MT012 Relative Homological Algebra

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

Information

Code MT012
Name Relative 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 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

Update Time: 09.05.2024 11:39