3088 East Hall, MWF 2-3

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.

- Macaulay2 webpage.
- Symbol index for Macaulay2.
- Downloading Macaulay2.
- The Macaulay2 Book.

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

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

- 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.)
- Macaualay2 related to your project. This must be clearly(!!) annotated so that someone else can read it.

- 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.
- Monday, April 1: Rough draft of full project due.
- Wednesday, April 17: Final draft of project due.

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