Prerequisites
A firm grasp of (commutative) rings, ideals, prime ideals and maximal
ideals, zero divisors, quotient rings, subrings and homomorphisms,
polynomial rings in several variables, finite field extensions, and
algebraically closed fields. This material is contained in many standard
books on algebra, for example in Chapters 7, 8, 9 (except 9.6), and 13
(except 13.3 and 13.6) and parts of 14.9 of the book 'Abstract algebra'
by Dummit and Foote (third edition), or in the book 'Algebra' by Serge
Lang (parts of Chapters 2, 3, 5, and 7 will be needed).
The 'Intensive Course on Categories and Modules' contains important
background material, and should be watched by all students not already
familiar with it. The course can be accessed through
https://elo.mastermath.nl/course/view.php?id=125
Aims of the course
Commutative algebra is the study of commutative rings and their modules,
both as a topic in its own right and as preparation for algebraic
geometry, number theory, and applications of these. We shall treat the
general theory, but also consider how to do explicit calculations.
We hope to cover the following topics:
- flatness
- Nakayama's lemma
- localisation of rings and modules
- Zariski topology and support
- integral extensions
- Nullstellensatz
- Noetherian and Artin rings and their modules
- associated primes and primary decomposition
- dimension theory and the dimension of fibres
- local rings and regularity
- completions
Other topics may be covered as well.
Lecturers
David Holmes (UL) & Marta Pieropan (UU)
- Docent: David Holmes
- Docent: Marta Pieropan