MT004 Introduction to Homological Algebra

6 ECTS - 3-0 Duration (T+A)- . Semester- 3 National Credit

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

Update Time: 09.05.2024 11:37