18.726 Lecture Plan

Last updated: 3 Feb 2005. Treat these like the 10-year budget predictions of the Congressional Budget Office, i.e., with a healthy dose of skepticism. In particular, there may be points of "buildup" where I've pushed things back but not all the way; I'll indicate these below. Section references are to Hartshorne. See below about the TBA topics.

Tue 2/1
Quick course overview; review of sheaves from 18.725: sheaves on topological spaces, glueing of sheaves; ringed spaces, locally ringed spaces
Thu 2/3
Spectra of rings as locally ringed spaces; affine schemes, schemes, examples
Tue 2/8
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)
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)
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)
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)
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
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)
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
Weil and Cartier divisors, class groups; line bundles, Picard group
Thu 3/10 (PS 5 due)
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)
Vanishing of the cohomology of an affine scheme; Serre's criterion for affinity

Tue 3/29
Cech cohomology, examples; comparison between Cech and sheaf cohomology on topological spaces, on schemes
Thu 3/31 (PS 7 due)
Cohomology of projective space; Serre's finiteness theorem; criterion for ampleness
Tue 4/5
Ext and sheaf Ext
Thu 4/7 (PS 8 due)
Duality on projective space; dualizing sheaves; existence of dualizing sheaves; Serre duality
Tue 4/12
Duality on curves via residues; higher direct image functors; quasi-coherence of higher direct images
Thu 4/14 (PS 9 due)
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)
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)
Cohomology and base change
Tue 5/10
Thu 5/12 (PS 12 due)

The TBA slots are left in partially to allow for a cushion in case I get seriously behind compared to this schedule. If they don't get used in that fashion, I will do a brief overview of some new topic in that time. Candidates include a bit of the geometry of surfaces (Chapter V), or the etale topology (as in Milne's Etale Cohomology).