Mondays and Wednesdays 01:00 PM - 02:15 PM on Zoom through Jan 26, and in MSC E406 after that
This is a graduate-level commutative algebra class. Topics will include Noetherian rings and modules, the Hilbert Basis Theorem, the spectrum of a ring, localization, Hilbert's Nullstellensatz, primary decomposition, Noether Normalization, systems of parameters, DVRs, dimension theory, Nakayama's Lemma, graded rings, and many more. I will be lecturing and posting notes on the website, but the only way to really learn the material is by doing lots of problems. For additional course details, see the syllabus.
Instructor: Brooke Ullery (bullery@emory, office hours Tuesdays 9-11 AM on Zoom -- link the same as the one for class.)
Text:We will not be following any one textbook, but our main sources will be Mel Hochster's notes as well as "Commutative algebra" by Atiyah and Macdonald. Time permitting, we may cover more advanced topics from Eisenbud's "Commutative algebra with a view towards algebraic geometry."
Homework: Problem sets will be assigned every 1-2 weeks. I encourage you to work on the problems together, but you must turn in your own solutions and list the names of your collaborators.
Final presentation: At the end of the semester, you will give a short (around 10 minutes) presentation on a topic of your choice. Some examples of potential topics: free resolutions and syzygies, Gröbner bases, Macaulay2 basics, Cohen-Macaulay rings, local cohomology. For more information and deadlines, see the rubric.
Grading: The final course grade will be calculated as follows: 85% homework, 15% final project. The "participation" portion of your grade can only help you. For example, if you turn in perfect homework assignments but never participate in class or office hours, you may still receive an A on the homework/participation portion. However, if you struggle with the homework assignments, but prove to me that you are working hard and engaging with the material (as evidenced by discussions in class or office hours), your participation may boost your homework grade.
Problem set 1: pdf file, tex file, due February 2
Problem set 2: pdf file, tex file, due February 16
Problem set 3: pdf file, tex file, due March 2
Problem set 4: pdf file, tex file, due March 23
Problem set 5: pdf file, tex file, due April 13
Problem set 6: pdf file, tex file, due April 27
I'll try to keep these up to date and post them right after we cover each topic in class. However, if we take multiple days to cover a topic, I may not post the notes until we are finished with that section.
Section 1: Introduction (Jan 10)
Section 2: Rings, modules, algebras (Jan 10, 12)
Section 3: Noetherian rings and the Hilbert basis theorem (Jan 12)
Section 4: Spec and the Zariski topology (Jan 12, 19)
Section 5: Algebraic sets and the Nullstellensatz (Jan 19, 24)
Section 6: Local rings and localization (Jan 24)
Section 7: Hom and tensor product (Jan 26, 31)
Section 8: Fibers of Spec maps (Jan 31, Feb 2)
Section 9: Support of a module and associated primes (Feb 2, 7)
Section 10: Prime avoidance (Feb 7, 9)
Section 11: Primary ideals and modules (Feb 9, 14)
Section 12: Primary decomposition (Feb 14, 16)
Section 13: Cayley-Hamilton Theorem and Nakayama's Lemma (Feb 16, 21)
Section 14: Integrality and the Nullstellensatz revisited (Feb 21, 23)
Section 15: Normalization (Feb 28)
Section 16: Lying over and going up theorems (Feb 28, Mar 2)
Section 17: The length of a module (Mar 2, 14)
Section 18: Flat families and the Tor functor (Mar 14, 16, 21)
Section 19: Dimension and Krull's Principal Ideal Theorem (Mar 21, 23)
Section 20: Systems of parameters and regular local rings (Mar 28, 30)
Section 21: The Going down theorem (Mar 30, Apr 4)
Section 22: Graded modules and Hilbert functions (Apr 4, 6)
Section 23: Associated graded rings and modules (Apr 6, 11)
Section 24: Artin-Rees and the Krull intersection Theorem (Apr 11)
Section 25: Completions of rings (Apr 11, 13, 18)
Section 26: Hensels lemma (Apr 18)
See details (including deadlines) on the final presentation here. The presentation schedule is below.
Wednesday, April 20:
Monday, April 25: