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SCHEDULE
- Class will meet on Tuesday and Thursday, 9:30-10:45 a.m. in LOV 200
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PREREQUISITES
- GRV I & II, or permission from the instructor.
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TEXT
- The “official” textbook is:
C. A. Weibel An introduction to homological algebra. Cambridge University Press.
There are many other excellent sources, either in book format, or available online. The following is a list of items (in
no particular order) I may refer to during the course:
- S. Mac Lane, Homology, Springer.
- Chapter XX in Lang’s Algebra, Springer.
- Chapter 1 in Kashiwara and Schapira, Sheaves on Manifolds.
- Schapira’s notes on Homological Algebra.
- B. Keller: Derived categories and their uses. Chapter of the Handbook of algebra, Vol. 1, edited by M.
Hazewinkel, Elsevier 1996.
Except for Mac Lane’s book, all are more or less slanted toward an approach centered on the concept of Derived
Category.
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COURSE OBJECTIVES
- The course objective is to show how a variety of algebraic constructions arising in topology,
algebra, and geometry, rather than merely comprising a collection of scattered examples, are in fact part of a unified
theory, and to show how the power tools of homological algebra can effectively address several important problems in
those areas.
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COURSE CONTENT
- The course will present the constructions leading to homology and cohomology in a systematic
fashion. Standard tools of the trade, such as sequences, diagrams, etc. will be introduced. Categories are an important
part of this toolbox, and an adequate part of the material will be devoted to their study. Sheaves will be touched
upon, but not introduced in a systematic manner.
In slightly more detail a list of topics is as follows:
- Rings, Modules, Complexes;
- Homology and Cohomology, Derived Functors;
- Classical constructions: Hom, Tensor product, Tor, Ext;
- Applications: Simplicial techniques, Extensions, Resolutions and cohomology for Groups and Algebras.
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HOMEWORK
- Homework will not be collected. Some problems will be assigned in due course for in-class presentation or
individual work to satisfy the grading requirements (see below).
In general, suggested homework problems will be posted on the homework page.
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EXAMS
- There will be no exams.
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GRADING
- Your grade will be based on attendance and class participation. Class participation will include in-class
presentation of at least one “homework” problem, or equivalent work as deemed necessary as the need arise.
Attendance determines the grade as follows: assuming the presentation requirements are met, the grade is determined
according to the following table:
| Absences | 0-5 | 6-10 | 11-15 | 16-20 | > 20 |
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| Grade | A | B | C | D | F |
Excuses will be considered if the student presents sufficient and verifiable documentation that absences are beyond
the student’s control (for example, health, or travel required by graduate work). In that case, a make-up project can
be used to raise the grade to up a full grade point.
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HONOR CODE
- A copy of the Academic Honor Code can be found in your current Student Handbook. You are bound by
this in all of your academic work. It is based on the premise that each student has the responsibility 1) to
uphold the highest standards of academic integrity in the student’s own work, 2) to refuse to tolerate
violations of academic integrity in the University community, and 3) to foster a high sense of integrity
and social responsibility on the part of the University community. You have successfully completed
many mathematics courses and know that on a “test” you may not give or receive any help from a
person or written material except as specifically designed acceptable. Out of class you are encouraged to
work together on assignments but plagiarizing of the work of others or study manuals is academically
dishonest.
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AMERICAN DISABILITIES ACT
- Students with disabilities needing academic accommodations should: 1) register
with and provide documentation to the Student Disability Resource Center (SDRC); 2) bring a letter to the
instructor from SDRC indicating you need academic accommodations. This should be done within the first week of
class.