Schemes, Spring 2017

Professor A.J. de Jong, Columbia university, Department of Mathematics.

This is the webpage of the graduate course "Spring 2017 Mathematics GR6262 ARITH and ALGEBRAIC GEOMETRY".

Tuesday and Thursday, 11:40 AM - 12:55 AM in Room 407 Math.

Grading will be based on homework and a final exam.

The TA is Remy van Dobben de Bruyn. He will be in the help room ??.

We will use the Stacks project as our main reference, but of course feel free to read elsewhere. If you see a four character alphanumeric code, like 0000, then this is a link to a chapter, section, exercise, or a result in the Stacks project.

Reading. Please keep up with the course by studying the following material as we go through it.

Part I: basics

  1. Locally ringed spaces 01HA 01HD (skip closed immersions)
  2. Affine schemes 01HR, 01HX
  3. Quasi-coherent modules on affines 01I6
  4. Schemes 01II
  5. Fibre products of schemes 01JO
  6. Quasi-compact morphisms 01K2
  7. Separation axioms 01KH
  8. Functoriality for quasi-coherent modules 01LA

Part II: cohomology

You can start with reading Hartshorne, Chapter III, Sections 1, 2. The Stacks project defines the derived category before discussing higher derived functors. Either first read a bit about the derived category and then read the links below or try reading the sections listed below anyway (without reading the definition of the derived category) and see what parts of them make sense with the definitions given in the lectures and see if you can prove the statements.

  1. Injective and projective resolutions 013G, 0643
  2. Derived functors 05T3, 05TB, 0156
  3. Spectral sequences 011M
    1. exact couples 011P
    2. differential objects 011U
    3. filtered differential objects 012A
    4. filtered complexes 012K
    5. double complexes 012X
  4. Cech cohomology 01ED
  5. Cech cohomology on presheaves 01EH
  6. Cech cohomology and cohomology 01EO
  7. Alternating Cech complex 01FG

Part III: cohomology of quasi-coherent modules on schemes

  1. Cohomology of quasi-coherent modules on schemes 01X8
  2. Cohomology of projective space 01XS
  3. Cohomology of coherent sheaves on Proj 01YR
  4. Quasi-coherence of higher direct images 01XH
  5. Euler characteristics 0BEI
  6. Hilbert polynomials 08A9
  7. FYI: Numerical intersections 0BEL

Part IV: coherent duality

You can start with reading Hartshorne, Chapter III, Sections 6, 7. Another place to read is the chapter "Proof of Serre duality" in Ravi's notes. The Stacks project has a discussion on the level of derived categories following ideas of Neeman and Lipman. This is probably impossible to grok without a serious effort, so I suggest you attend the lectures to help limit what you should read.

  1. Overview of a duality theory 0AU3
  2. Dualizing module on proper over "good" Noetherian local ring 0AWP

Part V: Curves

Please follow along by reading Hartshorne, Chapter IV. The following is more a list of topics.

  1. Varieties 020C
  2. Varieties and rational maps 0BXM
  3. Types of varieties 04L0
  4. Curves 0A22
  5. Degrees on curves 0AYQ
  6. The chapter on Algebraic Curves 0BRV

Exercises. Please do the exercises to keep up with the course:

  1. Due 1-24 in class: 028P, 028Q, 028R, 028W, 02E9 (be sure to open the enclosing section and read the definitions)
  2. Due 1-31 in class: 029Q, 029R, 029U (not part 2), 02FH (read definition preceding exercise), 02AW (answer as much as you can)
  3. Due 2-7 in class: Do 3 exercises from Section 0293, do 069S part 3, and do 0AAP
  4. Due 2-14 in class: Do at least 4 exercises from Section 0D8P
  5. Due 2-21 in class: Try at least 4 exercises from Section 0DAI
  6. Due 2-28 in class: Try at least 3 exercises from Section 0DB3
  7. Due 3-7 in class: Try at least 3 exercises from Section 0DCD
  8. Due 3-21 in class: Try at least 3 exercises from Section 0DD0
  9. Due 3-28 in class: Try at least 3 exercises from 02A7, 02AA, 02FK, Section 0DJ0

Background stuff. Most of this will be discussed in the lectures:

  1. Chapter on sheaves 006A
  2. Chapter on sheaves of modules 01AC
  3. Locally ringed spaces 01HA
  4. Bases and sheaves 009H