Commutative Algebra, Fall 2016

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

This is the webpage of the graduate course "Fall 2016 Mathematics GR6261 COMMUTATIVE ALGEBRA".

Tuesday and Thursday, 10:10 AM - 11:25 AM in Room 520 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 Mondays 5-6 and Wednesday 1-2 and 5-6.

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: dimension theory

  1. Noetherian graded rings 00JV (lecture 9-6 and 9-8)
  2. Noetherian local rings 00K4 (lecture 9-13, 9-15, and 9-20)
  3. Dimension 00KD (lecture 9-22 and 9-27)
  4. Background 16, 17, 18, 19 (lecture 9-29, 10-4, and 10-6)
  5. Depth 00LE (lecture 10-11)
  6. Cohen-Macaulay modules 00N2 (lecture 10-13)
  7. Cohen-Macaulay rings 00N7 (lecture 10-18)
  8. Catenary rings and spaces and dimension functions 00NH, 02I0, 02I8 (lecture 10-20)
  9. The dimension formula 02II (lecture 10-25)
  10. Chevalley's theorem 00F5 (lecture 10-27)
  11. Hilbert Nullstellensatz 00FS (lecture 10-27)
  12. Jacobson rings and Jacobson topological spaces 00FZ, 005T (lecture 11-1)
  13. Dimension of finite type algebras over fields 00OO, 07NB (lecture 11-1)

Part II: regular local rings and smooth rings

  1. Projective modules 05CD (lecture 11-3)
  2. Finite projective modules 00NV (lecture 11-10 and 11-15)
  3. What makes a complex exact 00MR (lecture 11-17)
  4. Regular local rings 00NN (lecture 11-22)
  5. Finite global dimension 00O2 (lecture 11-22 and 11-29)
  6. Regular rings and global dimension 065U (lecture 12-1)

Part III: Completion (not part of final exam)

  1. Completion 00M9
  2. Completion for Noetherian rings 0BNH
  3. Topological rings and modules 07E7 0AMQ
  4. Cohen structure theorem 0323

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

  1. Due 9-13 in class: Read about the spectrum of a ring 00DY and do 10 of the exercises from 027A
  2. Due 9-20 in class: 078G, 078H, 057Z, 0767, 0768, 0769, 076A
  3. Due 9-27 in class: 076F, 076G, 02DL, 02DM, 076I, 02EI, 09TZ
  4. Due 10-4 in class: 02CJ, 02LU, 02DS, 02DZ, 07DH
  5. Due 10-11 in class: 0CR8, 0CR9, 0CRA, 0CRC, 0CRD, 0CRE, 0CRF
  6. Due 10-18 in class: 0CS1, 0CS2, 0CS3, 0CS4, 07DL
  7. Due 10-25 in class: 0CT6, 0CT4, 0CT1, 0CT2, 07DK
  8. Due 11-1 in class: 0CVP, 0CVR, 0CVS
  9. Due Thursday 11-10 in class: 02CM, 02CN, 078J, 02CP (optional: prove Lemma 00P1 using the dimension function in the lectures)
  10. Due 11-15 in class: 078P, 02CR, 0CYH
  11. Due 11-22 in class: 078R, 06A3 (translated into algebra)
  12. Due 11-29 in class: no exercises this week.
  13. Due 1-6 in class: 02FE, 02FF, 0D1T

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

  1. Noetherian rings 00FM
  2. K-groups 00JC
  3. Localization 00CM
  4. Local rings 07BH
  5. Additive categories 09SE
  6. Abelian categories 00ZX
  7. Length 00IU
  8. Locally nilpotent ideals 0AMF
  9. Artin-Rees lemma 00IN
  10. Spectrum of a ring 00DY
  11. Dimension of topological spaces 0054
  12. Noetherian topological spaces 0050
  13. Artinian rings 00J4
  14. Prime avoidance 00DS
  15. Nakayama's lemma 00DV
  16. Support of a module 080S
  17. Dimension of a module 00KY
  18. Associated primes 00L9
  19. Ext groups 00LO
  20. Transcendence degree of fields 030D
  21. Hilbert Nullstellensatz 00FS
  22. Colimits 07N7