B2.2 Commutative Algebra - Material for the year 2019-2020

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2019-2020
Lecturer(s): 
Prof. Konstantin Ardakov
General Prerequisites: 

Rings and Modules is essential. Representation Theory and Galois Theory are recommended.

Course Term: 
Hilary
Course Lecture Information: 

16 lectures

Course Weight: 
1.00 unit(s)
Course Level: 
H

Assessment type:

  • Written Examination
Course Overview: 

Amongst the most familiar objects in mathematics are the ring of integers and the polynomial rings over fields. These play a fundamental role in number theory and in algebraic geometry, respectively. The course explores the basic properties of such rings.

Course Synopsis: 

Modules, ideals, prime ideals, maximal ideals.

Noetherian rings; Hilbert basis theorem. Minimal primes.

Localization.

Polynomial rings and algebraic sets. Weak Nullstellensatz.

Nilradical and Jacobson radical; strong Nullstellensatz.

Integral extensions. Prime ideals in integral extensions.

Noether Normalization Lemma.

Krull dimension; dimension of an affine algebra.

Noetherian rings of small dimension, Dedekind domains.

Reading List: 
  1. M. F. Atiyah and I. G. MacDonald: Introduction to Commutative Algebra, (Addison-Wesley, 1969).
  2. D. Eisenbud: Commutative Algebra with a view towards Algebraic Geometry, (Springer GTM, 1995).