HOMOLOGICAL ALGEBRA
| Title | ΟΜΟΛΟΓΙΚΗ ΑΛΓΕΒΡΑ / HOMOLOGICAL ALGEBRA |
| Code | Θ012 |
| Faculty | Sciences |
| School | Mathematics |
| Cycle / Level | 2nd / Postgraduate |
| Teaching Period | Winter/Spring |
| Coordinator | Chrysostomos Psaroudakis |
| Common | Yes |
| Status | Active |
| Course ID | 600025892 |
Programme of Study: PMS Tmīmatos Mathīmatikṓn (2025-2030)
Registered students: 0
| Orientation | Attendance Type | Semester | Year | ECTS |
|---|---|---|---|---|
| THEŌRĪTIKA MATHĪMATIKA | Elective Courses belonging to the selected specialization | Spring | - | 10 |
| Academic Year | 2025 – 2026 |
| Class Period | Spring |
| Faculty Instructors | |
| Weekly Hours | 3 |
| Total Hours | 39 |
| Class ID | 600268420
|
Type Of Offer
- Disciplinary Course
Course Type 2021
Specialization / Direction
Course Type 2016-2020
- Scientific Area
Mode of Delivery
- Face to face
Digital Course Content
- e-Study Guide https://qa.auth.gr/en/class/1/600268420
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
- Greek (Instruction, Examination)
- English (Instruction, Examination)
Learning Outcomes
The students will be able to
1. understand techniques of homological algebra with various applications
2. compute resolutions of modules
3. compute derived functors (of Hom and tensor)
4. compute the global dimension of algebras
5. realise the higher structure of triangulated categories
General Competences
- Work autonomously
- Work in teams
- Generate new research ideas
Course Content (Syllabus)
An Intorduction to Category Theory. Adjoint Functors and Limits. Abelian Categories. Hom and Tensor Functors. Projective and Injective Objects. Complexes and Homology. The Long Exact Sequence of Homology. Cones and Quasi-Isomophisms. Homotopy, Projective and Injective Resolutions. Derived Functors, Ext^1 and Yoneda Extenstions. Homological Dimensions. The Homotopy Category is Triangulated. Derived Categories. Interpretation of Ext as the Hom Space of Derived Category. Derived Equivalences.
Keywords
Category Theory. Abelian Categories. Complexes and Homology. Projective Resolutions. Derived Functors. Yoneda Extensions. The Homotopy Category. Derived Categories.
Educational Material Types
- Notes
- Book
Course Organization
| Activities | Workload | ECTS | Individual | Teamwork | Erasmus |
|---|---|---|---|---|---|
| Lectures | 39 | ||||
| Seminars | 58 | ||||
| Reading Assigment | 200 | ||||
| Exams | 3 | ||||
| Total | 300 |
Student Assessment
Description
The grading will be obtained in three parts:
(i) Exercise sheets through all semester, percentage of the final degree 20%.
(ii) First written exam in the first half of the course, percentage of the final degree 20%.
(iii) Second written exam in the other half of the course, percentage of the final degree 30%.
The final grade is the overall sum of the above intermediate exams.
Student Assessment methods
- Written Exam with Short Answer Questions (Formative)
- Written Exam with Extended Answer Questions (Formative)
- Written Assignment (Formative)
- Oral Exams (Formative)
- Written Exam with Problem Solving (Formative)
Bibliography
Additional bibliography for study
1)A. J. Berrick, Michael E. Keating, Categories and Modules with K-theory in view, Cambridge Studies in Advanced Mathematics 67, 1st Edition.
2) Sergei I. Gelfand, Yuri I. Manin, Methods of Homological Algebra, Springer Monographs in Mathematics, 2nd Edition.
3) Peter J. Hilton, Urs Stammbach, A Course in Homological Algebra, Graduate Texts in Mathematics 4, 2nd Edition.
4) Joseph J. Rotman, An Introduction to Homological Algebra, Universitext, 2nd Edition.
5) Charles A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Series Number 38.
Last Update
01-12-2024
