MAT 544: Commutative and Homological Algebra
Homework assignments
- Homework 1 (due Sep. 10): Atiyah-Macdonald, Ch.2, #2, #8, #14, #16, #17; Ch.3, #6, #9
- Homework 2 (due Sep. 22): Atiyah-Macdonald, Ch.6, #1i, #2, #9; Ch.7, #1, #4, #8, #11
- Homework 3 (due Oct. 1): see this PDF file
- Homework 4 (due Oct. 13): see this PDF file
- Homework 5 (due Oct. 29): see this PDF file
- Homework 6 (due Nov. 12): see this PDF file
- Homework 7 (due Dec. 1): Atiyah-Macdonald, Ch.10, #3, #4, #9
Lectures
Starting from the second week, all lectures are being recorded. You can access the recordings through Blackboard. Here is a list of topics and recommended reading; [AM] is Atiyah-Macdonald; [Eis] is Eisenbud; [Ma] is Matsumura; [We] is Weibel.
| Date | Topic | Reference |
|---|---|---|
| Aug 25 | Review of rings, ideals, modules | [AM] Ch.1-2 |
| Aug 27 | Tensor product, localization | [AM] Ch.2-3 |
| Sep 1 | Exact sequences, flatness of localization | [AM] Ch.3 |
| Sep 3 | Spectrum of a ring | [AM] Ch.1 exercises |
| Sep 8 | Noetherian rings | [AM], p.74-76, 80-81 |
| Sep 10 | Associated primes, definition and examples | [Eis] p.87-92 |
| Sep 15 | Associated primes, main theorem | [Eis] p.90-94 |
| Sep 17 | Primary decomposition | [AM], p.50-51, 82-84 |
| Sep 22 | Nullstellensatz | [AM], p.81-82, 85 |
| Sep 24 | Non-noetherian examples, artinian rings | [AM], Ch.8 |
| Sep 29 | Artinian rings | [AM], Ch.8 |
| Oct 1 | Local rings, integral dependence | [AM], p.21-22, 59 |
| Oct 6 | Integral dependence | [AM], p.59-62 |
| Oct 8 | Integral dependence, integral closure | [AM], p.61-63 |
| Oct 13 | Dedekind domains, DVRs | [AM], p.93-95 |
| Oct 15 | Fractional ideals, unique factorization | [AM], p.96-98 |
| Oct 20 | Categories and functors | [We], p.417-424, 429-431 |
| Oct 22 | Adjoint functors, injective and projective modules | [We], p.33-35, 38-39 |
| Oct 27 | Injective modules, chain complexes, Tor | [We], p.38-39, 1-4, 15-18 |
| Oct 29 | Tor, long exact sequence in homology | [We], p.10-14, 36, 53 |
| Nov 5 | Symmetry of Tor, checking for flatness | |
| Nov 10 | Topologies and completions | [AM], p.100-105 |
| Nov 12 | Graded rings and modules, Artin-Rees lemma | [AM], p.105-108 |
About the course
Summary. An introduction to the techniques of commutative and homological algebra useful in algebra, algebraic geometry, number theory, and related fields. Review of rings and modules, tensor products and localization. Spectrum of prime ideals, Noetherian and Artinian rings and modules, completion, dimension theory, local rings, discrete valuation rings and Dedekind domains, integral dependence. Chain complexes, projective and injective resolutions, examples of derived functors (Ext and Tor), basic category theory (adjoint functors, natural transformations, limits and colimits), abelian categories. Here is a more detailed syllabus (including suggested reading).
Grading. Grades will be based on weekly homework assignments, class participation, and on your contribution to a group project at the end of the semester. There is no final exam.
Homework. Most homework assignments will come from the book Introduction to Commutative Algebra by Atiyah and Macdonald. Here is a link to a scanned copy.
Time and location
We meet on Tuesday and Thursday from 9:45–11:05 am, in Frey Hall 305. My office is Math Tower 3–117; office hours will be held via Zoom on Fridays from 2:00pm to 4:00pm. Here is the link to my personal meeting room.
Policy Statements
Student Accessibility Support Services. If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Student Accessibility Support Center, ECC (Educational Communications Center) Building, Room 128, (631)632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential. Please see this website for more information.
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