Commutative Algebra

A.Y. 2019/2020
Overall hours
Learning objectives
The main task is to give an introduction to modern commutative algebra with a special regard to commutative ring theory, arithmetic, homological methods and algebraic geometry.
Expected learning outcomes
(first part) Theory and computations of primary decompositions, integral extensions, regular rings & a first step in dimension theory. (9 credits) The additional 3 credits course is providing the next key step in dimension theory.
Course syllabus and organization

Single session

Lesson period
First semester
Prerequisites for admission
We assume known the basic language of categories and functors up to the Yoneda Lemma. We also assume the standard notions: ideals, polynomial rings, multiplicatively closed subsets and localizations, tensor products of modules, Noetherian rings & modules. For example, with respect to M. Reid Undergraduate Commutative Algebra LMS student text series C.U.P. 1995 we will be dealing fast with Chapters 4, 5 & 7, 8 in the commutative algebra course assuming the other Chapters.
Assessment methods and Criteria
The final examination consists of three parts: a written exam, an oral exam and some homework.
Commutative Algebra (first part)
Course syllabus
Substitution principle, prime spectrum & points. Hilbert's Nullstellensatz
Primary decomposition & regular rings. Integral ring extensions & valuations
Noether's normalization. A first step in dimension theory. Derivations & Zariski tangent space.
Teaching methods
Lectures and exercises.
Teaching Resources
Course notes are available at the official ARIEL page. Moreover, the following are classical references:
- S. Bosch: Algebraic Geometry and Commutative Algebra. Universitext, Springer, 2013, 504 p.
- M.F. Atiyah & I.G. MacDonald: Introduction to Commutative Algebra.
Addison-Wesley 1969 (ed. Feltrinelli, 1981)
Commutative Algebra mod/2
Course syllabus
Next step in dimension theory. Primary decomposition of modules, support & associated primes. Filtered/graded modules & Artin-Rees. Hilbert-Samuel polynomial & the dimension theorem.
Teaching methods
Teaching Resources
S. Raghavan: Balwant Singh & R. Sridharan Homological Methods in Commutative Algebra Oxford Univ. Press/TIFR, 1975
Commutative Algebra (first part)
MAT/02 - ALGEBRA - University credits: 6
Practicals: 20 hours
Lessons: 28 hours
Commutative Algebra mod/2
MAT/02 - ALGEBRA - University credits: 3
Lessons: 21 hours
Educational website(s)
Meeting after email contact