Math 905 Fall 2024

Commutative Algebra I

Time and place:

MWF 11:30am–12:20pm, Burnett 203.

Worksheet previews — read before each class

Fall 2022 Lecture Notes by Grifo — highly recommended reading

Worksheets and solutions

• §1. Rings, Ideals, and Modules
• Monday 8/26: §1.1. Rings worksheet solutions
• Wednesday 8/28: §1.2. Ideals worksheet solutions
• Friday 8/30: §1.3. Algebras worksheet solutions
• Wednesday 9/4: §1.4. Modules worksheet solutions
• Friday 9/6: §1.5. Determinants worksheet solutions
• §2. Finiteness conditions
  • Monday 9/9: §2.6. Algebra-finite and module-finite maps worksheet solutions
  • Wednesday 9/11: §2.7. Integral extensions worksheet solutions
  • Friday 9/13: §2.8. UFDs and integral closure worksheet solutions
  • Monday 9/16: §2.9. Noetherian rings worksheet solutions
  • Wednesday 9/18: §2.10. Noetherian modules worksheet solutions
• §3. Graded rings
• Monday 9/23: § 3.11. Graded rings worksheet solutions
• Wednesday 9/25: §3.12. Graded modules worksheet solutions
• Friday 9/27: §3.13. Finiteness theorem for invariant rings worksheet solutions
• Monday 9/30: §3.14. Rees rings and Artin-Rees Lemma worksheet solutions
• §4. Nullstellensatz and spectrum
• Wednesday 10/2: §4.15. Noether normalization worksheet solutions
• Friday 10/4: §4.16. Nullstellensatz worksheet solutions
• Monday 10/7: §4.17. Strong Nullstellensatz worksheet solutions
• Wednesday 10/9: §4.18. Spectrum of a ring worksheet solutions
• Friday 10/11: §4.19. Spectrum and radical ideals worksheet solutions
• §5. Localization
• Monday 10/14: §5.20. Local rings and NAK worksheet solutions
• Wednesday 10/16: §5.21. Localization of rings worksheet solutions
• Wednesday 10/23: §5.22. Localization of modules worksheet solutions
• Friday 10/25: §5.23. Local properties worksheet solutions
• §6. Decompositions of ideals and modules
• Monday 10/28: §6.24. Minimal primes worksheet solutions
• Wednesday 10/30: §6.25. Associated primes worksheet solutions
• Friday 11/1: §6.26. More associated primes worksheet solutions
• Wednesday 11/6: §6.27. Primary decomposition: existence worksheet solutions
• Friday 11/8: §6.28. Primary decomposition: uniqueness worksheet solutions
• §7. Dimension and affine algebras
• Monday 11/11: §7.29. Dimension worksheet solutions
• Wednesday 11/13: §7.30. Krull-Seidenberg theorems—Applications worksheet solutions
• Friday 11/15: §7.31. Krull-Seidenberg theorems—Proofs worksheet solutions
• Monday 11/18: §7.32. Dimension and Noether Normalization worksheet solutions
• Wednesday 11/20: §7.33. Transcendence degree and dimension worksheet solutions
• §8. Local dimension theory
• Monday 11/25: §8.34. Length and simple modules worksheet solutions
• Monday 12/2: §8.35. Artinian Rings worksheet solutions
• Wednesday 12/4: §8.36. Krull height theorem worksheet solutions
• Friday 12/6: §8.37. Systems of parameters worksheet solutions
• Monday 12/9: §8.38. Local characterization of dimension worksheet solutions

Office hours

• Mondays 9:30am–11:00am (shared with Math 325 students)
• Thursdays 1:00pm–2:00pm (Math 905 only!)
• Week of Sept 9: no OH
• or by appointment

Problem sets

Links to the problem sets appear below.
• Problem Set #1 tex, due Friday, September 6
• Problem Set #2 tex due Friday, September 27
• Problem Set #3 tex due Friday, October 11
• Problem Set #4 tex due Friday, October 25
• Problem Set #5 tex due Friday, November 8
• Problem Set #6 tex due Friday, November 22
• Problem Set #7 tex due Friday, December 13. Gift added 12/8.

Syllabus