im dn−1 and Hn(X,OX(d)) =0 = H0(X,OX(−d− n− 1)).
47
Let d = −n − 1. Then l = 1 and deg f = 0. Thus Hn(X,OX(d)) = Cn(U,OX(d))/ im dn−1
is generated by (the coset of) the single element 1t0···tn . Hence Hn(X,OX(−n − 1)) ∼= k ∼=
H0(X,OX).
We will not prove (2) for d < −n− 1.
3. Note that we will implicitly use lemma 3.27 wherever needed. From the Cech complex weimmediately see that the cohomology vanishes for p > n. In particular, the statement is trivialfor n = 1. So assume that n ≥ 2.
Let Y be the closed subscheme of X defined by tn = 0. Then Y ∼= Pn−1k . Via the homomorphism
f 7→ ftn, we see that S(−1) ∼= (tn) (as graded S-modules). So there is an exact sequence
0→ S(−1)·tn−−→ S → S/(tn)→ 0.
Since ‡S(−1) = O(−1) (∼= IY ), S = OX and ‡S/(tn) ∼= Âk[t0, . . . , tn−1] ∼= π∗OY (where π : Y → Xis the closed immersion), the sequence of S-modules gives the following exact sequence ofOX -modules:
0→ OX(−1)→ OX → π∗OY → 0.
If we tensor the sequence with OX(d) we get the exact sequence
0→ OX(d− 1)→ OX(d)→ π∗OY (d)→ 0.
This gives a long exact sequence on cohomology:
0 H0(X,OX(d− 1)) H0(X,OX(d)) H0(X,π∗OY (d))
H1(X,OX(d− 1)) H1(X,OX(d)) H1(X,π∗OY (d))
Hn−1(X,OX(d− 1)) Hn−1(X,OX(d)) Hn−1(X,π∗OY (d))
Hn(X,OX(d− 1)) Hn(X,OX(d)) Hn(X,π∗OY (d))︸ ︷︷ ︸=0
0
α0
αn−1
Using parts (1) and (2) to count the dimensions of the k-vector spaces H0(X,OX(d − 1)),H0(X,OX(d)), H0(X,π∗OY (d)) and Hn−1(X,π∗OY (d)), Hn(X,OX(d−1)), Hn(X,OX(d)), wesee that α0 is the zero map and αn−1 is an injection.
By induction on n, we know that Hp(X,π∗OY (d)) = 0 for 0 < p < n− 1. Together with ourknowledge about α0 and αn−1, this implies that the maps
Hp(X,OX(d− 1))βp−→ Hp(X,OX(d))
are isomorphisms for 0 < p < n. The maps βp come from the maps
OX(d− 1)(Ui0,...,ip) OX(d)(Ui0,...,ip)
S(d− 1)(ti0 ···tip ) S(d)(ti0 ···tip )
·tn
(Recall that S(−1)→ S is given by multiplication with tn.) So βp is given by multiplicationwith tn.
48
We will prove that if ω ∈ Hp(X,OX(d)), then tln · ω = 0 for some l. This implies that βp is thezero map so that Hp(X,OX(d)) = 0 for 0 < p < n.
Set
F =⊕d∈Z
OX(d) =‚⊕d∈Z
S(d).
Now it turns out that
F (Ui0,...,ip) =
(⊕d∈Z
S(d)
)(ti0 ···tip )
∼= Sti0 ···tip .
Thus the Cech complex for F looks like
0 C0(U,F ) C1(U,F ) · · ·∏i0
Sti0
∏i0,i1
Sti0 ti1
If we localize this sequence at tn, we get
0→∏
Sti0 tn →∏
Sti0 ti1 tn → · · ·
This is the same as the Cech complex of F |Un given by the covering U0 ∩ Un, . . . , Un ∩ Un.Now Hp(Un,F |Un) = 0 for p ≥ 1 because Un is affine (theorem 3.15). Moreover Hp(X,F )tn =Hp(Un,F |Un) = 0 so that for any ω ∈ Hp(X,F ) we have tlnω = 0 for l sufficiently large.
3.4. The Riemann-Roch Theorem
Let k be a fixed algebraically closed field.
Definition 3.29. An integral quasi-projective scheme X over k of dimension d is smooth if for everyclosed point x ∈ X the local ring OX,x is a regular local ring of dimension d.
Remark 3.30. A smooth integral quasi-projective scheme over k is Noetherian.
Definition 3.31. Let X be a smooth integral quasi-projective scheme over k of dimension d. A Weildivisor on X is a formal sum
∑miMi where mi ∈ Z and Mi = µi with µi ∈ X such that OX,µi is
of dimension one (i.e. a DVR) and all but finitely many mi are 0.
Let f ∈ K(X)∗. The Weil divisor of f is
(f) =∑µ∈X
Ox,µ of dim 1
vµ(f)µ,
where vµ is the normalized valuation K(X)∗ → Z of the DVR OX,µ.
Remark 3.32. The divisor (f) is indeed a finite sum: Since X is Noetherian it is enough to check thefiniteness on open affine subschemes and then cover X with finitely many of them. So let U = SpecAbe an affine subscheme of X. Then K(X) is just the fraction field of A and hence we can write f = a
bwith a, b ∈ A. If vµ(f) 6= 0 for some µ ∈ U , then µ ∈ V ((a)) ∪ V ((b)) = V ((ab)). There are onlyfinitely many such µ in V ((ab)) (as in Noetherian rings there are only finitely many minimal primesover any ideal [?]), and hence only finitely many in U .
49
Definition 3.33. Let X be a smooth integral projective scheme over k of dimension 1. Then thedegree of a Weil divisor M =
∑miMi is degM =
∑mi ∈ Z.
Fact 3.34. Let X be as in the definition and f ∈ K(X)∗. Then deg(f) = 0.
Theorem 3.35. Let X be a smooth integral projective scheme over k of dimension 1. Then there isa one-to-one correspondence between Cartier and Weil divisors.
Proof. Suppose that D is a Cartier divisor given by the system (Ui, fi). On Ui consider the Weildivisor (fi). We can view (fi) as a Weil divisor on X by taking the closure of each component. On
any Ui ∩ Uj we have fifj
andfjfi
both in O∗X(Ui ∩ Uj). This means that (fi) and (fj) coincide on
Ui ∩ Uj , so that the whole system can be put together to give a single Weil divisor on X.Conversely let M =
∑mαMα (with Mα closed points) be a Weil divisor on X. Then define a
Cartier divisor as follows: Pick a closed point x ∈ X. The multiplicity of x in M is just mα if Mα = x(or 0 if x does not appear in M). Since Ox is a DVR, the maximal ideal mx ⊆ Ox is generated bya single element, say t. Set f = tmα . Then M = (f) in some open affine neighborhood of x (inany affine neighborhood there are only finitely many points where this could be false (Remark 3.32)and we can simply remove those points from the neighborhood). Now X is Noetherian, so it can becovered by finitely many such open affine schemes. This means that we get a system (Ui, fi) such
that Ui cover X and (fi) = M on Ui. Since (fi) = (fj) on Ui ∩Uj , fifj
andfjfi
are in O∗X(Ui ∩Uj). So
(Ui, fi) defines a Cartier divisor.
Definition 3.36. Let X be a smooth integral projective scheme over k of dimension 1. Let D be aCartier divisor on X. Then the degree of D, denoted degD, is the degree of the corresponding Weildivisor.
Remark 3.37. In the setting of the definition, if D ∼ D′, then degD = degD′.
Theorem 3.38. Let X be a smooth integral projective scheme over k. Let D be a Cartier divisor onX. Then all cohomology groups Hp(X,OX(D)) are finite dimensional k-vector spaces.
Proof. Not given.
Theorem 3.39 (Duality). Let X be a smooth integral projective scheme over k of dimension d.Then there exists a Cartier divisor KX such that for every Cartier divisor D on X,
dimkHp(X,OX(D)) = dimkH
d−p(X,OX(KX −D)).
Proof. Not given.
Theorem 3.40 (Riemann-Roch). Let X be a smooth integral projective scheme over k of dimension[Har77, IV.1.3]
1. Let D be any Cartier divisor on X. Then
dimkH0(X,OX(D))− dimkH
0(X,OX(KX −D)) = degD + 1− dimkH0(X,OX(KX)).
Proof. Let M =∑miMi be the Weil divisor corresponding to D. Let x ∈ X be a closed point.
Write Dx for the Cartier divisor associated to Weil divisor x. Let D be given by a system (Ui, fi)where Ui are open affine and let (Ui, gi) be a system for Dx. If W ⊆ X is any open subscheme, then
OX(D)(W ) =h ∈ K(X) : hfi ∈ OX(Ui ∩W ) ∀i
h7→h−−−→ OX(D +Dx)(W ) =e ∈ K(X) : efigi ∈ OX(Ui ∩W ) ∀i
is a well-defined homomorphism: since the Weil divisor of Dx is just x, all gi are in OX(Ui);so if hfi ∈ OX(W ∩ Ui), then hfigi ∈ OX(W ∩ Ui). These maps give an injective morphismOX(D)→ OX(D +Dx).
50
The quotient sheaf OX(D +Dx)/OX(D) is the skyscraper sheaf k(x) at x defined by k: if W ⊆ Xis an open affine and x /∈W , then gi ∈ O∗X(W ∩ Ui). Thus there is an exact sequence
0→ OX(D)→ OX(D +Dx)→ k(x)→ 0.
The corresponding long exact sequence of cohomology is
0 H0(X,OX(D)) H0(X,OX(D +Dx)) H0(X, k(x))
H1(X,OX(D)) H1(X,OX(D +Dx)) H1(X, k(x)) 0
Now H1(X, k(x)) = 0. So, by duality, we obtain the following sequence:
0 H0(X,OX(D)) H0(X,OX(D +Dx)) H0(X, k(x))
H0(X,OX(KX −D)) H0(X,OX(KX −D −Dx)) 0
Using linear algebra we obtain the formula
dimkH0(X,OX(D +Dx))− dimkH
0(X,OX(KX −D −Dx)) =
dimkH0(X,OX(D))− dimkH
0(X,OX(Kx −D)) + dimkH0(X, k(x))︸ ︷︷ ︸=1
.
Therefore the theorem holds for a divisor D if and only if it holds for D + Dx for any closedpoint x (note that deg(D + Dx) = degD + 1). So by removing and adding points to M , we canreduce to the case M = 0. But in this case dimkH
0(X,OX(D)) = 1 [exercise; hint: prove thatH0(X,OX(D)) = H0(X,OX) and that OX(X) is a finitely generated k-algebra which is a field] anddegD = 0, so that the theorem holds.
51
A. Example Sheets
A.1. Sheet 1
Exercise A.1. Give an example of a topological space X and a surjective morphism φ : F → G ofsheaves, such that F (X)→ G(X) is not surjective.
Solution. Consider the three-point-space X = a, b, c with the topology given by the open sets
X, U = a, b, V = b, c, U ∩ V = b, ∅.
On X consider the sheaves F , G and the map F → G as indicated in the following diagram (whereA is any nontrivial Abelian group):
F (U) = A
F (X) = A F (U ∩ V ) = A F (∅) = 0
F (V ) = A
G(U) = A
G(X) = A×A G(U ∩ V ) = 0 G(∅) = 0
G(V ) = A
id
id
id
id
p1
p2
a7→
(a,a)
id
id
where p1 and p2 are the projections on the first resp. second coordinate. The map F (X)→ G(X) isnot surjective, but the morphism F → G is because the maps on the stalks are
Fa = Aid−→ Ga = A
Fb = A −→ Gb = 0
Fc = Aid−→ Gc = A
A different (maybe more natural) counterexample is the following: Let X = C \ 0 and consider assheaves F and G both the sheaf of non-vanishing holomorphic functions. Let the map φ : F → G bedefined by φU : f 7→ f2 on each open subset U . By complex analysis (e.g. [Con78, Theorem VIII.2.2]):An open connected subset G of C is simply connected if and only if for all holomorphic functions g onG there exists a holomorphic function f on G such that f2 = g. Since X is has a simply connectedtopological base, this implies that φ is surjective. On the other hand X itself is not simply connected.Hence φX : F (X)→ G(X) is not surjective.
Exercise A.2. Let F be a sheaf on a topological space X, and let U ⊆ X be an open subset. Foreach open subset V ⊆ U put F |U (V ) = F (V ). Show that F |U is a sheaf on U which is called therestriction of F to U .
Solution. TBA
52
Exercise A.3. Let F , G be sheaves on a topological space X. For any open set U ⊆ X let
Hom(F , G)(U) = Hom(F |U , G |U ) = morphisms φ : F |U → G |U.
Show that Hom(F , G) is a sheaf on X.
Solution. TBA
Exercise A.4. Let X be a topological space and let X =⋃Ui be an open cover. Suppose that for
each i we have a sheaf Fi in Ui. Moreover, assume that for each pair i, j we have an isomorphismφij : Fi|Ui∩Uj → Fj |Ui∩Uj such that: for each i, φii is the identity morphism, and for each i, j, k wehave φik = φjkφij on Ui ∩ Uj ∩ Uk. Show that there exists a unique sheaf F on X and isomorphismsψi : F |Ui → Fi such that for each pair i, j we have ψj = φijψi on Ui ∩ Uj . The sheaf F is said to bethe sheaf obtained by gluing the sheaves Fi.
Solution. TBA
Exercise A.5. Let X be a topological space, and F , G sheaves on X. For any open subset U ⊆ X letF ⊕G(U) = F (U)⊕G(U). Show that F ⊕G is a sheaf on X which is called the direct sum of F , G .
Solution. TBA
Exercise A.6. A topological space X is said to be Noetherian if for any sequence U1 ⊆ U2 ⊆ · · · ofopen subsets, there is n such that Ui = Ui+1 for any i ≥ n. Now assume that X is a Noetheriantopological space. Let Fi be a direct system of sheaves on X, and for any open subset U ⊆ X let(lim−→Fi)(U) = lim−→(Fi(U)). Show that lim−→Fi is a sheaf. Prove the same statement replacing directsystem with inverse system and replacing direct limit with inverse limit (in this case you do not needthe Noetherian property).
Solution. TBA
Exercise A.7. Give an example of a locally ringed space which is not of the form (X = SpecA,OX)for any ring A.
Solution. TBA
Exercise A.8. Let A be a ring and X a scheme. Show that there exists a 1-1 correspondence betweenthe set of ring homomorphisms A → OX(X) and the set of morphisms of schemes X → SpecA.Determine these sets when A = Z, the ring of integers.
Solution. Any morphism X → SpecA induces a homomorphism on global sections A→ OX(X).Conversely, suppose we are given a homomorphism A→ OX(X). For any open affine SpecB =
U ⊆ X the restriction homomorphism OX(X) → OX(U) = B induces a homomorphism A → B.This in turn induces a morphism of schemes U → SpecA. We have to check that we can glue all themorphisms together. Let V = SpecC be another affine open subset of X and W = SpecD ⊆ U ∩ V .Then we have a commutative diagram of rings
B
A OX(X) D
C
53
which induces a commutative diagram of schemes
SpecB
SpecA SpecD
SpecC
Exercise A.9. Let X be a scheme, and L a field. Show that to give a morphism SpecL→ X is thesame as giving a point x ∈ X and a field extension k(x) := Ox
mx→ L where mx is the maximal ideal
of the local ring Ox. The field k(x) is called the residue field of x on X.
Solution. TBA
Exercise A.10. Let X be a scheme and x ∈ X. Show that there is a natural morphism Spec Ox → X.
Solution. TBA
Exercise A.11. Let Fp be the finite field with p elements. Describe SpecFp[t]. What are the residuefields of its points?
Solution. TBA
Exercise A.12. Let X be a scheme. The Zariski tangent space Tx at x is defined to be the dual ofthe k(x)-vector space mx
m2x
where mx is the maximal ideal of Ox. Assume that X is defined over a fieldL, i.e., there is a given morphism f : X → SpecL. Show that to give a morphism SpecL[t]/(t2)→ Xover L, i.e., to give a commutative diagram
Spec L[t]t2 X
SpecL
f
is the same as giving a point x ∈ X with k(x) = L and an element of Tx.
Solution. We will write Y = Spec(L[t]/(t2)
). Note that Y consists of the single point y = (t).
Suppose we are given a morphism g : Y → X of L-schemes. Set x = g(y). Then f and g induceinclusions L ⊆ k(x) ⊆ k(y) = L and so k(x) = L. The local homomorphism Ox → Oy restricts to anL-homomorphism α : mx → my. The ideal my = (t) ⊆ L[t]/(t2) is isomorphic as an L-vector space toL (via at 7→ a). Also m2
y = 0, so that m2x ⊆ kerα. Thus we get an L-linear map mx/m
2x → L, i.e. an
element in Tx.
Conversely, suppose we are given x ∈ X with k(x) = L and an L-linear map mx/m2x → L. From
the latter we obtain a map α : mx → L. Let β : Ox → Ox/mx = L be the projection map. The
composition L→ Oxβ−→ L is the identity. So for any element a ∈ Ox, β(a) is the unique element such
that a− β(a) ∈ mx. Now define a map
γ : Ox → L[t]/(t2), a 7→ β(a) + α(a− β(a))t.
54
Then γ is clearly a homomorphism of Abelian groups and is L-linear. To check that is a ringhomomorphism we compute
γ(a)γ(b) = (β(a)− α(a− β(a))t) (β(b)− α(b− β(b))t) =
β(a)β(b) + α(β(a)b+ aβ(b)− 2β(a)β(b))t = β(ab) + α(ab− β(ab)) = γ(ab)
sinceab− β(a)b− aβ(b) + β(a)β(b) = (a− β(a))(b− β(b)) ∈ m2
x.
Hence γ induces an L-morphism Y → X.The two constructions are inverses.
Exercise A.13. Let α : A→ B be a homomorphism of rings and f : Y = SpecB → X = SpecA theinduced morphism of schemes.
1. Show that α is injective if and only if the morphism φ : OX → f∗OY is injective.
2. Show that α is surjective if and only if f is a homeomorphism onto a closed subset of X andφ : OX → f∗OY is surjective.
Solution. TBA
Exercise A.14. Consider a homomorphism α : S → T of graded rings which preserves degrees, and letU = P ∈ ProjT : α−1P ∈ ProjS. Show that we naturally get an induces morphism f : U → ProjS.
Find an example of α which is not a isomorphism but such that the induced morphism f is anisomorphism.
Solution. TBA
Exercise A.15. Let A be a ring. Prove that the following are equivalent:
1. SpecA is not connected as a topological space.
2. There are nonzero element e, e′ ∈ A such that ee′ = 0, e2 = e, e′2 = e′, and e+ e′ = 1.
3. There are nonzero rings A′ and A′′ such that A is isomorphic to the product A×A′.
Solution. TBA
Exercise A.16. Describe SpecZ[t] and the morphism SpecZ[t]→ SpecZ induced by the homomor-phism Z→ Z[t].
Solution. TBA
Exercise A.17. Describe SpecR[t1, t2] and the morphism SpecC[t1, t2] → SpecR[t1, t2] induced bythe homomorphism R→ C.
Solution. TBA
Exercise A.18. Formulate and prove a statement similar to A.4 for gluing schemes (see [Har77,exercise II.2.12]).
Solution. TBA
Exercise A.19. Let Ai be a direct system of rings and let A = lim−→Ai. Show that the inverse limitlim←− SpecAi exists in the category of affine schemes and it is isomorphic to SpecA.
Solution. TBA
55
A.2. Sheet 2
Exercise A.20. Let X be an integral scheme. Show that X has a generic point η, i.e. η is dense inX. Show that the local ring Oη is a field which is called the function field of X denoted by K(X).Moreover show that this field is the fraction field of OX(U) for any open affine subscheme U ⊆ X.
Solution. TBA
Exercise A.21. Show that every closed subscheme of X = SpecA is uniquely determined by an ideala of A (we show this in corollary 2.87, however, for an elementary proof you might like to consult[Har77, exercise II.3.11]).
Solution. TBA
Exercise A.22. Let α : S → T be a surjective graded morphism of graded rings. Show that thisinduces a morphism ProjT → ProjS which is a closed immersion.
Solution. TBA
Exercise A.23. Separatedness is local on the base, i.e. a morphism f : X → Y of Noetherian schemesis separated if and only if Y can be covered by open subschemes Ui such that the induced morphismsf−1Ui → Ui are separated.
Solution. TBA
Note that the same is true for properness (with an analogous proof).
Exercise A.24. Let f : X → Y be a separated morphism where Y is affine. Show that if U, V ⊆ X areopen affine subschemes, then U ∩ V is also affine. Show that this is not true without the separatedcondition.
Solution.
Exercise A.25. A morphism f : X → Y is said to be finite if Y can be covered with open affinesubschemes Ui = SpecAi such that for each i the inverse image f−1Ui is affine, say SpecBi, andsuch that Bi is a finitely generated Ai-module.
Show that a finite morphism is proper.
Solution. Finite morphisms are of finite type, so that, like in exercise A.23, we only need to showproperness for an affine cover. Let Ui be a cover as in the definition. Then we need to show that forevery valuation ring R with fraction field K and every diagram
SpecK f−1Ui
SpecR Ui
there exists exactly one morphism from SpecR to f−1Ui fitting into the diagram. Since all schemesare affine, we can go to commutative algebra:
K Bi
R Ai
β
g
α
56
Uniqueness of g is clear, so we only need to prove existence. Since Bi is a finitely generated Ai-module,β(Bi) is a finitely generated α(Ai)-module. Hence it is integral over α(Ai) [Eis95, Corollary 4.6]. ButR is integrally closed in K [Eis95, Exercise 11.1b] and contains α(Ai). So β(Bi) ⊆ R as required.
Exercise A.26. Let Y = SpecA. Show that PnY ∼= ProjA[t0, . . . , tn].
Solution. TBA
A.3. Sheet 3
Exercise A.27. Show that a schemes X is affine if and only if there are b1, . . . , bn ∈ OX(X) such thatD(bi) is affine and the ideal generated by all the bi is OX(X). (see [Har77, exercise II.2.17])
Solution. TBA
Exercise A.28. Let (X,OX) be a ringed space and F and G be OX -modules. Show that (F ⊗OX G)x ∼=Fx ⊗Ox Gx.
Solution. TBA
Exercise A.29. Let X = SpecA be an affine scheme. Show that HomA(M,N) ∼= HomOX (M, ‹N) forany A-modules M , N .
Solution. TBA
Exercise A.30. Let X be a scheme and let 0→ F → G → E → 0 be an exact sequence of OX -modules.Show that of two of the sheaves are quasi-coherent, then the third one is also quasi-coherent.
Solution. If G and E are quasi-coherent, we can proceed as in Theorem 2.80. For the other two weneed the following observation: If F is quasi-coherent and U ⊆ X is affine, then
0→ F (U)→ G(U)→ E(U)→ 0
is exact. This can be proven either elementarily as in [Har77, Proposition II.5.6] or from Theorem3.15.
Suppose F and G are quasi-coherent. Then the cokernel E is quasi-coherent by the exact sameargument as in Theorem 2.80 (using the above observation).
If F and G are quasi-coherent, we get a commutative diagram (for U ⊆ X affine, using Lemma2.76)
0 flF (U) flG(U) flE(U) 0
0 F |U G |U E |U 0
∼= ∼=
By the five lemma, flG(U)→ G |U is an isomorphism. Hence G is quasi-coherent.
Exercise A.31. Let X = ProjS, where S =⊕
d≥0 Sd is a graded ring that is generated by S1 overS0. Prove that for every integer n, OX(n) is an invertible sheaf.
Solution. TBA
Exercise A.32. Let X = Pnk = Proj k[t0, . . . , tn] where k is a field, and let Y be the closed sub-kscheme defined by the ideal generated by a homogeneous polynomial F of degree d. Show that theideal sheaf IY of Y is isomorphic to OX(−d).
57
Solution. Let S = k[t0, . . . , tn]. Then the map p → pF is an isomorphism of graded S-modules
S(−d)→ (F ) ⊆ S. Thus the ideal sheaf (F ) defining Y is isomorphic to ‡S(−d) = OX(−d).
A.4. Sheet 4
Exercise A.33. Let f : X → Y be a continuous map of topological spaces. Let F be a flasque sheafin Sh(X). Show that f∗F is flasque.
Proof. Let V ⊆ U ⊆ Y be open subsets. Then the restriction map f∗F (U) → f∗F (V ) is justF (f−1(U))→ F (f−1(V )), which is surjective by assumption.
Exercise A.34. Let X be a closed subset of a topological space Y and f : X → Y the inclusion. Provethat for any sheaf F on X we have Hp(X,F ) = Hp(Y, f∗F ) for every p.
Proof. Let I • be a flasque resolution of F on X. Then f∗I • is a flasque resolution of f∗F : forp ∈ X, (f∗F )p = Fp and for p /∈ X, sheaf (f∗F )p = 0, and hence the complex stays exact. SinceF (X) = f∗F (Y ), the cohomology groups are the same.
Exercise A.35. Let X be a Noetherian scheme and let Q(X) be the category of quasi-coherentschemes on X. Show that Q(X) has enough injectives.
Proof. Let F be a quasi-coherent sheaf on X. We need to embed it into a quasi-coherent injective(in Q(X)) sheaf I . Fix a finite cover of X by open affine subschemes Ui and let ιi be the inclusionsUi → X.
There exist OX(Ui)-modules Mi such that F |Ui =›Mi. Since the category of OX(Ui)-modules has
enough injectives, we can find injective modules Ii such that Mi → Ii. Set I =⊕
i ιi∗Ii. We have
injections ›Mi → Ii and thus morphisms F → ιi∗Ii. Taking the direct sum over i gives an injectivemorphism F → I .
We need to show that I is injective. Let 0 → G → H be an injection of quasi-coherent sheaves
on X and assume that we have a morphism G → I . Any morphism G → ιi∗Ii corresponds to a
morphism G |Ui → Ii. As Ii is injective, we can extend this morphism to a morphism H |Ui → Ii.Taking the direct sum over i of these morphisms, we obtain a morphism H → I which extends G → I .Thus I is indeed injective.
Exercise A.36. A morphism f : X → Y is said to be affine if for any open affine subscheme U ⊆ Y ,f−1U is affine. Now let f : X → Y be an affine morphism of separated Noetherian schemes. Provethat for any quasi-coherent sheaf F on X, we have Hp(Y, f∗F ) ∼= Hp(X,F ) for every p.
Solution. We can compute sheaf cohomology of F and f∗F using Cech cohomology. So let U be afinite open affine cover of Y . The U′ = f−1U : U ∈ U is a cover of X by open affine subschemes.Note that Cp(U′,F ) = Cp(U, f∗F ). Hence the Cech cohomology groups of F on X and of f∗F on Yagree.
Exercise A.37. Let X be a closed subscheme of P2k defined by the ideal of a homogeneous polynomial F
of degree d where k is a field. Show that dimkH0(X,OX) = 1 and dimkH
1(X,OX) = 12 (d−1)(d−2).
Solution. By exercise A.32, we have an exact sequence
0→ OP2k(−d)→ OP2
k→ i∗OX → 0,
58
where i : X → P2k is the closed immersion. The corresponding long exact sequence starts with
0 H0(P2k,OP2
k(−d)) H0(P2
k,OP2k) H0(P2
k, i∗OX(d))
H1(P2k,OP2
k(−d)) H1(P2
k,OP2k) H1(P2
k, i∗OX(d))
H2(P2k,OP2
k(−d)) H2(P2
k,OP2k) H2(P2
k, i∗OX(d))
We have already calculated most of these groups in Theorem 3.28. The only two non-vanishingsegments of the sequence are
0→ H0(P2k,OP2
k)→ H0(P2
k, i∗OX)→ 0,
0→ H1(P2k, i∗OX)→ H2(P2
k,OP2k(−d))→ 0.
So (using Exercise A.34 and again Theorem 3.28) we have
H0(P2k,OX) ∼= H0(P2
k,OP2k) ∼= k
and
H1(P2k,OX) ∼= H2(P2
k,OP2k(−d)) ∼= H0(P2
k,OP2k(d− 3)).
The last cohomology group is isomorphic to the k-vector space of monomials of degree d − 3 in 3
variables, which has dimension(d−3+2
2
)= (d−1)(d−2)
2 .
Exercise A.38. Let X be a topological space, U = (Ui)i∈I a finite open cover, and F a sheaf on X.
Assume that for each p ≥ 0 and each i0 < · · · < ip we have H l(Ui0,...,ip ,F |Ui0,...,ip ) = 0 whenever
l > 0. Show that there are isomorphisms H l(X,F ) ∼= H l(U,F ) for any l ≥ 0.
Solution. We will work along the lines of the proof of theorem 3.26.
For each i ∈ I, there exists an injective sheaf Ii on Ui such that F |Ui injects into Ii. Let Gi be thedirect image of Ii under the inclusion map Ui → X. Set G =
∏Gi. Then G is a flasque sheaf on X
and F injects into G . Let H = G/F . So we have a short exact sequence
0→ F → G → H → 0.
Consider any sequence i0 < i1 < · · · < ip. The long exact sequence of cohomology gives
0→ F (Ui0,...,ip)→ G(Ui0,...,ip)→ H (Ui0,...,ip)→ 0
since H1(Ui0,...,ip ,F |Ui0,...,ip ) = 0. Since also all higher cohomology groups of the flasque sheaf Gvanish, the same long exact sequence shows that H l(Ui0,...,ip ,H |Ui0,...,ip ) = 0 for all l ≥ 1, i.e. that
the assumption of the theorem also holds for H . The direct sum of all these sequences gives an exactsequence
0→ C•(U,F )→ C•(U,G)→ C•(U,H )→ 0
By theorem 1.37 we get a long exact sequence.
· · · → H l(U,F )→ H l(U,G)→ H l(U,H )→ H l+1(U,F )→ · · ·
Since G is flasque, H l(U,G) = 0 = H l(X,G) for all l ≥ 1 (theorems 3.9 and 3.25). In particular, we
59
have
H0(U,F ) H0(U,G) H0(U,H ) H1(U,F ) 0
H0(X,F ) H0(X,G) H0(X,H ) H1(X,F ) 0
We already know that H0(U,F ) = H0(X,F ) = F (X) (Theorem 3.17; the same is true for G and H ).The maps on the 0-th cohomology groups are just the maps on global sections. So the cokernels arethe same, i.e. H1(U,F ) ∼= H1(X,F ) and the analogous statement holds for H (since it also fulfillsthe assumptions). Parts of the long exact sequence look like
0 H l(U,H ) H l+1(U,F ) 0
0 H l(X,H ) H l+1(X,F ) 0
This allows us to do induction to show the result for all l.
60
B. Some Mathematicians
The information here is mainly taken from Wikipedia. I tried to add German pronunciations whenthey where not available in Wikipedia. However, my knowledge of the IPA is very limited, so theremight be errors.
• Pierre Cartier [French pronunciation] (1932 (Sedan, France) –)
• Eduard Cech ["Eduart "tSEx] (1893 (Stracov, Bohemia; now in Czech Republic) – 1960 (Prague))
• Alexander Grothendieck [alEk"sand5 göotEn"di:k] (1928 (Berlin, Germany) – )
• Amalie Emmy Noether ["nø:t5] (1882 (Erlangen, Germany) – 1935 (Bryn Mawr, Pennsylvania,USA))
• Charles Emile Picard [French pronunciation] (1856 (Paris, France) – 1941 (Paris))
• Georg Friedrich Bernhard Riemann ["öi:man] (1826 (Breselenz, Germany) – 1866 (Selasca,Italy))
• Gustav Roch [öox] (1839 (Leipzig, Germany) – 1866 (Venice, Italy))
• Jean-Pierre Serre [French pronunciation] (1926 (Bages, France) – )
• Andre Weil [adöe vEj] (1906 (Paris, France) – 1998 (Princeton, USA))
• Oscar Zariski (born Oscher Zaritsky) [?] (1899 (Kobrin, Russia; today Belarus) – 1986 (Brookline,Massachusetts, USA))
61
C. Changes
1.0.1 – 2010-08-10
• Small error in the proof of Theorem 2.23.
• Typos.
1.0 – 2010-06-11
• Added lemma 2.5 (quasi-compactness off principal open subsets), therefore changing thenumbering in chapter 2 by one.
• Corrected many errors. In particular the statement of Theorem 3.15 was not entirely as inthe lectures.
0.3.2 – 2010-06-04
• Some solutions for the example sheets.
• Some typographical improvements.
0.3.1 – 2010-05-08
• Some solutions for the example sheets.
• Some small corrections an improvements (removed an incorrect remark).
0.3 – 2010-04-24
• Added the missing assumption “Noetherian” to Theorem 3.13.
• Split out two lemmas from Theorem 3.9. Note that this changed all theorem numbers inChapter 3.
• Added a missing sheafification in the proof of Theorem 3.6.
• Minor corrections and improvements.
0.2 – 2010-02-26
• Changed the license to CC BY-SA (somehow forgot to add the Share Alike part).
• Construction of the fibred product for affine schemes (theorem 2.33). Examples for fibres.
• One example of a separated morphism from lectures.
• Added information about some of the mathematicians involved (if you can provide betterpronunciation guides or want to write biographies, please write me an email).
• Added the remaining part of the proof of theorem 2.91.
62
Bibliography
[Con78] John B Conway. Functions of One Complex Variable, volume 11 of Graduate Texts inMathematics. Springer, New York, 1978.
[EH00] David Eisenbud and Joe Harris. The Geometry of Schemes. Springer, New York, 2000.
[Eis95] David Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry, volume150 of Graduate Texts in Mathematics. Springer, 1995.
[GD60] Alexander Grothendieck and Jean Dieudonne. Elements de geometrie algebrique: I. Lelangage des schemas. Publications mathematiques de l’I.H.E.S., 1960. Available from:http://www.numdam.org/item?id=PMIHES_1960__4__5_0.
[GD61a] Alexander Grothendieck and Jean Dieudonne. Elements de geometrie algebrique: II. Etudeglobale elementaire de quelques classes de morphismes. Publications mathematiques del’I.H.E.S., 1961. Available from: http://www.numdam.org/item?id=PMIHES_1961__8__5_0.
[GD61b] Alexander Grothendieck and Jean Dieudonne. Elements de geometrie algebrique: III. Etudecohomologique des faisceaux coherents, Premiere partie. Publications mathematiques del’I.H.E.S., 1961. Available from: http://www.numdam.org/item?id=PMIHES_1961__11_
_5_0.
[Har77] Robin Hartshorne. Algebraic Geometry, volume 52 of Graduate Texts in Mathematics.Springer, New York, 1977.
[Liu02] Qing Liu. Algebraic Geometry and Arithmetic Curves. Oxford University Press, New York,2002.
63
List of Notation
∼ linear equivalence of divisors 33
|U restriction map 5
AnA affine n-space over A 16
Ab category of Abelian groups 9
C•(U,F ) “sheafified” Cech complex 45
C•(U,F ) Cech complex 42
∆ diagonal morphism 22
D(f) principal open set 13
D+(b) 20
degM degree of the Weil divisor M 50
deg d degree of a Cartier divisor 50
Div(X) group of Cartier divisors (modulo principal divisors) 33
(f) Weil divisor of f ∈ K(X) 49
F ⊗OX G tensor product of OX -modules 9
F sheaf 5
F + sheaf associated to the presheaf F 6
f∗F direct image sheaf 7
F ⊕ G direct sum of sheaves 53
f∗G inverse image of an OY -module 9
f−1G inverse image sheaf 8
F (n) F ⊗OX OX(n) 31
Fq finite field with q elements 54
Fx stalk at x 6
hi(A•) cohomology of the complex A• 10
Hi(X,F ) i-th cohomology group of F 38
Hom(F , G) sheaf of morphisms 53
HomOX (F ,G) OX -module morphisms F → G 8
64
H•(U,F ) Cech cohomology groups 42
IY ideal sheaf of Y 30
k(x) residue field at x 8
K(X) function field of X 18
mx maximal ideal at x 8
M sheaf associated to the module M (on ProjS) 21
M sheaf associated to the module M 14
M(b) degree zero elements in Mb 4
M(p) degree zero elements in Mp 4
M(X) category of OX -modules 9
ΩB/A module of relative differential forms of B over A 34
OX(D) sheaf associated to the divisor D 33
OX(n) fiS(n) for X = ProjS 31
ΩX/Y sheaf of relative differential forms 35
PnA projective n space over A 21
Pic(X) Picard group of X 9
ProjS 20
PnY projective n-space over Y 21
Q(X) category of quasi-coherent sheaves on X 58
RiF right derived functors of F 11
M(b) degree zero elements in Sb 4
Sh(X) category of sheaves of Abelian groups on X 9
M(p) degree zero elements in Sp 4
SpecA spectrum of a ring 14
Tx Zariski tangent space at x 54
V (a) closed set of the ideal a 13
V+(a) 20
Ws x ∈ X : s /∈ mx ⊆ (OX)x 17
X ×S Y fibred product of X and Y over S 19
Xy fibre over y 20
65
Index
Abelian category, 9with enough injectives, see enough injec-
tivesacyclic, 11
resolution, 12additive functor, 10affine
morphism, 58scheme, 15, 40, 57space, 16
canonical divisor, 36canonical sheaf, 36Cartier divisor, 33–34, 50
degree, see degree of a Cartier divisorgroup of, 33linear equivalence, see linear equivalenceprincipal, 33
Cech cohomology, 42Cech complex, 42, 45closed immersion, 18, 23, 55
is proper, 26closed subscheme, 18
induced reduced, see induced reduced struc-ture
coherent, 28–31, 40cohomology
Cech, see Cech cohomologygroup of a sheaf, 38–42of a complex, 10
complex, 10Cech, see Cech complex, 45morphism, 10of sheaves, 7short exact sequence, 10
constant sheaf, 7
degreeof a Cartier divisor, 50of a Weil divisor, 50
derivation, 34derived functor, 11, 38diagonal morphism, 22differential form, see also sheaf of relative dif-
ferential forms
differential forms, 35dimension, 40direct image sheaf, 7, 8direct sum of sheaves, 53divisor, see Cartier divisor, 33
Weil, see Weil divisorduality theorem, 50
enough injectives, 11, 37, 38, 58
fibre, 20fibred product, 19
projection map, see projection mapfinite morphism, 56
is proper, 56finite type, see morphism of finite typeflasque, 38
implies acyclic, 39resolution, 40
function field, 18, 56functor
additive, 10left exact, 10right derived, see derived functor
generic fibre, 20generic point, 56genus
of a plane curve, 58gluing
schemes, 55sheaves, 53
graded module, 4, 21graded ring, 4, 20group
of Cartier divisors, see Cartier divisorPicard, see Picard group
Hausdorff, 22
ideal sheaf, 30of a closed immersion, 30
image of a morphism of sheaves, 6, 30immersion
closed, see closed immersionopen, see open immersion
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induced reduced structure, 18injective
object, 10resolution, 11sheaf, 37
is flasque, 38injective sheaf, 38integral scheme, 17inverse image sheaf, 8, 30
of OX -modules, 9invertible sheaf, 9, 34, 57irreducible scheme, 17isomorphism, 8
of presheaves, 6of ringed spaces, 8of schemes, 16of sheaves, 6
kernel of a morphism of sheaves, 6, 30
left exact functor, 10linear equivalence, 33local homomorphism, 4locally ringed space, 8, 15, 16
morphism, 8long exact sequence, 10, 11
morphismaffine, see affine morphismfibre, 20finite, see finite morphismof complexes, 10of finite type, 22, 24of locally ringed spaces, 8of OX -modules, 8of presheaves, 6of ringed spaces, 8of schemes, 16, 53of schemes over S, 18of sheaves, 6
image, see image of a morphism of sheavesinjective, 6kernel, see kernel of a morphism of sheavessurjective, 6, 52
projective, see projective morphismproper, see proper morphismquasi-compact, see quasi-compact mor-
phismquasi-projective, see quasi-projective mor-
phismseparated, see separated morphism
universally closed, see universally closedmorphism
Noetherianscheme, 18, 24topological space, 53
Noetherian scheme, 22
open immersion, 18open subscheme, 18OX -module, 8–9, 37
associated to a module, 14, 27coherent, see coherentdirect image, see direct image sheafinverse image, see inverse image sheafinvertible, see invertible sheafmorphism, see morphism of OX -modulesquasi-coherent, see quasi-coherenttensor product, 9
Picard group, 9, 34plane curve, 58presheaf, 5
associated sheaf, 6image, 6kernel, 6morphism, see morphism of presheaves
prime divisor, 33prime spectrum, 13principal divisor, 33principal open set, 13, 57projection map, 19projective morphism, 21, 26
is proper, 26projective space, 21, 47
over a scheme, 21proper morphism, 24–27
composition, 26properness is local on the base, 56
push-down sheaf, 7, 30
quasi-coherent, 28–31, 40, 58quasi-compact, 13quasi-compact morphism, 27quasi-projective morphism, 21
reduced scheme, 17reduction mod p, 20relative differential form, 34, see also sheaf of
relative differential formsrelative differential forms, 35residue field, 8, 54resolution
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acyclic, see acyclic resolutionflasque, see flasque resolutioninjective, see injective resolution
restriction homomorphism, 5Riemann-Roch theorem, 50right derived functor, 11ringed space, 8–9, 14, 21
morphism, 8
S-scheme, see scheme over Sscheme, 16
affine, see affine schemeintegral, see integral schemeirreducible, see irreducible schememorphism, see morphism of schemesNoetherian, see Noetherian schemeover S, 18reduced, see reduced scheme
section, 5separated morphism, 22–24
separatedness is local on the base, 56sequence
of sheaves, 7, 28sheaf, 5–8, 38
associated to a module, 14, 21associated to a presheaf, 6cohomology, see cohomology group of a
sheafcondition, 5constant, 7direct image, see direct image sheafdirect sum, 53flasque, see flasquegluing, 53inverse image, see inverse image sheafinvertible, see invertible sheafmorphism, see morphism of sheavesof modules, see OX -moduleof morphisms, 53of relative differential forms, 35, 43push-down, see direct image sheafrestriction, 5, 8, 52skyscraper, 8stalk, 6twisted, 31
short exact sequenceof complexes, 10of sheaves, 7
skyscraper sheaf, 8smooth, 49spectrum, 13stalk, 6
structure sheaf, 8, 14, 21subscheme
closed, see closed subschemeopen, see open subscheme
tangent space, 54tensor product of OX -modules, 9terminal object, 19twisted sheaf, 31twisting sheaf of Serre, 31
universally closed morphism, 24
valuation ring, 22Valuative Criterion
for properness, 24for separatedness, 22
Weil divisor, 49, 50degree, see degree of a Weil divisorof a function, 49
Zariski tangent space, 54Zariski topology, 13
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