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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. Minimal pure-injective resolution of flat modules. 8 Midterm 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

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 *** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

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

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 13 3 39 Preparations before/after weekly lectures 13 5 65 Preparation for midterm exam 1 15 15 Preparation for final exam 1 25 25 Preparing assignments 10 5 50 Final 1 3 3 Midterm 1 3 3 TOTAL WORKLOAD (hours) 200

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 LO.1 4 3 3 4 4 4 3 3 LO.2 4 3 3 4 4 4 3 3 LO.3 4 3 3 4 4 4 3 3 LO.4 4 3 3 4 4 4 3 3 LO.5 4 3 3 4 4 4 3 3