COURSE UNIT TITLE

: RELATIVE HOMOLOGICAL ALGEBRA

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 6058 RELATIVE HOMOLOGICAL ALGEBRA ELECTIVE 3 0 0 8

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ENGIN MERMUT

Offered to

Mathematics (English)
Mathematics (English)

Course Objective

The aim of this course is to introduce the main techniques and methods in relative homological algebra.

Learning Outcomes of the Course Unit

1   Will be able to generalize the idea behind the proof of the existence of torsion free covers of modules over commutative domains.
2   Will be able to understand the definitions of covers and envelopes in general for a class of modules.
3   Will be able to understand how cotorsion theories have been used in proving the existence of flat covers of modules for an arbitrary ring.
4   Will be able to use the properties of Iwanaga-Gorenstein and Cohen-Macaulay rings and their modules.
5   Will be able to analyze some different kinds of Gorenstein covers and envelopes.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 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.
8 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.

Recomended or Required Reading

Textbook(s):
[1] Edgar E. Enochs and Overtoun M. G. Jenda. Relative Homological Algebra. Walter de Gruyter, 2000.
[2] Jinzhong Xu. Flat covers of modules. Springer, 1996.

Planned Learning Activities and Teaching Methods

Lecture notes, presentation, problem solving, discussion.

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 ASG ASSIGNMENT
2 MTE MIDTERM EXAM
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE ASG * 0.30 + MTE * 0.30 + FIN * 0.40
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.30 + MTE * 0.30 + RST * 0.40


Further Notes About Assessment Methods

Homework
1 Midterm Exam
Final Exam

Assessment Criteria

%30 (Homework) + %30 (Midterm examination) +%40 (Final examination)

Language of Instruction

English

Course Policies and Rules

You can be successful in this course by studying from your textbooks and lecture notes on the topics to be covered every week, coming to class by solving the given problems, establishing the concepts by discussing the parts you do not understand with your questions, learning the methods, and actively participating in the course.

Contact Details for the Lecturer(s)

Engin Mermut
e-mail: engin.mermut@deu.edu.tr
Phone: (232) 301 85 82

Office Hours

To be announced later.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 3 42
Preparations before/after weekly lectures 14 5 70
Preparation for midterm exam 1 15 15
Preparation for final exam 1 25 25
Preparing assignments 8 5 40
Final 1 3 3
Midterm 1 3 3
TOTAL WORKLOAD (hours) 198

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.143344433
LO.243344433
LO.343344433
LO.443344433
LO.543344433

CONTACT INFORMATION
Dokuz Eylül Üniversitesi Cumhuriyet Bulvarı No: 144 35210 Alsancak / İZMİR
Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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