COURSE UNIT TITLE

: COMPUTATIONAL COMMUTATIVE ALGEBRA II

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 4066 COMPUTATIONAL COMMUTATIVE ALGEBRA II ELECTIVE 4 0 0 7

Offered By

Mathematics (English)

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ENGIN MERMUT

Offered to

Mathematics (Evening)
Mathematics (English)

Course Objective

The aim of this course is to introduce methods of commutative algebra with a view towards algebraic geometry and with the computational methods using Groebner basis. We shall continue our study from the course MAT4065 Computational Commutative Algebra I.

Learning Outcomes of the Course Unit

1   Will be able to find finitely many homogeneous generators for the ring of all invariant polynomials under a given finite matrix group G using Emmy Noether's theorem and the Reynolds operator when the order of the group is small.
2   Will be able to interpret the ring of invariants under a given finite matrix group G as a quotient ring of the polynomial ring and as an orbit space under the action of the group G on the affine space.
3   Will be able to interpret the relation between homogeneous ideals and projective varieties using the related correspondence and the projective Nullstellensatz when the field is algebraically closed.
4   Will be able to find the projective closure of an affine variety.
5   Will be able to use the projective elimination and extension theorems to understand the missing points in the geometric interpretation of elimination theory in the affine case.
6   Will be able to compute the dimension of an affine variety over an algebraically closed field using the Hilbert function.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Invariant theory of finite groups. Symmetric polynomials. Finite matrix groups and rings of invariants.
2 Generators for the ring of invariants.
3 Relations among generators and the geometry of orbits.
4 The projective plane. Projective space and projective varieties.
5 The projective algebra-geometry dictionary. The projective closure of an affine variety.
6 Projective elimination theory.
7 The geometry of quadric hypersurfaces.
8 Bezout's theorem.
9 The variety of a monomial ideal.
10 The complement of a monomial ideal.
11 The Hilbert function and the Dimension of a Variety.
12 Elementary properties of dimension. Dimension and algebraic independence.
13 Dimension and nonsingularity.
14 The tangent cone.

Recomended or Required Reading

Textbook(s): Cox, D., Little, J. and OShea D. Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Third Edition, Springer, 2007.
Supplementary Book(s): 1) Reid, M. Undergraduate Algebraic Geometry, Cambridge University Press, 1998.
2) Reid, M. Undergraduate Commutative Algebra, Cambridge University Press, 1995.
3) Greuel, G.-M. and Pfister, G. A Singular Introduction to Commutative Algebra, With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann, Second, Extended Edition, Springer, 2008.
References: 1) Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry, Springer, 1995.
2) Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra, Addison Wesley, 1994.
3) Sharp, R. Y. Steps in Commutative Algebra, Second edition, Cambridge University Press, 2004.
4) Matsumura, H. Commutative Ring Theory, Cambridge University Press, 1989.

Planned Learning Activities and Teaching Methods

Lecture notes, presentation, problem solving, discussion.

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 VZ Vize
2 FN Final
3 BNS BNS VZ * 0.40 + FN * 0.60
4 BUT Bütünleme Notu
5 BBN Bütünleme Sonu Başarı Notu VZ * 0.40 + BUT * 0.60


Further Notes About Assessment Methods

1 Midterm Exam
Final Exam

Assessment Criteria

%40 (Midterm examination) +%60 (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 4 56
Preparations before/after weekly lectures 14 5 70
Preparation for final exam 1 20 20
Preparation for midterm exam 1 25 25
Final 1 2 2
Midterm 1 2 2
TOTAL WORKLOAD (hours) 175

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12PO.13
LO.1535333
LO.2535333
LO.3535333
LO.4535333
LO.5535333
LO.6535333

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

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