Homological and Computational Commutative Algebra

Math 615

Daniel Erman
3088 East Hall, MWF 2-3




Course Description
This course will cover topics in homological and computational commutative algebra. The essential homological topics include: the Koszul complex, Cohen- Macaulayness, and the Auslander-Buchsbaum formula. The essential computational topics include: Grobner bases, free resolutions, and Castelnuovo-Mumford regularity. Further topics will depend on the interests of the class, and may include: syzygies in algebraic geometry, spectral sequences, or derived categories.

There will be a heavy emphasis on computing examples in Macaulay2.

Eisenbud's "Commutative Algebra with a view towards Algebraic Geometry" will be a good reference for the course.

Here is the syllabus for the course.

Useful links


Grade
The grade will be determined 1/3 by the exercises and 2/3 by the final project.

Exercises
  1. HW 1, due January 16. Here is sample Macaulay2 code to get you started.
  2. HW 2, due January 25. Here is sample Macaulay2 code to get you started.
  3. HW 3 (updated), due February 6. Here is sample Macaulay2 code.
  4. HW 4, due February 13 Here is sample Macaulay2 code.
  5. HW 5, due February 20. This week's homework is to starting thinking about your project.

Project Here is the the handout with details about final projects for this course. As a summary, every final project will consist of:
  1. A short expository paper outlining the topic you investigated, and includes a summary of the examples and computations you considered. Include references. (This must be typed and should be like 3-5 pages.)
  2. Macaualay2 related to your project. This must be clearly(!!) annotated so that someone else can read it.
There will be opportunities to give short in-class presentations on these projects during April. Here are the relevant upcoming due dates (note that these are slightly updated from what was written in the handout):
  1. Friday, March 15: Progress report due. This should be a write-up that demonstrates progrees on both the expository and computational sides of the project. Should be at least 2 paragraphs, and at most 2 pages.
  2. Monday, April 1: Rough draft of full project due.
  3. Wednesday, April 17: Final draft of project due.



Lectures
  1. Jan 9: Overview [E, A3.1].
  2. Jan 11: Free and projective resolutions [E, A3.2 and A3.3]. Jan11.m2.
  3. Jan 14: Koszul complex, I [E, 17.1] Jan14.m2.
  4. Jan 16: Koszul complex, II [E, 17.2]
  5. Jan 18: Koszul complex, III [E, 17.3]
  6. Jan 21: MLK Jr Day
  7. Jan 23: Cancelled (family conflict)
  8. Jan 25: Koszul complex wrap-up and monomial orders [Cox, Little, O'Shea Chapter 2]
  9. Jan 28: Division algorithm [Cox, Little, O'Shea Chapter 2] Jan25.m2.
  10. Jan 30: Groebner bases [Cox, Little, O'Shea Chapter 2]
  11. Feb 1: Applications to algorithms [Cox, Little, O'Shea Chapter 2 and Eisenbud 15.10] Feb1.m2.
  12. Feb 4: Graded rings, modules, and Hilbert functions
  13. Feb 6: Flatness and Groebner bases [Eisenbud 15.8].
  14. Feb 8: Minimal free resolutions
  15. Feb 11: Minimal free resolutions in local case
  16. Feb 13: Minimal free resolutions in graded case; Hilbert's syzygy theorem
  17. Feb 15: Project overview; Hilbert function becomes polynomial
  18. Feb 18: Auslander--Buchsbaum theorem and consequences, I
  19. Feb 20: Auslander--Buchsbaum theorem and consequences, II
  20. Feb 22:
  21. Feb 25: Depth and Cohen--Macaulay rings [Eisenbud, 18]
  22. Feb 27: Consequences of Cohen--Macaulayness Feb27.m2.
  23. Mar 1: Freeness for Cohen--Macaulay rings
  24. Mar 11: Two proofs that the ideal of generic maximal minors is prime, I
  25. Mar 13: Two proofs that the ideal of generic maximal minors is prime, II
  26. Mar 15: Free resolutions and fitting invariants, I [Eisenbud 20]
  27. Mar 18: Free resolutions and fitting invariants, II
  28. Mar 20: Castelnuovo--Mumford regularity, I
  29. Mar 22: Sarah Mayes guest lecture on generic initial ideals.
  30. Mar 25: Castelnuovo--Mumford regularity, II
  31. Mar 27: Derived functors
  32. Mar 29: Mapping cone and double complexes
  33. Apr 1: Spectral sequences, I
  34. Apr 3: Spectral sequences, II
  35. Apr 5: Spectral sequences, III
  36. Apr 8: Derived category, I
  37. Apr 10: Derived category, II
  38. Apr 12: Derived Category, III
  39. Apr 15: Geometry of Hilbert functions and Betti tables, I
  40. Apr 17: Geometry of Hilbert functions and Betti tables, II
  41. Apr 19: Student presentations
  42. Apr 22: Student presentations
  43. Apr 24: Student presentations
If some topics take less time than expected, then I will try to include one of the following: canonical modules; Schreyer's algorithm for computing free resolutions; Hilbert--Burch theorem.