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Information
| Code | MT012 |
| Name | Relative Homological Algebra |
| Term | 2022-2023 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 | Doktora Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. YILMAZ DURĞUN |
Course Goal / Objective
The aim of this course is to introduce the main techniques and methods in relative homological algebra.
Course Content
Complexes of modules and homology. Direct and inverse limits. I-adic topology and completions. 2 Torsion free covering modules. Examples. 3 F-precovers and covers. Direct sums of covers. Projective, flat and injective covers. 4 F-preenvelopes and envelopes. Direct sums of envelopes. Flat and pure-injective envelopes. 5 Fibrations, cofibrations and Wakamatsu lemmas. Set theoretic homological algebra. Cotorsion theories. 6 Left and right F-resolutions. Derived functors and balance. 7 F-dimensions. Minimal pure-injective resolution of flat modules. 9 Iwanaga-Gorenstein rings. The minimal injective resolution of a commutative Noetherian ring that is Gorenstein. 10 Torsion products of injective modules. Local cohomology and the dualizing module. 11 Gorenstein injective, Gorenstein projective and Gorenstein flat modules. 12 Gorenstein injective covers and envelopes. 13 Gorenstein projective and Gorenstein flat covers. Gorenstein flat and projective preenvelopes. Kaplansky classes. 14 Balance over Gorenstein and Cohen-Macaulay Rings.
Course Precondition
none
Resources
Relative Homological Algebra Edgar E. Enochs and Overtoun M. G. Jenda
Notes
Introduction to Homological Algebra C. Weibel
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Will be able to generalize the idea behind the proof of the existence of torsion free covers of modules over commutative domains |
| LO02 | Will be able to understand the definitions of covers and envelopes in general for a class of modules. |
| LO03 | Will be able to understand how cotorsion theories have been used in proving the existence of flat covers of modules for an arbitrary ring. |
| LO04 | Will be able to use the properties of Iwanaga-Gorenstein and Cohen-Macaulay rings and their modules. |
| LO05 | Will be able to analyze some different kinds of Gorenstein covers and envelopes. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Knows the results of previous research in a special field of mathematics | 3 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. | 4 |
| 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. | 4 |
| PLO06 | Bilgi - Kuramsal, Olgusal | Effectively uses the technical equipment needed to express mathematics | 5 |
| PLO07 | Bilgi - Kuramsal, Olgusal | Sets up original problems in her field and offers different solution techniques | 5 |
| PLO08 | Bilgi - Kuramsal, Olgusal | It carries out original and qualified scientific studies on the subject related to its field. | 4 |
| PLO09 | Bilgi - Kuramsal, Olgusal | Analyzes existing mathematical theories and develops new theories. | |
| 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. | |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | To have foreign language knowledge at a level to be able to follow foreign sources related to the field and to communicate verbally and in writing with foreign stakeholders. | |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | It presents and publishes its original works within the framework of scientific ethical rules for the benefit of its stakeholders. | |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Adheres to the ethical rules required by its scientific title |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Complexes of modules and homology. Direct and inverse limits. I-adic topology and completions. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 2 | Torsion free covering modules. Examples. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 3 | F-precovers and covers. Direct sums of covers. Projective, flat and injective covers. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 4 | F-preenvelopes and envelopes. Direct sums of envelopes. Flat and pure-injective envelopes. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 5 | Fibrations, cofibrations and Wakamatsu lemmas. Set theoretic homological algebra. Cotorsion theories. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 6 | Left and right F-resolutions. Derived functors and balance. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 7 | F-dimensions. Minimal pure-injective resolution of flat modules. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 8 | Mid-Term Exam | Review of the relevant pages from lecture books | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Iwanaga-Gorenstein rings. The minimal injective resolution of a commutative Noetherian ring that is Gorenstein. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 10 | Torsion products of injective modules. Local cohomology and the dualizing module. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 11 | Gorenstein injective, Gorenstein projective and Gorenstein flat modules. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 12 | Gorenstein injective covers and envelopes. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 13 | Gorenstein projective and Gorenstein flat covers. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 14 | Gorenstein flat and projective preenvelopes. Kaplansky classes. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 15 | Balance over Gorenstein and Cohen-Macaulay Rings. | Review of the relevant pages from lecture books | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 16 | Term Exams | Review of the relevant pages from lecture books | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | Review of the relevant pages from lecture books | Ölçme Yöntemleri: Yazılı Sınav |
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 | ||