# Commutative Algebra Autumn 2017

 Lecturer Prof. Paul Nelson Lectures Tue 15-17, HG D 1.1 Thu 13-15, HG D 1.1 Coordinator Paul Steinmann Exercise classes Mon 15-16, HG D 1.1 Mon 15-16, ML F 36

Starting dates
First lecture: Tue, September 19, 2017
First exercise classes: Mon, September 25, 2017

This course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry. We shall cover approximately the material from most of the textbook by Atiyah-MacDonald or the first half of the textbook by Bosch.

Topics include:

• Basics about rings, ideals and modules
• Localization
• Primary decomposition
• Integral dependence and valuations
• Noetherian rings
• Completions
• Basic dimension theory

Prerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory).

Notes from Prof. Nelson about dimension: "Commutative algebra: some basics on Krull dimension"

Specific references for each topic:
• Basics on rings, modules, localization [A-M, 1-3]
• Noetherian rings and the Nullstellensatz [A-M, 7]
• Primary decomposition [A-M, 4]
• Artin rings [A-M, 8; Bosch, 2.2]
• Krull dimension theory [Bosch, 2.3-2.4; Eisenbud, 8-10 and 13]
• Integral extensions [A-M, 5; Bosch, 3]
• Valuation rings [A-M, 5; Eisenbud, 11.1]

The new exercise sheet will be posted here on Thursdays. We expect you to look at the problems over the weekend and to prepare questions for the exercise class on Monday.

Please hand in your solutions by the following Thursday at 13:00 in your assistant's box in HG J68. Your solutions will usually be corrected and returned in the following exercise class or, if not collected, returned to the box in HG J68.

exercise sheet due by solutions
Exercise sheet 1 September 28 Solution sheet 1
Exercise sheet 2 October 5 Solution sheet 2
Exercise sheet 3 October 12 Solution sheet 3
Exercise sheet 4 October 19 Solution sheet 4
Exercise sheet 5 October 26 Solution sheet 5
Exercise sheet 6 November 2 Solution sheet 6
Exercise sheet 7 November 9 Solution sheet 7
Exercise sheet 8 November 16 Solution sheet 8
Exercise sheet 9 November 23 Solution sheet 9
Exercise sheet 10 November 30 Solution sheet 10
Exercise sheet 11 December 7 Solution sheet 11
Exercise sheet 12 December 14 Solution sheet 12
Exercise sheet 13 December 21 Solution sheet 13
timeroomassistantlanguage
Mo 15-16HG D 1.1Dante Bonolisen
Mo 15-16ML F 36Seraina Regina Wachteren

Notes from Prof. Nelson about dimension: "Commutative algebra: some basics on Krull dimension"

Primary Reference:
1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969)
Secondary Reference:
1. "Algebraic Geometry and Commutative Algebra" by S. Bosch (Springer 2013)
Tertiary References:
1. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995)
2. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989)
3. "Commutative Algebra" by N. Bourbaki (Hermann, Masson, Springer)