Schemes
Professor A.J. de Jong,
Columbia university,
Department of Mathematics.
In the Spring of 2020 I am will teach the course on schemes.
Stay tuned!
Tuesday and Thursday, 10:10 -- 11:25 AM in room 507 math.
The TA is Carl Lian.
Grades will be based on the weekly problem sets and a final exam.
FINAL EXAM I will email these to you on the first day
of the exam period and you need to email back the solutions to me
before the end of the last day of the exam period.
Lectures
- First lecture was about prerequisites, the spectrum of a ring as a locally ringed space, affine schemes, schemes. We gave two examples of non-affine schemes. We discussed the structure sheaf on affine 2 space A^2_k over a field k.
- Second lecture discussed the universal property of an affine scheme in the category of locally ringed spaces. We proved it. We discussed an easy case of glueing of topological spaces / ringed spaces / locally ringed spaces, namely the case where we are just glueing two pieces. We discussed the affine line with 0 doubled and the affine plane with 0 doubled.
- We discussed Proj of a graded ring and the sheaf of modules associated
to a graded module. We defined projective space over a ring and we defined
projective schemes over a ring.
- We discussed quasi-coherent modules on affine schemes and on general
schemes. The category of quasi-coherent modules on Spec(A) is equivalent
to the category of A-modules via the global sections functor. For a ring
map A → B the pushforward and pullback functors on quasi-coherent
modules correspond to the (adjoint) restriction and tensor product functors
on modules. For a morphism of schemes pullback preserves quasi-coherent
modules but in general pushforward does not. However, at the very end of
the lecture we proved that pushforward along a quasi-compact and quasi-separated
morphism of schemes does preserve quasi-coherency.
- We discussed fibre products of schemes, affine morphisms
as well as closed immersions, finite morphisms, and integral morphisms.
- We discussed separation axioms for schemes. We discussed properties
of morphisms of schemes which are "local on the base",
"preserved by base change", or satisfy some "permanence" property.
We defined universally closed morphisms. We proved that projective space
P^n_S is universally closed over its base S. We defined a proper morphism
as one which is of finite type, separated, and universally closed. Thus
P^n_S is proper over S. We finished with a proof of the fact that a
morphism A^1_S → P^n_S over S is never a closed immersion.
- We discussed invertible modules, the Picard group, the open defined
by a section of an invertible module, the graded ring associated to an
invertible module, the graded module associated to a sheaf of modules,
the comparison map being an isomorphism for quasi-coherent modules, and
the fact that quasi-coherent modules on Proj(A) are always associated
to the graded module you get from them. References:
- Invertible modules:
0AFW
01CR
- Graded ring and graded module
01CV
- Picard group
01CX
- Open X_s associated to a section of invertible module
01CY
- Sections of quasi-coherent modules over X_s
01PW
- The isomorphism of a quasi-coherent module F over X = Proj(A)
with the module associated to the graded module associated to F
0AG5
- We discussed ample invertible modules.
- We defined them as in the Stacks project, see
01PR
- We defined finite type modules on ringed spaces, see
01B4
- We stated a proposition that an invertible module L is ample
if and only if for every finite type quasi-coherent mdoule F there
is an n > 0 such that F ⊗ L^n is globally generated, see
01Q3
- We sketched ingredients in the proof of the proposition
- We defined Noetherian schemes, see
01OU
- We defined coherent modules on Noetherian schemes, see
01XY
01Y7
- We formulated a goal wrt cohomology of coherent modules
on projective schemes over Noetherian rings.
- We defined a scheme X over a ring R to be quasi-projective over R
if there exists a quasi-compact immersion of X into P^n_R for some n.
We discussed how the existence of an ample invertible sheaf is related
to being quasi-projective: namely, for a scheme X over a ring R
the following are equivalent
- X is quasi-projective over R,
- X is of finite type over R and has an ample invertible module
- X is a quasi-compact open subscheme of a scheme projective over R
We discussed the functorial characterization of P^n_R. Then we started
discussing cohomology of sheaves of modules on ringed spaces. We stated and
proved a lemma on locality of cohomology.
- Lemma: injective objects of abelian categories are preserved
under functors having an exact left adjoint. Example 1: restriction of
an injective O_X module to an open. Example 2: an injective O_X module
is an injective object of the category of presheaves of modules. We
discussed Cech cohomology. We proved that higher Cech cohomology of an
injective presheaf of O_X modules vanishes. We deduced that the higher
Cech cohomology groups are the right derived functors of the 0th
Cech cohomology on the category of presheaves of modules. We stated
and proved 01EW.
We deduced that the higher cohomology of a quasi-coherent module over an
affine scheme is zero (using a commutative algebra result from last
semester, namely 01X9).
- We proved that Cech cohomology for an open covering U of X
computes cohomology of a sheaf of modules F if the cohomology of F
over the finite intersections of opens in U is zero. This in particular
applies to quasi-coherent modules over a separated scheme X and any
affine open covering of X. As an application we proved that the
higher direct images of a quasi-coherent sheaf by a quasi-compact
and separated morphism are quasi-coherent. We explained the relationship
between the Cech complex and the alternating (or ordered) Cech complex.
Finally, we computed the cohomology of the twists of the structure sheaf
on P^n_R. We did this using the following relation between
the alternating Cech complex and the Koszul complex: Let A be a ring.
Let I = (f_1, ..., f_r) be an ideal generated by r elements. Let
U = Spec(A) - V(I) which is covered by the opens D(f_i). Then
the corresponding "extended" alternating Cech complex for the structure sheaf
is the colimit of the Koszul complexes K_*(A, f_1^n, ..., f_r^n) over n.
See 0913.
- We talked about the cohomology of coherent sheaves on projective space
and on projective schemes over Noetherian rings. We finished with a very
general version of the existence of dualizinng modules, i.e., modules
representing the functor of "duals of top cohomology" on the category
of quasi-coherent modules.
- We revisited the Yoneda lemma. We discussed the universal element
over P^n_R if you think of P^n_R as the scheme representing the functor
of tuples (L, s_0, ..., s_n) as usual. We proved that any contravariant
functor F : Mod_R → Sets which transforms colimits into limits is
representable, in fact by F(R). Lemma: quasi-coherent modules
on Noetherian schemes are filtered colimits of coherent modules.
Lemma: there is a set of isomorphism classes of coherent modules
on any given Noetherian scheme. We discussed again the general
existence of 'dualizing modules' on finite type schemes over
Noetherian rings. We discussed why O(-n - 1) is a dualizing module
on projective space over a field.
- We proved that for X = P^n_k where k is a field we have a
canonical functorial duality between the k-linear dual of H^{n - i}(X, F)
and the vector space Ext^i(F, O(-n - 1)) for coherent modules F on X.
The proof used that both
form universal delta functors (with index i = 0, 1, 2, ...) on
the category of coherent modules on X. We discussed how the pairings
one gets can be viewed as composition of Ext classes follows
by "integration" by some map t : H^n(X, O(-n - 1)) → k.
Or in case F is finite locally free, we can view the pairings
as first using the cup product, then the evalution map, and then
using t. Finally, we defined dualizing sheaves or
dualizing modules for proper schemes over k and we
stated a theorem that they always exist for projective schemes
over k and how to compute them.
- See the first set of lecture notes.
- See the second set of lecture notes
- Here are the lecture notes:
number-1
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- Here are the lecture notes:
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- Here are the lecture notes
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- Here are the lecture notes
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- Here are the lecture notes
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Homework: (make sure you refresh the page)
- Due Thursday, January 30 in lecture: do 8 exercises from Section Tag 0280
- Due Thursday, February 6 in lecture: do 4 of the exercises from
Section Tag 0280
and try to do
Exercise Tag 02A1
- Due Thursday, February 13 in lecture:
029Q
069T
028Z and
do 2 exercises from Section
0293
- Due Thursday, February 20 in lecture:
02A3
02AL
0DT4
029U
0D8V
- Due Thursday, February 27 in lecture:
0D8Q
0D8R
0D8S
0D8T
0DAK
- Due Thursday, March 5 in lecture:
0DB9
(hard; try to just show it holds for the dimensions although I am
not sure that this is easier)
0DBA
0DCF
0DCG
0DCH
- Due Thursday, March 12 in lecture:
0DD1
0EEN
0DT3
(you may need to look up properties of dualizing modules in Hartshorne
in order to answer this and this is encouraged)
0EEP
0EEQ
- Due Thursday, March 26 in Carl's mailbox.
Carefully prove Lemma 2.1 from
the second set of lecture notes.
Try to do
0DCI
0DCJ
- Due Thursday, April 2 in Carl's mailbox.
- Find a property P of ring maps which satisfies conditions
(1)(a) and (1)(c) of
Definition 01SR
as well as the condition that P(R → A) implies P(R → A_a)
for all a in A but P is not local in the sense of the definition.
(I haven't tried this exercise myself.)
- In each of the following cases find the open subset of
Spec(A) where the morphism Spec(A) → Spec(R) is smooth.
- A = R[x, y]/(x^2 y + x y^2 - xy)
- A = R[x, y]/(x^2 - y^3)
- A = R[x, y, z, w]/(xz, yz, wz)
- Exercise 0AAS
- Due Thursday, April 9 in Carl's mailbox:
02FE
02FF
0EGP (this may be
an impossible exercise). Final question:
Let X be a closed subscheme of P^1 x P^1 over
a field given by a nonzero bihomogeneous polynomial of bidegree (a, b).
Compute the dualizing sheaf of X in terms of the invertible
modules O_X(m, n) = pullback of O_{P^1 x P^1}(n, m) by X → P^1 x P^1.
- Due Thursday, April 16 in Carl's mailbox:
- Compute the divisor class group of A^n over a field.
- Compute the divisor class group of P^n over a field.
- Aside and elucidation for the next 3 questions:
if X is smooth projective over a field K, then
the condition H^0(X, O_X) = K is equivalent to asking
X to be geometrically irreducible. You may use
this in the questions above if you wish to do so.
- Find a smooth projective variety X over Q_2, the field of 2-adic numbers,
with H^0(X, O_X) = Q_2 which has no Q_2-rational point.
- Find a smooth projective variety X over C(t), the field of
rational functions in one variable over the complex numbers,
with H^0(X, O_X) = C(t) which has no C(t)-rational point.
- Does there exist a nonalgebraically closed field K such
that every smooth projective variety X over K with H^0(X, O_X) = K
has a K-rational point? (This is an impossible question; please skip.)
- Let k be a field. Let C be a closed subscheme of P^2_k. Assume
C is smooth and every irreducible component of C has dimension 1.
Show that C is geometrically integral. (Of course you may use results
from Hartshorne and/or the Stacks project.)
- Due Thursday, April 23 in Carl's mailbox:
- Let A → B → C be injective local homomorphisms of
discrete valuation rings. Show that as directly as possible that
length(C/m_A C) is the product of length(B/m_A B) and length(C/m_B C).
Motivation: This is the algebra fact that shows that pullbacks of
divisors is functorial for nonconstant morphisms of nonsingular curves.
- Let A → B be an injective local homomorphism of discrete
valuation rings which is also finite. Let L/K be the induced (finite)
extension of fraction fields. Denote ord_A : K^* → Z the discrete
valuation of A and similarly for ord_B : L^* → Z.
Let l/k be the induced (finite) extension
of residue fields. Let g ∈ L be a nonzero element with
norm h = Nm_{L/k}(g) ∈ K. Show that [l : k] ord_B(g) is equal
to ord_A(h). Hints: first show that the norm of a unit is a unit,
then show that if g is the image of a uniformizer of A the result is
true, then conclude in general because both sides are additive.
Motivation: This is the algebra fact that shows
that the pushforward of a principal divisor by a finite morphism
of nonsingular curves is the principal divisor associated to the
norm of the function upstairs.
- Let f : X → Y be a finite locally free morphism of degree d,
i.e., a finite morphism such that f_*O_X is a finite locally free
O_Y-module of rank d. Show that for L in Pic(X) the pushforward f_*L
is a finite locally free O_Y-module of rank d.
- Same assumption and notation as previous exercise. Show that
for L, M in Pic(X) we have ∧(f_*L) ⊗ ∧(f_*M) =
∧(f_*(L ⊗ M)) ⊗ ∧(f_*O_X) where ∧
means top exterior power over O_Y and the tensor products are taken
over O_X or O_Y depending on whether the modules live on X or on Y.
This is not easy! Motivation: This shows that the map sending L to
∧(f_*L) ⊗ ∧(f_*O_X)⊗ -1
defines a homorphism of abelian groups Pic(X) → Pic(Y).
- Fix an algebraically closed ground field k. Let C be a smooth
plane curve of degree d > 3. Show that there does not exist
a degree 2 morphism from C to P^1. (A plane curve means a curve
which is isomorphic to a closed subscheme of P^2.)
- Choose any field k you like. Find explicitly a smooth projective curve
over k which is not a plane curve.
As usual please feel free to make additional assumptions to make
the exercises easier. Also, please don't do all of these.
- Due Thursday, April 30 in Carl's mailbox: Try to to as much as
possible of Exercise
0EGR
where the curve is supposed to be a smooth projective curve of genus
7 and you get to pick the algebraically closed ground field.
Books to read/browse. Some of these are more advanced.
- Algebraic Geometry by Hartshorne
- EGA by Grothendieck
- the Stacks project online
- Basic Algebraic Geometry by Shavarevich
- Algebraic Geometry by Goertz and Wedhorn
- Algebraic Geometry and Arithmetic Curves by Qing Liu
- Red Book by Mumford
- Curves and their Jacobians by Mumford
- Abelian Varieties by Mumford
- Lectures on Curves on an Algebraic Surface by Mumford
- ACGH: Geometry of Algebraic Curves by Arbarello, Cornalba, Griffiths, and Harris
- Algebraic Curves by Fulton
- Algebraic Varieties by Kempf
- Undergraduate algebraic geometry by Reid
- Algebraic Geometry by Harris
- Principles of Algebraic Geometry by Griffiths and Harris
- Introduction to Algebraic Geometry by Cutkosky
- The Arithmetic of Elliptic Curves by Silverman
- Rational curves on algebraic varieties by Kollar
- Neron Models by Bosch, Lutkebohmert, and Raynaud
