12/13/17: Application of our description of morphisms from a scheme to
an affine scheme: every scheme has a unique morphism to Spec
ℤ. Relative projective
n-spaces ℙnS. [Class notes. Vakil:
Sections 4.4.9-10]
12/13/17: Application of our description of morphisms from a scheme to
an affine scheme: every scheme has a unique morphism to Spec
ℤ. Relative projective
n-spaces ℙnS. [Class notes. Vakil:
Sections 4.4.9-10]
12/11/17: Homework 5 is due. Morphisms from a scheme to an affine
scheme are determined by homomorphisms of rings of global
sections. The spectrum of a quasi-coherent algebra, relative affine
n-spaces 𝔸nS. [Class
notes. Hartshorne: Exercises II.2.4 and II.5.17(c)]
12/06/17: Examples: an affine line with a doubled point and the
projective line. The proof of the theorem on scheme gluing. Gluing
morphisms of schemes. [Class notes. Vakil: Sections
4.4.6-8. Hartshorne: Example II.2.3.6, Exercise II.2.12, Step 3 in the
proof of Theorem II.3.3]
12/04/17: The proof of the theorem on the coherence of direct
image. Construction of schemes by gluing: generalities. Example: an
affine line with a doubled point. [Class notes. Vakil: Sections
4.4.4-5. Hartshorne: Exercise II.1.22, Sections II.2.3.5-6, Exercise
II.2.12]
12/1/17: Irregular class meeting. The canceled class is made up
at regular time on Friday in regular classroom. Direct and inverse
images of quasi-coherent sheaves. [Class notes. Vakil: Sections
16.1-3. Hartshorne: Sections II.5.2 and II.5.8, Exercise II.5.3]
11/29/17: Class canceled: the instructor is sick.
11/27/17: Direct and inverse images
of OX-modules. Exactness properties of direct and
inverse image. [Class notes. Vakil: Section 16.3.4-5. Hartshorne: The
beginning of Section II.5 (pp. 109-110)]
11/22/17: Class canceled for Wednesday evening before
Thanksgiving. Happy holiday!
11/20/17: A remark on locally finitely presentable sheaves on a
general scheme. Direct and inverse images of sheaves. [Class
notes. Vakil: Sections 2.2.H, 2.6, 2.7. Hartshorne: P. 65, Exercise
II.1.18]
11/15/17: Homework 4 is due. Quasi-coherent sheaves on a general
scheme. [Class notes. Vakil: Section 13.4. Hartshorne: Section II.5
from 5.6 through 5.7]
11/13/17: Quasi-coherent and coherent sheaves on a general scheme,
continued. [Class notes. Vakil: Sections 13.1-3, 13.6, skipping
13.3.4. Hartshorne: Section II.5 through 5.5, skipping direct and
inverse images f* and f*]
11/8/17: Quasi-coherent and coherent sheaves on a general
scheme. [Class notes. Vakil: Sections 2.5, 13.2-3. Hartshorne:
pp. 63-64, Section II.5 through 5.6, skipping direct and inverse
images f* and f*]
11/6/17: Abelian properties of sheaves: kernels, cokernels, exact
sequences. Being associated with a module is a local property of a
sheaf on an affine scheme. [Class notes. Vakil: Sections 2.5,
13.2-3. Hartshorne: pp. 63-64, Section II.5 through 5.6, skipping
quasi-coherent sheaves on nonaffine schemes and direct and inverse
images f* and f*]
11/1/17: Other forms of Hilbert's Nullstellensatz. Quasi-coherent
sheaves. [Class notes. Vakil: Sections 3.2.3-I, 4.1.D-4. Hartshorne:
Section II.5 through 5.1, skipping direct and inverse
images f* and f*, and II.5.5]
10/30/17: The homework is due. The affine n-space and another form of
Hilbert's Nullstellensatz. [Class notes. Vakil: Sections 3.2.3-I]
10/25/17: Affine schemes, general schemes and their morphisms. [Class
notes. Vakil: Sections 4.3, 6.3 through 6.3.D. Hartshorne: Section
II.2 before graded rings and Proj]
10/23/17: Ringed spaces, their morphisms. Locally ringed
spaces. [Class notes. Vakil: Sections 2.3.A, 2.2.13, 6.2-3 before
6.3.2. Hartshorne: pp. 71-72, Section II.2 through p. 73]
10/18/17: Proving that the extension of the sheaf OX
from basic open sets in X = Spec A to all opens sets is a
sheaf: the case of a general covering. Sheaves associated to
A-modules. The stalk. The étalé space of a
presheaf. Sheafification. [Class notes. Vakil: Sections 2.7, 2.2.3-6,
2.2.11, 2.4. Hartshorne: Section II.5, skipping morphisms, graded
rings and Proj]
10/16/17: The homework is due. Discussion of homework. Extending the
sheaf OX from basic open sets in X = Spec A
to all opens sets. Proving it is a sheaf (the case of a covering of
an open set by basic open sests). [Class notes. Vakil: Section
2.7. Hartshorne: pp. 70-72]
10/11/17: Inductive and projective limits, a.k.a. colimits and
limits. [Class notes. Vakil: Section 1.4]
10/9/17: Constructing the structure sheaf on the spectrum of a ring:
sheaf Axiom 2 (gluing) of the functor D(f) ↦ Af on
the basic open sets. Construction of the structure sheaf on Spec
A. [Class notes. Hartshorne: Section II.2 (pp. 70-72). Vakil: Section
4.1 after Exercise 4.1.B]
10/4/17: Constructing the structure sheaf on the spectrum of a ring:
sheaf Axiom 1 (locality) of the functor D(f) ↦ Af
on the basic open sets. [Class notes. Hartshorne: Section II.2
(pp. 70-72). Vakil: Section 4.1 through Exercise 4.1.B]
10/2/17: Maps of spectra, continued. Sheaves. [Class
notes. Hartshorne: Section II.1. Vakil: Sections 2.1, 2.2, 3.2.9-10]
9/27/17: The first homework is due. Discussion of homework. Maps of
spectra. [Class notes. Hartshorne: Section II.2 before graded rings,
ignoring sheaves. Vakil: Sections 3.2.9-10]
9/25/17: Zariski topology: compactness. Irreducibility. Maps of
spectra under quotients. The generic point of an irreducible
set. [Class notes. Hartshorne: Section II.2 before graded rings,
ignoring morphisms and sheaves for the time being. Vakil: Sections
3.3, 3.6]
9/20/17: More on spectra and their topology. The ideal-closed set
correspondence. Kolmogorov separation. [Class notes. Hartshorne:
Section II.2 before graded rings, ignoring morphisms and sheaves for
the time being. Vakil: Sections 3.2 skipping 3.2.9-10, 3.7]
9/18/17: The spectrum of a ring: basic closed and open sets, the
Zariski topology and its properties. [Class notes. Hartshorne: Section
II.2 before graded rings, ignoring morphisms and sheaves for the time
being. Vakil: Sections 3.1, 3.2 through 3.2.8, 3.4, 3.5]
9/15/17: Reminder: no class meetings on Fridays anymore.
9/13/17: Coordinate interpretation of morphisms of solution
functors. Gluing solution functors. The idea of a scheme. [Class
notes. Hartshorne: pp. 58-59. Vakil: Intro to Section 3.1.]
9/11/17: Maps between solution sets of different polynomial
systems. Yoneda's Lemma. The equivalence of the category of solution
functors of polynomial systems to the opposite of the category of
finitely presentable k-algebras. [Class notes. Vakil: Sections
1.3.10-11. Compare our theorems to Proposition 3.5 and its corollaries
in Hartshorne's Chapter I.]
9/8/17: Class meeting canceled and replaced by a talk by Professor
Yuri Tschinkel on Rationality Problems (regular classroom, regular
time). [Preprint https://arxiv.org/abs/1708.05699.]
9/6/17: Introduction. Syllabus handed out. Motivation (a modern twist
on Classical Algebraic Geometry): solution sets of a polynomial
system V over a ring k as a functor of points on the
category of k-algebras k'. (To brush up on categories
and functors, read Section 1.2 of
Vakil's Rising
Sea.) The equivalence of this functor to the functor
Homk(O(V), –) of k-algebra
homomorphisms O(V) → k'.
[Syllabus.
Class notes. Vakil: Section 1.2. Hartshorne: Pages 1-2 before the
definition of an algebraic set.]
Last modified: (2018-02-07 19:49:19 CST)