# PMH2 Commutative Algebra

## General Information

This page relates to the Pure Mathematics Honours course "Commutative Algebra".

Lecturer for this course: Michael Ehrig.

For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.

## Organisational Matters

**Office:** Carslaw 526

**Email:** michael.ehrig@sydney.edu.au

**Timetable for Lectures:** *Monday 11am - 12pm* and *Tuesday 11am - 12pm* in Room **830**.

**Consultation Hours:** Monday 2 - 4pm, Friday 10am - 12pm, or by appointment.

**Exam:** Monday 26 June, 2pm - 4:10pm, AGR (Room 829).

## Overview

The goal of this course is to study commutative rings and algebras and cover some of the most fundamental theorems in the development of the field: Noether normalisation, Hilbert Nullstellensatz, and the theory of localisation and local rings, among others. Throughout the course we will discuss how to interprete commutative algebra from the view point of affine algebraic geometry and category theory.

## Assessment

1. Two assignments worth 20% each; they will be posted here two weeks before the due date. More details will be posted soon.

- The first assignment is be due Wednesday, April 12. Assignment 1

(Solutions: Assignment 1)

- The second assigment is due Friday, June 2. Assignment 2

(Solutions: Assignment 2)

2. A written exam worth 60% to be held at the end of semester 1, covering the whole content of the course. The exam time and venue are listed above.

## References

The material will overlap with a number of references, but two that cover many of the subjects are:

M.F. Atiyah and I.G. Macdonald, *Introduction to Commutative Algebra*, Addison-Wesley 1969.

Miles Reid, *Undergraduate Commutative Algebra*, Cambridge University Press 1995.

## Exercises

Exercises will be posted here regularly every week. These will not be collected and marked, but are a good way of learning the material. Many of them are are also announced in the lecture and reproduced here.

- Exercise sheet 1 (rings and ideals)

- Exercise sheet 2 (affine algebraic geometry)

- Exercise sheet 3 (tensor products)

## Notes

Notes for the lectures will be posted here on an irregular basis.

- Chapter 0 - Basic notions for rings

- Outlook - Algebraic Geometry - Part I

- Chapter 1 - Modules

- Chapter 2 - Localization and Fractions

- Chapter 3 - Noetherian & Artinian

## Timetable

Show timetable / Hide timetable.