Homological Algebra (ΑΛ5)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΑΛ5
Semester	2
Course Title	Homological Algebra
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The main purpose of the course is to introduce the main concepts, tools and methods of Homological Algebra, and to indicate some direct applications in various areas of Mathematics and other related sciences. At the end of the course, we expect the student to understand the basic concepts and the main theorems that are analysed in the course, to understand how these are applied to concrete examples arising from different thematic contexts, to be able to apply them to derive new elementary consequences in various areas, and finally to be able to perform some (not so obvious) calculations related to the thematic core of the course.

General Competences

The course aims at enabling the graduate to acquire the ability to analyse and synthesize basic knowledge of Homological Algebra, which is an important part of modern Mathematics with numerous applications in other sciences. When the graduate comes in for the first time in connection with the basic notions of Homological Algebra, (s)he strengthens her/his creative, analytical and inductive thinking, and her/his ability to apply abstract knowledge in different thematic fields.

Syllabus

• Basic concepts and results from Ring Theory.
• Introduction to Module Theory.  
• Fundamental constructions of modules.
• Introduction to the basic elements of Category Theory.
• Projective, injective and flat modules.
• Complexes and (Co)Homology.
• Projective and Injective Resolutions.
• Derived Functors.
• Ext and Tor. 
• Homological Dimension. 
• Applications of Homological Algebra.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Student's study Hours 78 Exercises: Problem Solving 70.5 Course total 187.5
Student Performance Evaluation	The evaluation is based on the combined performance of the graduate student in: Weekly homeworks, Presentations during the semester, Major Homework at the end of the course, Written examination at the end of the courses in Greek with questions and problems of development of theoretical topics and problem solving.

Attached Bibliography

• P. Hilton and U. Stammbach: "A Course in Homological Algebra", Springer-Verlag, (1971).
• J. Rotman: "An Introduction to Homological Algebra", Springer, Second Edition, (2009).
• M. Scott Osborne: "Basic Homological Algebra", Springer, (2000).
• Ch. Weibel: "An Introduction to Homological Algebra", Cambridge University Press, (1994).
• S.I. Gelfand and Yu. Manin: "Methods of Homological Algebra", Springer, Second Edition, (2003).
• P. Bland: "Rings and their Modules", De Gruyter, (2011).