2020/2021

# Introduction to Commutative Algebra

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Type:
Optional course (faculty)

Delivered by:
Faculty of Mathematics

Where:
Faculty of Mathematics

When:
3, 4 module

Instructors:
Alexander Pavlov

Language:
English

ECTS credits:
6

Contact hours:
72

### Course Syllabus

#### Abstract

At the most basic level, algebraic geometry is the study of geometry of solution sets of polynomial systems of equations. Classically, the coefficients of polynomial equations are assumed to lie in an algebraically closed field. Considering more general coefficient rings, in particular rings of integers in number fields, one arrives at modern algebraic geometry and algebraic number theory. Commutative algebra provides the tools for answering basic questions about solution sets of polynomial systems, such as finite generation of the system, existence of solutions in some extension of the coefficient ring, dimension and irreducible components, and smoothness and singularities.

#### Learning Objectives

- Learn proofs of fundamental theorems of commutative algebra.
- Understand relations between commutative algebra and other areas of mathematics, such as algebraic geometry and algebraic number theory.
- Apply theorems to provide proofs of statements about commutative rings and modules over them (problem solving).
- Apply knowledge of commutative algebra to analyze examples of particular rings and modules, for example given in terms of generators and relations.

#### Expected Learning Outcomes

- At the end of the course students are expected to be able to state fundamental theorems of commutative algebra, provide proofs and apply the theorems to solve problems and analyze examples.

#### Course Contents

- Ideals and radicals
- Modules over commutative rings
- Localization
- Chain conditions for rings and modules
- Primary decomposition
- Integral extensions
- Flatness
- Completions
- Dimension theory
- Discrete valuation rings and Dedekind rings

#### Interim Assessment

- Interim assessment (4 module)0.4 * Final exam + 0.3 * Midterm exam + 0.1 * Participation in the tutorial + 0.2 * Tests

#### Bibliography

#### Recommended Core Bibliography

- Atiyah, M. F., & Macdonald, I. G. (1969). Introduction To Commutative Algebra. Reading, Mass: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=421131
- Atiyah, M. F., & Macdonald, I. G. (2016). Introduction To Commutative Algebra, Student Economy Edition (Vol. Student economy edition). Boca Raton, FL: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1802204

#### Recommended Additional Bibliography

- Altman, A., & Kleiman, S. (2013). A term of Commutative Algebra. United States, North America: Worldwide Center of Mathematics. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.55CA89AB