Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Commutative Algebra» для направления
01.04.01 «Математика» подготовки магистра
Правительство Российской Федерации
Федеральное государственное автономное образовательное учреждение высшего профессионального образования
"Национальный исследовательский университет
"Высшая школа экономики"
Факультет Математики
Программа дисциплины
Commutative Algebra
для направления 010100.68 «Математика» подготовки магистра
магистерская программа «Математика» (англоязычная)
Автор программы:
PhD, Кондо Сатоши, satoshi.kondo@gmail.com
Рекомендована секцией УМС по математике «___»____________ 2015 г
Председатель С.М. Хорошкин
Утверждена УС факультета математики «___»_____________2015 г.
Ученый секретарь Ю.М. Бурман_______________________________
Москва, 2015
Настоящая программа не может быть использована другими подразделениями университета
и другими вузами без разрешения кафедры-разработчика программы.
Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Commutative Algebra» для направления
01.04.01 «Математика» подготовки магистра
1
Scope of use and legal references
This course program sets minimal requirements to the knowledge and skills of students and determines
the contents and kinds of lectures and reporting. The program is intended for lecturers teaching this course,
course assistants and students of 01.04.01 specialization «Mathematics» who study the course “Infinite
dimensional Lie algebras and vertex operator algebras”.


2
The program has been elaborated in accordance with the Educational standard of HSE for training
area 010100.68 «Mathematics» (Master level);
In accordance with the working studying plan of the university for training area of 01.04.01 specialization «Mathematics» (Master level), master program “Mathematics”, approved in 2014.
Learning objectives
The course “Commutative Algebra” aims at making the students familiar with the basics on commutative
ring theory. The students are to learn the basic concepts and notions that may be applicable in algebraic
geometry.
Course goals:
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
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3
To make students familiar with the main examples of rings, algebras, and modules.
To describe the main purposes in commutative ring theory.
To study the basic constructions such as tensor products, localization, and normalization.
To familiarize students with categorical notions.
To educate students about the application to algebraic geometry of commutative ring theory.
Learning outcomes
By the end of the course student is supposed to
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
4
Know: The basic concepts of the commutative ring theory and its connection to algebraic geometry.
Be able to: Understand the fundamental theorems such as Hilbert nullstellensatz and the notion of
primary decomposition in Noetherian ring theory.
Place of Discipline in MA program structure
This course is a professional one. This is an elective course for the “Mathematics” specialization.
This course is based on knowledge and competences that were provided by the following disciplines:

Алгебра (Algebra)
Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Commutative Algebra» для направления
01.04.01 «Математика» подготовки магистра
5
Course plan
№
1
2
3
4
6
Total
hours
Название раздела
Categories and direct limits
Rings and modules
Nullstellensatz
Primary decomposition
32
32
32
32
Total:
128
32
32
Independent
student’s work
16
16
16
16
64
Requirements and Grading
Type of grading
Type of work
Running
(week)
Problem
solving
1 year
1 2 3
4
2
2
midterm
Final
6.1
Contact hours
Lecture
Seminars
s
8
8
8
8
8
8
8
8
Examination
Examination
V
V
Characteristics
Use blackboard
8
wtitten, 120 min.
wtitten, 120 min
Knowledge and skills grading criteria
All work is graded on the scale from 1 to 10.
6.2
Calculating the grades for the course
The resulting grade for the running check evaluates the results of student’s work and is calculated according to the following formula:
Отекущий = 0.9·Ок/р + 0.1·Одз ;
The students are required to participate in the problem sessions in order to be able to take the examinations.
The resulting grade for the final evaluation in the form of examination is calculated according to this formula (where Оэкзамен is the evaluation of the student’s performance at the exam itself):
Оитоговый = 0,8·Оэкзамен + 0,2·Отекущий
This grade which is the resulting grade for the course is written down in the student’s certificate (diploma).
7
Content of the subject
Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Commutative Algebra» для направления
01.04.01 «Математика» подготовки магистра
Section 1. Categories and direct limits
№
Topic
1
Categories, functors, natural transformations
2
Adjoint functors, adjoint pairs
3
Direct limits, universality
4
Filtered category, filtered direct limits
Total:
Total
hour
s
Lectures
Seminars
Independent
student’s work
8
2
2
4
8
2
2
4
8
2
2
4
8
2
2
4
32
8
8
16
Section 2. Rings and modules
№
1
2
3
4
Topic
Rings and modules (definitions)
Tensor product of modules and of algebras
Localization of rings and of modules
Integrality, normality
Total:
Total
hours
8
8
8
8
32
Contact hours
Lecture
Seminars
s
2
2
2
2
2
2
2
2
8
8
Independent
student’s work
4
4
4
4
16
Section 3. Hilbert Nullstellensatz
№
1
2
3
4
Topic
Support of an ideal
Going-up, going-down theorems
Noether normalization
Nullstellensatz
Total:
Total
hours
8
8
8
8
32
Contact hours
Lecture
Seminars
s
2
2
2
2
2
2
2
2
8
8
Independent
student’s work
4
4
4
4
16
Section 4. Primary decomposition
№
Topic
Total
hours
Contact hours
Lecture
Seminars
s
Independent
student’s work
Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Commutative Algebra» для направления
01.04.01 «Математика» подготовки магистра
1
2
3
4
8
Chain conditions
Notherian rings
Associated primes
Primary decomposition
8
8
8
8
2
2
2
2
2
2
2
2
4
4
4
4
Total:
32
8
8
16
Grading estimation for the running check and the final assessment of students
8.1
Topics for Current Control
Question for the test:
1. Compute the tensor product of explicitly given modules.
2. Give an example of a filtered colimit.
3. Verify the fact the tensoring is not left exact.
4. Compute the support of an ideal.
5. Compute the primary decomposition of an ideal.
6. Give an example of a finitely generated, non finitely presented module.
8.2
Questions for evaluating student’s performance
1. Compute the normalization of a ring.
2. Show that taking the tensor product and taking the localization commute.
3. Compute the tensor product of two nonzero vector spaces.
4. Show that a module is a direct limit of its finitely generated submodules.
5. Compute the spectrum of the polynomial ring over a field.
6. Compute the set of minimal primes of a ring.
7. Describe the set of maximal ideals of the polynomial ring over an algebraically closed field.
8. Give the proof of the Hilbert Basis theorem.
9
Readings and materials for the course
9.1 Fundamental textbook
A Term of Commutative Algebra Allen Altman, Steven Kleiman
ISBN-10: 0-9885572-1-5 ISBN-13: 978-0-9885572-1-5 237 Pages
Required reading
Atiyah, M. F.; Macdonald, I. G. Introduction to commutative algebra. Addison-Wesley Publishing
Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp
9.2
Further reading
Matsumura, Hideyuki Commutative ring theory. Translated from the Japanese by M. Reid. Second
edition.Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1989.
xiv+320 pp. ISBN: 0-521-36764-6 13-01
9.3
Matsumura, Hideyuki Commutative algebra. Second edition. Mathematics Lecture Note Series,
56.Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN: 0-8053-7026-9
Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Commutative Algebra» для направления
01.04.01 «Математика» подготовки магистра
Eisenbud, David Commutative algebra. With a view toward algebraic geometry. Graduate Texts in
Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. ISBN: 0-387-94268-8; 0-387-94269-6
Bourbaki, Nicolas Commutative algebra. Chapters 1–7. Translated from the French. Reprint of the
1989 English translation. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. xxiv+625 pp.
ISBN: 3-540-64239-0