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MATH 632: ALGEBRAIC GEOMETRY II
COHOMOLOGY ON ALGEBRAIC VARIETIES
˘
LECTURESBYPROF.MIRCEAMUSTAT¸A;NOTESBYALEKSANDERHORAWA
These are notes from Math 632: Algebraic geometry II taught by Professor Mircea Musta¸ta˘
A
in Winter 2018, LT X’ed by Aleksander Horawa (who is the only person responsible for any
E
mistakes that may be found in them).
This version is from May 24, 2018. Check for the latest version of these notes at
http://www-personal.umich.edu/~ahorawa/index.html
If you find any typos or mistakes, please let me know at ahorawa@umich.edu.
The problem sets, homeworks, and official notes can be found on the course website:
http://www-personal.umich.edu/~mmustata/632-2018.html
This course is a continuation of Math 631: Algebraic Geometry I. We will assume the
material of that course and use the results without specific references. For notes from the
classes (similar to these), see:
http://www-personal.umich.edu/~ahorawa/math_631.pdf
and for the official lecture notes, see:
http://www-personal.umich.edu/~mmustata/ag-1213-2017.pdf
Thefocusofthepreviouspartofthecoursewasonalgebraicvarietiesanditwillcontinuethis
course. Algebraic varieties are closer to geometric intuition than schemes and understanding
them well should make learning schemes later easy. The focus will be placed on sheaves,
technical tools such as cohomology, and their applications.
Date: May 24, 2018.
1
˘
2 MIRCEAMUSTAT¸A
Contents
1. Sheaves 3
1.1. Quasicoherent and coherent sheaves on algebraic varieties 3
1.2. Locally free sheaves 8
1.3. Vector bundles 11
1.4. Geometric constructions via sheaves 13
1.5. Geometric vector bundles 16
1.6. The cotangent sheaf 19
2. Normal varieties and divisors 26
2.1. Normal varieties 26
2.2. Geometric properties of normal varieties 28
2.3. Divisors 35
2.4. Weil divisors 35
2.5. Cartier divisors 40
2.6. Effective Cartier divisors 44
3. Cohomology 45
3.1. Derived functors 45
3.2. Cohomology of sheaves 54
3.3. Higher direct images 56
3.4. Cohomology of quasicoherent sheaves on affine varieties 59
3.5. Soft sheaves on paracompact spaces 61
3.6. De Rham cohomology and sheaf cohomology 62
3.7. Introduction to spectral sequences 66
3.8. The Grothendieck spectral sequence 69
˘
3.9. Cech cohomology 72
3.10. Coherent sheaves on projective varieties 74
3.11. Cohomology of coherent sheaves on projective varieties 81
n
4. Morphisms to P 88
4.1. Linear systems 92
4.2. Ample and very ample line bundles 93
MATH 632: ALGEBRAIC GEOMETRY II 3
5. Ext and Tor functors 96
6. Depth and Cohen-Macaulay rings 101
6.1. Depth 101
6.2. The Koszul complex 106
6.3. Cohen–Macaulay modules 110
7. More on flatness and smoothness 113
7.1. More on flatness 113
7.2. Generic flatness 116
7.3. Generic smoothness 118
8. Formal functions theorem 121
8.1. Statement and consequences 121
8.2. Proof of the Formal Functions Theorem 8.1.2 125
9. Serre duality 128
9.1. Preliminaries 128
9.2. Examples of Serre duality 129
10. Algebraic curves 133
10.1. Riemann–Roch Theorem 134
10.2. Classification of curves 137
10.3. Morphisms between algebraic curves 137
11. Intersection numbers of line bundles 139
11.1. General theory 139
11.2. Intersection numbers for curves and surfaces 143
12. Introduction to birational geometry 145
12.1. Preliminaries 145
12.2. Birational maps 146
12.3. Smooth blow-ups 148
12.4. Picard group of a smooth blow-up 148
˘
4 MIRCEAMUSTAT¸A
1. Sheaves
1.1. Quasicoherent and coherent sheaves on algebraic varieties. The object we will
consider is a ringed space (X,O ) where X is an algebraic variety and O is the sheaf of
X X
regular functions on X.
Recall that OX-module is a sheaf F such that for any open subset U ⊆ X, F(U) is an
OX(U)-module and if U ⊆ V is an open subset, then
F(V)→F(U)
is a morphism of OX(V)-modules, where the OX(V)-module structure on F(U) is given by
the restriction map OX(V) → OX(U).
Fact 1.1.1. If X has a basis of open subsets U, closed under finite intersection, giving an
OX-module on X is equivalent to giving OX(U)-modules F(U) for any U ∈ U with restriction
maps between these which satisfy the sheaf axiom.
Example 1.1.2. If X is an affine variety and A = O(X), M is an A-module, then we obtain
f f
an O -module M such that for any f ∈ A, Γ(D (f),M) = M with the restriction maps
X X f
induced by localization. (The sheaf axiom was checked in Math 631.)
Weget a functor
{A-modules} → {O -modules}
X
f
M7→M.
Definition 1.1.3. Suppose X is affine. An O -module F is quasicoherent (coherent) if
X
∼f
F =M for some (finitely-generated) A-module M.
∼ e
Example 1.1.4. The sheaf of regular functions O on X is a coherent sheaf with O =A.
X X
If V ⊆ X is irreducible and closed with p = IX(V),
f f
M = lim Γ(U,M)= limM =M .
V −→ −→ f p
U open f6∈p)
U∩V6=∅
Remarks 1.1.5.
(1) Given any OX-module M, we have a canonical morphism of OX-modules:
^
Φ : Γ(X,M)→M
M
given on D (f) by the unique morphism of A -modules
X f
Γ(X,M)f →Γ(DX(f),M)
induced by the restriction map.
Then the following are equivalent:
• Mis quasicoherent,
• ΦM is an isomorphism,
• for any f ∈ A, the canonical map
Γ(X,M)f →Γ(DX(f),M)
is an isomorphism.
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