Commutative Algebra

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

This is the webpage of the graduate course "Fall 2017 Mathematics GR6261 COMMUTATIVE ALGEBRA".

Tuesday and Thursday, 11:40 AM - 12:55 PM in Room 507 Math.

Grading will be based on homework and a final exam on December 15 at 9AM -- 12PM in Room 507.

The TA is Raymond Cheng. 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. We will go through material at a rapid pace and the assumption is that you'll spend a lot of time with the theory by yourself in order to understand and keep up with the course.

Part I: tiny bit of dimension theory

  1. Spectrum of a ring 00DY (lecture 9-5)
  2. Prime avoidance 00DS (lecture 9-5)
  3. Noetherian topological spaces 0050 (lecture 9-5)
  4. Noetherian rings 00FM (lecture 9-5 and 9-7)
  5. Noether normalization 00OY (lecture 9-7)
  6. Transcendence degree 030D (lecture 9-12)
  7. Norm and Trace 0BIE (lecture 9-12)
  8. Dimension and dimension functions 0054 02I8 (lecture 9-12)
  9. Gauss' lemma and company 0BC1, 0AFV, 00H7 (lecture 9-12 and 9-14)
  10. Hilbert Nullstellensatz 00FV (lecture 9-14)
  11. Dimension theory 0054, 07NB (lecture 9-14)
  12. Conclusion handout (pdf) (lecture 9-14)

Part II: tiny bit of varieties

For background, please read along in Hartshorne, Chapter I. The Stacks project defines the category of schemes first, and then a variety is an irreducible separated scheme of finite type over a given base field. As in Hartshorne, in this part of the course we will fix an algebraically closed ground field k.

  1. Comparing closed subsets 005Z (lecture 9-19)
  2. Affine varieties (lecture 9-19)
  3. Products of varieties 05P3 (lecture 9-19)
  4. Regular functions (lecture 9-19)
  5. Morphisms of varieties (lecure 9-19)
  6. Rational maps of varieties (lecture 9-21)
  7. Varieties and rational maps 0BXM (lecture 9-21)

Part III: tiny bit on sheaves and cohomology

You can start with reading Hartshorne, Chapter II, Section 1 and Chapter III, Sections 1, 2. Also, please read the very slick discussion in Ravi Vakil's lecture notes on Cech cohomology. 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. Sheaves 006A (lecture 9-26 and 9-28)
  2. Sheaves of modules 01AC (lecture 9-26 and 9-28)
  3. Injective and projective resolutions 013G, 0643 (lecture 10-3)
  4. Derived functors 05T3, 05TB, 0156 (lecture 10-3)
  5. The spectral sequences of a double complex 012X (lecture 10-5)
  6. Locality of cohomology 01E0 (lecture 10-5)
  7. Cech cohomology 01ED (lecture 10-10)
  8. Cech cohomology on presheaves 01EH (lecture 10-10)
  9. Cech cohomology and cohomology 01EO (lecture 10-10 and 10-12)
  10. Alternating Cech complex 01FG (lecture 10-10)
  11. Vanishing cohomology on Noetherian spaces 02UU (lecture 10-10)

Part IV: sheaves on a spectrum

  1. The structure sheaf on spec of a ring
  2. The sheaf of modules associated to a module on spec of a ring
  3. Vanishing of higher cohomology for these sheaves
  4. See handout (pdf) (lecture 10-12)

Part V: tiny bit about schemes

  1. Locally ringed spaces 01HA 01HD (skip closed immersions, lecture 10-17)
  2. Affine schemes 01HR, 01HX (lecture 10-12, 10-17)
  3. Quasi-coherent modules on affines 01I6
  4. Cohomology of quasi-coherent on affine 01X8
  5. Schemes 01II
  6. Fibre products of schemes 01JO
  7. Quasi-compact morphisms 01K2
  8. Separation axioms 01KH
  9. Functoriality for quasi-coherent modules 01LA

Part VI: 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 VII: 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

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

  1. Due 9-12 in class: do 10 of the exercises from 027A
  2. Due 9-19 in class: 02DO 02DR 02CJ 076I
  3. Due 9-26 in class: 0E9D 0E9E 0E9F 0E9G 078W
  4. Due 10-3 in class: 02EU 02EW 028N 078Z 0E9H
  5. Due 10-10 in class: 0CRC 0CRD 0CRE 0D8Q 0D8S 0D8T
  6. Due 10-17 in class: 057Z 0CYH 0D8Y
  7. Due 10-24 in class: do 5 of the following 028P 028Q 028R 028W 02E9 (be sure to open the enclosing section and read the definitions) 029Q 029R 029U (not part 2) 02FH (read definition preceding exercise) 02AW (answer as much as you can)
  8. Due 10-31 in class: Do 3 exercises from Section 0293, do 069S part 3, and do 0AAP
  9. Due 11-9 in class: Do at least 4 exercises from Section 0DAI
  10. Due 11-14 in class: Try to do 1 exercise from Section 0DB3. If you don't have enough time, then don't worry and just skip this week.
  11. Due 11-21 in class: Do 3 exercises picked from either Section 0DB3 or Section 0DCD.
  12. Due 12-5 in class: Do 2 exercises picked from Section 0DD0 and do exercise 0AAS.

These exercises are not always doable purely with the material discussed in the course. Sometimes you'll have to look up things online or in books and use what you find.

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

  1. Irreducible components 004U (lecture 9-5)
  2. Finite type ring maps 00F2 (lecture 9-7)
  3. Finite ring maps 0562 (lecture 9-7)
  4. Integral ring maps 00GH (lecture 9-7)
  5. Finite and integral ring extensions 00GH (lecture 9-7)
  6. Jacobson topological spaces 005T (lecture 9-21)
  7. Jacobson rings 00FZ (lecture 9-21)
  8. Abelian categories 00ZX (lecture 9-26)
  9. Adjoint functors 0036 (lecture 9-26)
  10. Yoneda Lemma 001L (lecture 9-26)
  11. Ext groups 00LO (lecture 10-3)
  12. Tor groups + zig-zags 00LY (lecture 10-5)
  13. Sheaves and bases 009H (lecture 10-12)
  14. Localization 00CM
  15. Normal rings 037B
  16. Spectral spaces 08YF
  17. Stuff about fields 09FA
  18. Simplicial methods 0162