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Information
| Code | MT004 |
| Name | Introduction to Homological Algebra |
| Term | 2024-2025 Academic Year |
| Semester | . Semester |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 6 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Yüksek Lisans Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. DOĞAN DÖNMEZ |
Course Goal / Objective
To grasp and use the concepts of category and functor.
Course Content
Categories and functors. Mophisms. Natural transformations. Category of modules and its properties. Exact sequences. Projective and injective modules. Hom and tensor products. Complexes and homology. Exact functors. Derived functors: Ext and Tor.
Course Precondition
Pre-requisites None
Resources
J. Rotmann: Homological Algebra
Notes
Lecture Notes
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Understands the concepts of category and functor. |
| LO02 | Understands natural transformations. |
| LO03 | Fully understands the category of modules. |
| LO04 | Understands exact sequences and exact functors. |
| LO05 | Understands complexes and homology. |
| LO06 | Understands exact and half exact functors. |
| LO07 | Understands derived functors. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in his area of expertise and other areas of mathematics. | 5 |
| PLO03 | Bilgi - Kuramsal, Olgusal | Establishes new mathematical models with the help of the knowledge gained in the field of specialization. | 3 |
| PLO04 | Bilgi - Kuramsal, Olgusal | Has basic knowledge in all areas of mathematics. | 4 |
| PLO05 | Bilgi - Kuramsal, Olgusal | It presents the knowledge gained in different fields of mathematics and their relations with each other in the simplest and most understandable way. | 3 |
| PLO06 | Bilgi - Kuramsal, Olgusal | Effectively uses the technical equipment needed to express mathematics. | 5 |
| PLO07 | Bilgi - Kuramsal, Olgusal | poses original problems related to field and presents different solution techniques. | 4 |
| PLO08 | Bilgi - Kuramsal, Olgusal | carries out original and qualified scientific studies on the subject related to its field. | |
| PLO09 | Bilgi - Kuramsal, Olgusal | Analyzes existing mathematical theories and develops new theories. | 4 |
| PLO10 | Beceriler - Bilişsel, Uygulamalı | Knows the teaching-learning techniques in areas of mathematics that require expertise and uses these techniques effectively at every stage of education. | 2 |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | To have knowledge of a foreign language at a level to be able to follow foreign sources related to the field and to communicate verbally and in writing with foreign stakeholders. | 2 |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | presents and publishes its original works within the framework of scientific ethical rules for the benefit of its stakeholders. | 3 |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Adheres to the ethical rules required by its scientific title | 5 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Categories, morphisms and isomorphisms. Examples | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 2 | Covariant and contravariant functors. Examples. Natural transformation between functors | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 3 | Equivalence between functors. Category of modules. Module homomorphisms. Submodule, quotient module. Free modules. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 4 | Direct sum and direct product. Properties of direct sum and direct product. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 5 | Hom functors. Their properites. Projective and injective modules. Their properties. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 6 | Tensor product of modules. Properties of tensor product. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 7 | Properties of the tensor product functor. Adjoint isomorphism. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 8 | Mid-Term Exam | Solving homework problems | Ölçme Yöntemleri: Ödev |
| 9 | Baer s criterion. Existence of injective modules. Injective resolution. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 10 | Category of complexes. Properties of the category of complexes. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 11 | Homology. Properties of the homology functors. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 12 | Short Exact sequences of complexes. Connecting homomorphisms. Long exact sequence. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 13 | Derived functors. Derived functors of an additive half exact functor. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 14 | Derived functors. Derived functors of an additive half exact functor 2 | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 15 | Some applications of homological algebra. | Studying the relevant parts in references | Öğretim Yöntemleri: Anlatım |
| 16 | Term Exams | Solving homework problems | Ölçme Yöntemleri: Ödev |
| 17 | Term Exams | Solving homework problems | Ölçme Yöntemleri: Ödev |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 14 | 3 | 42 |
| Out of Class Study (Preliminary Work, Practice) | 14 | 5 | 70 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 0 | 0 | 0 |
| Mid-term Exams (Written, Oral, etc.) | 1 | 15 | 15 |
| Final Exam | 1 | 30 | 30 |
| Total Workload (Hour) | 157 | ||
| Total Workload / 25 (h) | 6,28 | ||
| ECTS | 6 ECTS | ||