Math 771- Homological Algebra

IMPORTANT:

 Attendance of lectures is required.!


Course Code: 2360771

Schedule: Mondays:       10:40-12:30  at room M-215
                     Wednesdays:   09:40-10:30  at room M-215

Content: Categories, functors, derived functors, extensions, resolutions, homology and cohomology of
complexes. Some applications depending on the consent of the instructor such as modular
representation theory or cohomology of groups or Lie algebras, algebraic topology.

Course Objectives:
1-To introduce standart concepts of homological algebra such as Ext and Tor and apply these in the setting of specific areas chosen
depending the consent of the instructor.
2-To prepare students to a current research area, introduce them open questions.
3-To enable students with the computational skills such as computation of homology groups, cohomology groups which are needed in
algebraic topology, algebraic geometry, etc...

Course Learning Outcomes:
Obtain knowledge of the facts and techniques from homological algebra e.g. diagram chasing etc.
Apply homological algebra methods to problems in algebra and topology.
Compute homology and cohomology groups of spaces and groups

Tentative Weekly Outline:

1 Introduction; Simplicial, singular homology
2 Categories and Functors
3 Modules, Hom and Tensor
4 Special modules; projective, injective and flat modules
5 Categorical constructions and limits
6 Abelian Categories and Complexes
7 Homology functors, left and right derived functors
8 Tor functors
9 Ext Functors
10 Universal Coefficient Theorems for Homology and Cohomology, The Künneth Theorem
11 Extension of Modules
12 Group Extensions
13 Group cohomology and homology
14 Bar resolution

Course Textbook(s):
An Introduction to Homological Algebra, J.J. Rotman
A Course in Homological Algebra, P.J. Hilton, U. Stammbach
An Introduction to Homological Algebra, C.A. Weibel

Supplementary Readings :
Homological Algebra, H. Cartan, S. Eilenberg
Homology, S. Mac Lane

Assessment of Student Learning/ Grading::
Assignments %30: Homework assignments will be given every two weeks.
Midterm Exam %30: Take home exam
Final Exam %40: Take home exam