Lecturer

Rahul Pandharipande

Coordinator

Lycka Drakengren

Content

The topics presented in the course will include:

• Basics facts about rings, ideals, and modules
• Constructions of rings: quotients, polynomial rings, localization
• The prime spectrum of a ring
• Chain conditions, Noetherian/Artinian rings and modules
• The tensor product of modules over commutative rings
• Some homological algebra
• Integral extensions, going up, going down
• Finitely generated algebras over fields, including the Noether Normalization Theorem and the Nullstellensatz
• Discrete valuation rings and some applications
• Dimension theory

Prerequisites

Review of groups, rings and fields

Lecture notes

Exercises

Exercise sheets will be given out on Thursdays, from 18 Sep onwards. Solutions should be handed in at the exercise class the week after.

Hand in solutions to the full problem set, indicating any questions you have. Solutions should be handwritten or printed. Exercises marked with an asterix will be corrected. The corrected solutions will be given back the subsequent exercise class.

Other

Proof of the Snake Lemma

History of homological algebra

An introduction to Tor, Ext, and homological algebra in general

More homological algebra

Exercise classes

Literature

Primary Reference:

• "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969)

Secondary References:

• "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995)
• "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989)