Math 771- Homological Algebra

IMPORTANT:

!The classes will be done online via the Zoom platform. Before each class, the Zoom link will be sent to your METU email address. These lectures are intended for Math 771 students and sharing the link with third parties is strictly forbidden
and lecture notes will be posted in ODTUclass. !
!Attendance of lectures is required.!

!If you want to follow the course as a guest student, please send me an email: pasemra@metu.edu.tr!

Course Code: 2360771

Schedule:
Tuesdays:       10:40-12:30
                     Wednesdays:  16:40-18:30


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:
Oral exam