MAT 544  Algebra III: Homological algebra, representation theory and commutative algebra   Fall 2018 

TuTh 10-11:20am Math 4-130

Instructor: Ljudmila Kamenova
Office: Math Tower 3-115
Office hours: Wednesdays 1-3pm, MLC on Fridays 1-2pm

The Instructor may be reached by e-mail at
Feel free to send me an e-mail or drop by outside of office hours.

The course will introduce three components. First we will give an overview of commutative algebra. Next we will review homological algebra from MAT 535, Algebra II, and we will survey more advanced topics. Finally, we will outline the structure theory of finite dimensional complex representations of finite groups and of semisimple Lie algebras.


1. Modules, Nakayama's lemma, Noetherian rings.

2. The Nullstellensatz and Spec A.

3. Rings of fractions and localization.

4. Primary decomposition.

5. DVRs and normal integral domains.

6. Chain complexes.

7. Derived functors.

8. Spectral sequences.

9. The Derived category.

10. Representations of finite groups.

11. Representations of symmetric groups.

12. Classification of semisimple Lie algebras.

13. The Weyl Character Formula.

Recommended books: There is no required textbook. The lectures will be based on the following references.

Grading: The final course grade will be determined based on class participation - 30%, homework sets - 40%, and a final oral report by each student - 30%.

Every student will give an oral in-class presentation of approximately 30 minutes in the last few weeks of the semester. The topic of the presentation is to be determined by the student and the instructor jointly. It may be on anything related to the course but not directly covered during lectures or on the problem sets, e.g., a skipped section of the textbooks, an interesting sequence of (unassigned) textbook problems, related material from another algebra textbook, etc. The topic should be discussed with the instructor and approved no later than November 1st.


HW 1 (due 9/6/18): Click here for the problems.

HW 2 (due 9/20/18): Click here for the problems.

HW 3 (due 10/4/18): Click here for the problems.

HW 4 (due 10/18/18): Click here for the problems.

HW 5 (due 11/1/18): Click here for the problems.

HW 6 (due 11/15/18): Click here for the problems.

SASC Statement. If you have a physical, psychological, medical or learning disability that may impact your course work, please contact the 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. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and the Student Accessibility Support Center. For procedures and information go to the following website.

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