Professor A.J. de Jong,
Department of Mathematics.
In the Spring of 2020 I am will teach the course on schemes.
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.
- 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:
- Graded ring and graded module
- Picard group
- Open X_s associated to a section of invertible module
- Sections of quasi-coherent modules over X_s
- The isomorphism of a quasi-coherent module F over X = Proj(A)
with the module associated to the graded module associated to F
- We discussed ample invertible modules.
- We defined them as in the Stacks project, see
- We defined finite type modules on ringed spaces, see
- 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
- We sketched ingredients in the proof of the proposition
- We defined Noetherian schemes, see
- We defined coherent modules on Noetherian schemes, see
- We formulated a goal wrt cohomology of coherent modules
on projective schemes over Noetherian rings.
- We will discuss quasi-projectivity and the relationship
of (quasi-)projectivity with the existence of ample invertible modules.
Then we will start talking about cohomology of sheaves of modules.
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:
do 2 exercises from Section
- Due Thursday, February 20 in lecture:
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