Aim 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. There is very little formal training in commutative algebra within the local Dutch mathematics curricula. This Mastermath course aims to remedy this. We shall treat the general theory, but also consider how to do explicit calculations.
We aim to cover the following topics: - flatness - Nakayama’s lemma - localisation of rings and modules - Zariski topology and support - integral extensions - Nullstellensatz - Noetherian and Artinian 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.
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) 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).
Rules about Homework / Exam
The final grade will be based 10% on assignments and 90% on the final written examination or resit. The number of assignments will be determined at the beginning of the course.