Department of Mathematical Sciences
Northern Illinois University

MATH 522, Fall 1999

10:00-10:50, M W F, DU 328

Professor John Beachy, Watson 355, 753-6753

Office Hours: 11:00-12:00 MWF (in Watson 355), or by appointment

email: beachy@math.niu.edu | My faculty homepage | My personal homepage

Syllabus (printable) | Lecture Schedule | Resources on the WEB

Assignments | Class notes | Homework hints

SYLLABUS

COURSE: HOMOLOGICAL ALGEBRA (3)
Categories and functors, projective and injective modules, complexes and homology, Ext, Tor, and dimensions. Applications to cohomology of groups and ring theory.

PREREQUISITE: MATH 521 or consent of department.

TEXT: An Introduction to Homological Algebra, by Rotman, Academic Press, 1979

SYLLABUS:

Chapter 1: Introduction
Some history and motivating problems; categories and functors; tensor products
Chapter 2. Hom and tensor
Direct sums and products; exact sequences; adjoint functors; direct and inverse limits
Chapter 3. Projectives, injectives, and flats
Modules: free, projective, injective, and flat
Extensions of modules
Notes on n-fold extensions of modules
Chapter 6: Homology
Homology functors; derived functors
Chapter 7: Ext
The functor Ext and its relation to extensions of modules
Chapter 8: Tor
The functor Tor and its relation to torsion in modules

Unfortunately, it seems to be impossible to also cover several important chapters: Chapter 4: Specific Rings; Chapter 9: Son of Specific Rings; and Chapter 11: Spectral Sequences.

GRADING: Semester grades will be based on 400 points: 100 points for the midterm exam; 200 points for homework and other classwork; and 100 points for the final exam.

FINAL: The final exam is scheduled for Monday, Dec 6, 10:00-11:50 a.m.

LECTURE SCHEDULE

Week    Pages     Topic                                 S  M Tu  W Th  F  S
                                                                    
 1       1-22     Categories, tensor products     Aug  22 23 24 25 26 27 28

 2  23-34, 57-62  Sums, products, free modules         29 30 31  1  2  3  4
                  
 3      34-39     Exactness, adjoints             Sep   5  6  7  8  9 10 11
                                 meet on Wed and Fri 
 4      39-49     Direct limits                        12 13 14 15 16 17 18

 5      49-56     Inverse limits                       19 20 21 22 23 24 25

 6      62-75     Projectives and injectives           26 27 28 29 30  1  2

 7      84-94     Flat modules                    Oct   3  4  5  6  7  8  9

 8     166-178    Hom functors                         10 11 12 13 14 15 16

 9     178-193    Derived functors                     17 18 19 20 21 22 23

10      Notes     Extensions of modules                24 25 26 27 28 29 30

11     194-202    Ext                             Nov  31  1  2  3  4  5  6

12     202-211    Ext and extensions                    7  8  9 10 11 12 13

13     211-219    Axioms                               14 15 16 17 18 19 20

14     220-223    Tor                                  21 22 23 24 25 26 27

15     224-227    Tor and torsion                 Dec  28 29 30  1  2  3  4

ASSIGNMENTS


DUE     PAGE      PROBLEMS

CLASS NOTES

Here is a diagram showing the construction of the connecting homomorphism in the long exact homology sequence.

RESOURCES ON THE WEB

History of Homological Algebra, by Chuck Weibel, 40 pages, in .dvi format

The Mathematical Atlas: Category theory, homological algebra, by Dave Rusin

A Course in Homological Algebra, by Lee Lady, University of Hawaii

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