Date
|
Section(s)
|
Topics
|
Tue 2/1
|
II.1-2
|
Quick course overview; review of
sheaves from
18.725: sheaves on topological spaces, glueing of sheaves; ringed
spaces, locally ringed
spaces
|
Thu 2/3
|
II.2
|
Spectra of rings as locally
ringed spaces; affine
schemes, schemes, examples |
Tue 2/8
|
II.2-3
|
The Proj functor, projective
space; the
functor
from abstract algebraic varieties to schemes; morphisms of
schemes; the functor of points;
properties of schemes: reduced
|
Thu 2/10 (PS 1 due)
|
II.3
|
Properties of schemes:
connected, irreducible, integral, locally noetherian, noetherian;
dimension and codimension; open immersions, closed immersions,
examples; construction of fibred products
of schemes |
Tue 2/15 (KSK away)
|
II.3-4
|
Fibres, the philosophy of base
change; properties of morphisms: of
finite type, locally of finite type, finite, quasi-finite,
quasi-compact; |
Thu 2/17 (PS 2 due;
KSK away)
|
II.4
|
Separated, quasi-separated
morphisms; morphisms between affine schemes are separated;
the diagonal morphism is always a locally closed immersion;
image of a quasi-compact map is closed iff it is stable under
specialization; a non-quasi-separated morphism |
Tue 2/22
|
|
NO LECTURE (MIT on Monday
schedule)
|
Thu 2/24 (PS 3 due)
|
II.4-5
|
Properness; valuative criterion
for properness; projective spaces are proper;
closed immersions are proper; "strongly" projective and
quasi-projective
morphisms; quasi-coherent sheaves, their adjointness property |
Tue 3/1
|
II.5
|
Facts about quasi-coherent
sheaves (sheaf property, exactness, pullback, pushforward); ideal
sheaves and closed
immersions; coherent sheaves; sheaf associated to a graded module
[buildup]
|
Thu 3/3 (PS 4 due;
KSK away)
|
II.5
|
Relationship between sheaves on
a Proj and graded modules; twisting sheaves; Serre's theorem on
coherence of pushforwards (also in the proper case); very ample,
relatively globally generated, ample, relatively ample sheaves
|
Tue 3/8
|
II.6
|
Weil and Cartier divisors, class
groups; line bundles, Picard group |
Thu 3/10 (PS 5 due)
|
II.6-7
|
Divisors on curves, statement of
Riemann-Roch; linear systems; line bundles and morphisms to projective
spaces; relative Proj; blowings up
|
Tue 3/15
|
II.8; III.1-2
|
Kahler differentials; sheaves of
differentials;
regular schemes; Cohen-Macaulay schemes; review of homological algebra:
abelian categories, injective resolutions, derived functors,
(universal) delta-functors, effaceability; categories of sheaves have
enough injectives; definition of sheaf cohomology; Grothendieck's
vanishing theorem [buildup]
|
Thu 3/17 (PS 6 due)
|
III.3
|
Vanishing of the cohomology of
an affine scheme; Serre's criterion for affinity |
|
|
SPRING BREAK
|
Tue 3/29
|
III.4
|
Cech cohomology, examples;
comparison between Cech and sheaf cohomology on topological spaces, on
schemes |
Thu 3/31 (PS 7 due)
|
III.5
|
Cohomology of projective space;
Serre's finiteness theorem; criterion for ampleness |
Tue 4/5
|
III.6
|
Ext and sheaf Ext |
Thu 4/7 (PS 8 due)
|
III.7
|
Duality on projective space;
dualizing sheaves; existence of dualizing sheaves; Serre duality |
Tue 4/12
|
III.7-8
|
Duality on curves via residues;
higher direct image functors; quasi-coherence of higher direct images |
Thu 4/14 (PS 9 due)
|
III.9
|
Flat morphisms; cohomology
commutes with flat base extension; flat families; flatness and
constancy of Hilbert polynomials; examples of flat limits |
Tue 4/19
|
|
NO LECTURE (Patriots Day)
|
Thu 4/21 (PS 10 due)
|
III.10
|
Smooth morphisms, etale
morphisms; generic smoothness; Bertini's theorem |
Tue 5/3
|
II.9; III.11
|
Formal schemes, Zariski's Main
Theorem, Stein
factorizations |
Thu 5/5 (PS 11 due)
|
III.12
|
Cohomology and base change |
Tue 5/10
|
TBA
|
TBA
|
Thu 5/12 (PS 12 due)
|
TBA
|
TBA
|