COURSE UNIT TITLE

: HOMOLOGICAL ALGEBRA

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 5022 HOMOLOGICAL ALGEBRA ELECTIVE 3 0 0 7

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSOCIATE PROFESSOR SALAHATTIN ÖZDEMIR

Offered to

Mathematics (English)
Mathematics (English)

Course Objective

The aim of this course is to introduce the basic methods in homological algebra.

Learning Outcomes of the Course Unit

1   Will be able to use the properties of the adjoint functors Hom and tensor product in the categories of modules.
2   Will be able to find Ext(C,A) using projective or injective resolutions.
3   Will be able to find Tor(A,B) using projective or injective or flat resolutions.
4   Will be able to use the properties of the rings characterized by their homological dimensions.
5   Will be able to understand how the homological methods for modules are generalized to homology in abelian categories.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Motivation and history. Categories and functors.
2 The adjoint pair of functors Hom and the tensor product in the categories of modules.
3 Projective, injective and flat modules.
4 Homology of complexes of modules and related long exact sequences.
5 Ext and Tor using projective, injective and flat resolutions. Elementary properties.
6 Derived functors. Long exact sequences.
7 Projective dimension, injective dimension, flat dimension of modules.
8 The left global dimension, the right global dimension, the weak dimension of a ring.
9 Global dimension of matrix rings and polynomial rings. Hilbert s Syzygy Theorem. Localization and global dimension.
10 Ext and extensions: the Baer sum of short exact sequences. Pullbacks and pushouts.
11 Some special rings characterized homologically-1: Semisimple rings and von Neumann regular rings.
12 Some special rings characterized homologically-2: Hereditary rings and Dedekind domains, semihereditary rings and Prüer domains.
13 Abstract homological algebra: Homology in abelian categories.
14 Limits (inverse limits) and colimits (directed limits). Lazard s Theorem.

Recomended or Required Reading

Textbook(s):
[1] M. Scott Osborne. Basic homological algebra. Springer-Verlag, 2000.
[2] Joseph J. Rotman. An introduction to homological algebra. Second edition. Springer, 2009.
[3] Refail Alizade ve Ali Pancar. Homoloji Cebire Giriş. 19 Mayıs Üniversitesi, Samsun, 1999.
Supplementary Book(s):
[1] L. R. Vermani. An elementary approach to homological algebra. Chapman & Hall/CRC, 2003.
[2] Henri Cartan and Samuel Eilenberg. Homological algebra. Princeton Landmarks in Mathematics. Princeton University Press, 1999. With an appendix by David A. Buchsbaum, Reprint of the 1956 original.
[3] S. Maclane. Homology. Springer-Verlag, 1963.
[4] P. J. Hilton and U. Stammbach. A course in homological algebra. Second edition. Springer-Verlag, 1997.
[5] Charles A.Weibel. An introduction to homological algebra. Cambridge University Press, 1994.
[6] Sergei I. Gelfand and Yuri I. Manin. Methods of homological algebra. Second edition. Springer-Verlag, 2003.

Planned Learning Activities and Teaching Methods

Lecture Notes, Presentation, Problem Solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 ASG ASSIGNMENT
2 FIN FINAL EXAM
3 FCG FINAL COURSE GRADE ASG * 0.50 + FIN * 0.50
4 RST RESIT
5 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.50 + RST * 0.50


*** 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: salahattin.ozdemir@deu.edu.tr
Office: B 351/1 - (232) 301 86 08

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 3 39
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) 174

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.14334434153
LO.24334434153
LO.34334434153
LO.44334434153
LO.54334434153

CONTACT INFORMATION
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Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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