Introduction to Commutative Algebra, Spring 2016

# Introduction to Commutative Algebra, Spring 2016

## General Information

• Introduction to Commutative Algebra, 201-2-0371, is a first course in modern commutative algebra that provides the background for further study of commutative and homological algebra, algebraic geometry, algebraic combinatorics, algebraic methods in cryptography, etc.
• The prerequisites for the course are: Linear algebra (Algebra 1 and 2), and Algebraic structures (strongly recommended, but not strictly necessary).

#### Lectures:

Tuesday, 8-10, Building 58, room 201
Thursday, 8-10, Building 58, room 201

#### Office hours:

Tuesday, 18-20, Building 58, room 213. Tel. 08-6461694

## Syllabus

1. Rings, ideals, and homomorphisms
2. Modules, Cayley-Hamilton theorem, and Nakayama’s lemma
3. Noetherian rings and modules, Hilbert basis theorem
4. Integral extensions, Noether normalization lemma, and Nullstellensatz
5. Affine varieties and schemes
6. Localization of rings and modules
7. Primary decomposition theorem
8. Discrete valuation rings
9. Selected topics
syllabus in Hebrew

## Home assignments, exams, and grades

• Every week I will upload a problem set, and every second week we will have a one-hour long problem solving session, where the students will present their solutions to the problem sets on the blackboard.
• It will be possible to gain up to 20 points for presenting your solutions of the homeworks.
• There will be a final exam.
• The grade in the course will be calculated by the following formula: min{100, 0.9*F+H}, where F is the grade for the final exam, and H is the number of points gained for presenting your solutions of the homeworks in class.

## Important rules

• Solve home assignments every week! Keep in mind that it is very important to try to solve the problems by yourself. Knowing the solution of somebody else will not help you to solve the final exam problems!
• Come prepared to the class.
• Don't hesitate to ask questions during the lectures.
• If you don't understand something or did not succeed to solve a problem in a home assignment then attend the office hours and ask questions.

## Recommended books

• Miles Reid, Undergraduate Commutative Algebra;
• Miles Reid, Undergraduate Algebraic Geometry;
• Altman, Kleiman, A Term of Commutative Algebra.

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